Supercell method for the computation of energies of crystals

Transcription

1 Supercell method for the computation of energies of crystals David Gontier CEREMADE, Université Paris-Dauphine Warwick EPSRC Symposium: Density Functional Theory and Beyond: Analysis and Computation Warwick, July 5, 2017 Joint work with Salma Lahbabi David Gontier Simulation of crystals 1 / 33

2 Goals: Define and compute numerically the energy per cell of a crystal. Define and compute numerically the energy of a local defect inside a crystal. What is a crystal? Remarks A periodic arrangement of (fixed) nuclei on a lattice. Modeled by a periodic density charge µ per, or by a periodic potential V per. An arrangement of moving electrons around these nuclei. The number of electrons is infinite. There is no Schrödinger-like equations a priori. What is the energy (per cell) of the electrons? David Gontier Simulation of crystals 2 / 33

3 The thermodynamic limit David Gontier Simulation of crystals 3 / 33

4 Fact: There exist good models for finite systems. Thermodynamic limit: See the infinite periodic crystal as the limit of finite systems. L = 1 L = 2 L = 3 L = Roadmap: Choose a model for finite systems (Schrödinger equation, Hartree(-Fock), DFT, ). For each L N, calculate the ground state energy E L for the charge density µ L. Take the limit L. Questions: Does the sequence L 3 E L (energy per cell) converge as L? If yes, can we characterise the limit? David Gontier Simulation of crystals 4 / 33

5 The thermodynamic limit for different models Model Existence of the limit Characterisation of the limit Schrödinger 1,2 yes no non-interacting Schrödinger yes yes DFT no (yes?) reduced-hf (rhf) 3 yes yes Hartree-Fock (HF) 3 maybe (yes?) Remarks The main difficulty for the Existence/Characterisation is to prove that the final solution is periodic. This is the case for the rhf/non-interacting model: the model is convex. Otherwise, symmetry breaking may happen (periodic problem, non periodic minimiser). Speed of convergence? We expect a very slow speed of convergence: the edge effects only vanish at the limit. Idea If the final solution is periodic, perform a periodic thermodynamic limit. 1 C. Fefferman, Commun. Math. Phys. 98 (1985), no X. Blanc, C. Le Bris, and P.-L. Lions, Comm. Part. Diff. Eq. 28 (2003). 3 I. Catto, C. Le Bris, and P.-L. Lions, Ann. Inst. H. Poincaré (C) 18 (2001), no. 6. David Gontier Simulation of crystals 5 / 33

6 The non-interacting (linear) case David Gontier Simulation of crystals 6 / 33

7 Notations for the crystal Lattice: R = az 3. Unit cell: Γ = [ a/2, a/2) 3. Reciprocal lattice: R = (2π/a) Z 3. Reciprocal unit cell: Γ = [ π/a, π/a) 3. Supercell: Γ L := LΓ. The Hamiltonian we wish to study (mean field) potential: V per L 2 per (Γ). Non-interacting Hamiltonian: H per = V per, acting on L 2 (R 3 ). Number of electrons per cell: N N. Questions How to define the energy per cell? How to compute it? Idea: Perform a periodic thermodynamic limit. David Gontier Simulation of crystals 7 / 33

8 The supercell model Full model Supercell model L Laplacian with periodic boundary conditions: L. Supercell Hamiltonian H L := 1 2 L + V per acting on L 2 per(γ L ). Number of electrons = NL 3. Roadmap Compute the total energy E L of the supercell. Define the energy per cell as the limit of L 3 E L when L. David Gontier Simulation of crystals 8 / 33

9 1-body Hamiltonian h 1 := 1 2 L + V per self-adjoint operator acting on H 1 = L 2 (Γ L ). Fact: h 1 is compact resolvent (= discrete spectrum). ε 1... ε N the smallest eigenvalues. u 1,..., u N H 1 the corresponding normalised eigenvectors. N-body non-interacting Hamiltonian N H N := h 1 (x i ) i=1 Ground state energy of H N acting on the fermionic space N H1. E 0 N = ε ε N = u 1, h 1 u u N, h 1 u N = Tr H1 [h 1 ( u 1 u u N u N )] (cyclicity of the trace) = inf { Tr H1 [h 1 γ], γ is a projector of rank N } (min-max principle) = inf { Tr H1 [h 1 γ], γ P N } (min of linear function on convex set), where P N is the convex hull of the set of projectors of rank N: P N := { γ S(H 1 ), 0 γ 1, Tr H1 (γ) = N }. The minimiser is γ N := 1 (H N ε N ), and satisfies Tr H1 (γ N ) = N. David Gontier Simulation of crystals 9 / 33

10 Supercell minimiser γ L := 1 ( ) H L < ε L F Supercell energy per unit cell Exact minimiser In our case where ε L F is chosen so that 1 [ L 3 Tr L 2 per (Γ γ L] = N. L) I L := 1 L 3 EL = 1 L 3 Tr L 2 per (Γ L) [ H L γ L]. In the thermodynamic limit, we expect γ per of the form γ per := 1 (H per < ε F ) where ε F is chosen so that Tr (γ) = N. Exact energy per cell I per = Tr [H per γ per ]. Trace per cell 1 Tr := lim L L 3 Tr L 2 per (Γ L). Questions Does the sequence L 3 E L converge to I per as L? What is the speed of convergence? Lemma In the insulating case, there exists C R + and α > 0 such that, for all L N, I L I per Ce αl. David Gontier Simulation of crystals 10 / 33

11 The Bloch-Floquet transform David Gontier Simulation of crystals 11 / 33

12 L 2 per([0, L]) = Vect { e 2iπ0x, e 2iπ( L 1 )x, e 2iπ( ( ) L 2 )x,..., e 2iπ L 1 x L, e 2iπ( ( ) ( ) ( ) L L )x, e 2iπ L+1 x 2iπ L+2 x 2iπ 2L 1 x L, e L,..., e L, } = L 2 0 L 2 2π L 2 4π L 2 2π(L 1). L L L Fibers L 2 q = Vect { e ik x e iq x, k R } = { ψ L 2 loc (R3 ), R R, ψ( + R) = e iq R ψ( ) }. L 2 q does not depend on L, q Γ. 3D case L 2 per(γ L ) = Q Λ L L 2 Q. Λ L Γ David Gontier Simulation of crystals 12 / 33

13 Supercell set of translations : R L := R Γ L. (Supercell) Bloch-Floquet transform for functions f L 2 (Γ L ) = Q Λ L f Q with f Q L 2 Q. (Supercell) Bloch fibers f Q (x) := e iq R f(x R) = e iq R τ R f(x). R R L R R L Bloch-Floquet for operators Lemma Let A L : L 2 per(γ L ) L 2 per(γ L ) be such that τ R A L = A L τ R for all R R. Then, the operator A L is block-diagonal with respect to the L 2 Q decomposition. Proof [ A L f ]Q (x) = ( ) e iq R τ R A L f (x) = e iq R A L (τ R f)(x) R R L R R L = A L e iq R τ R f (x) = A L ( ) f Q (x). R R L David Gontier Simulation of crystals 13 / 33

14 In other words, writing ÃL Q : L2 Q L2 Q, we have A L Q A L 0 A L Q = (Bloch transform = block-decomposition) Ã L Q L 3 Covariance ) L 2 q = S q (L 2 per(γ) with S q [f](x) = e iq x f(x) and (S q ) 1 = S q. Twisted Bloch transform A L Q := S QÃL Q S Q, : L 2 per(γ) L 2 per(γ). Basic properties Spectrum Trace per cell σ(a L ) = Q Λ L σ Tr L (A L ) := 1 L d Tr L 2 per (Γ L) (AL ) = 1 L d (ÃL Q ) = Q Λ L σ ( ) A L Q. Tr L 2 per (Γ) (ÃL Q ) = 1 L d Q Λ L Q Λ L Tr L 2 per (Γ) (AL Q ). David Gontier Simulation of crystals 14 / 33

15 Example: The periodic Hamiltonian Bloch transform H q := H L q = 1 2 H L = 1 2 L + V per acting on L 2 per(γ L ). i 1 + q 2 + V per = q (i 1 ) + q2 2 + V per acting on L 2 per (Γ). Supercell thermodynamic limit (L ) Γ Γ Λ = Γ Λ L=4 Λ L=8 Trace per unit volume H L = Q Λ L H L Q Tr L (A) = 1 L 3 L ˆ H per = H qdq. Γ Q Λ L A Q L Tr (A) := Γ A q dq. Supercell model Regular sampling of Γ. David Gontier Simulation of crystals 15 / 33

16 Remark: The operator H q (acting on L 2 per (Γ)) is compact resolvent: H q = ε nq u nq u nq, ε 1,q ε 2,q, u nq, u mq L 2 per (Γ) = δnm. n=1 ε nq (ev) ε F N Band diagram of the silicon q Γ Fermi energy: ε F R s.t. n=1 Γ 1(ε nq ε F ) dq = N. Insulating system: The system is insulating if there exists a gap g > 0 such that q Γ, ε N,q + g 2 ε F ε N+1,q g 2. David Gontier Simulation of crystals 16 / 33

17 Full Hamiltonian One-body density ˆ γ per = 1(H per ε F ) = 1(H q ε F ) dq. Γ }{{} γ q Energy per unit cell (update) I per = Tr (H perγ per) = Supercell model Supercell one-body density Supercell energy per unit cell (update) I L = Tr (H L γ L) = 1 L 3 Tr L 2 Γ per (Γ) (Hqγq) dq = Γ γ L = 1(H L ε F ) = Q Λ L Tr L 2 per (Γ) Q Λ L γ Q. ( HQ γ Q ) = 1 L 3 ( N ) ε nq dq n=1 Q Λ L ( N ) ε nq. n=1 Error for the energy ( I L I per = 1 N ) L 3 ε nq Q Λ L n=1 ( N ) ε nq dq. Γ n=1 This is the difference between a Riemann sum and an integral. David Gontier Simulation of crystals 17 / 33

18 Tools in complex analysis David Gontier Simulation of crystals 18 / 33

19 Integrand Properties K(q) := N ε nq. n=1 The function K is R -periodic. The functions q ε nq are not smooth in general (conic singularities). However, the sum of the eigenvalues is smooth. Band diagram of graphene. Lemma ( 4,5,6 ) If the system is an insulator, then the (R -periodic) function K admits an analytic extension on some complex strip S A := R 3 + i[ A, A] 3 with A > 0. Link to the fact that γ per (x, y) Ce α x y. 4 W. Kohn, Phys. Rev. 115 (1959). 5 J. Des Cloizeaux, Phys. Rev. 135 (1964). 6 C. Brouder, G. Panati, M. Calandra, C. Mourougane, and N. Marzari, Phys. Rev. Lett. 98 (2007). David Gontier Simulation of crystals 19 / 33

20 Elements of the proof Cauchy residual formula γ q = 1(H q ε F ) = 1 2iπ Analytic continuation C dλ λ H q. C ε F σ(h) H q = q (i 1 ) + q2 2 + Vper = Hz = z (i 1 ) + zt z 2 + Vper Lemma There exists A > 0 such that z S A, γ z := 1 2iπ C is well-defined. Moreover, the map z γ z is analytic on S A. Integrand K(q) := dλ λ H z N ε nq = Tr L 2 per (Γ) (H qγ q ) = K(z) := Tr L 2 per (Γ) (H zγ z ). n=1 K(z) is R -periodic and analytic on S A. David Gontier Simulation of crystals 20 / 33

21 Lemma (classical) If f : S A C is analytic on S A for some A > 0 and satisfies f(z + k) = f(z) for all k R, then there exists C R + and α > 0 such that L N, 1 L 3 f(q) f(q)dq Q Λ Γ L Ce αl. Conclusion Lemma (DG, Salma Lahbabi) In the insulating case, there exists C R + and α > 0 such that, for all L N, L 3 E L I per Ce αl. (Convergence of the energy per cell) ρ γper ρ γ L Ce αl. (Convergence of the electronic density) L per David Gontier Simulation of crystals 21 / 33

22 Numerical illustration (Linear model for silicium taken in M.L. Cohen and T.K. Bergstresser, Phys. Rev. 141 (1966). Speed of convergence for the silicium (linear model). David Gontier Simulation of crystals 22 / 33

23 The non-linear case (reduced Hartree-Fock case) David Gontier Simulation of crystals 23 / 33

24 How to define an interacting model? Take an effective model (DFT, HF, ), and perform a (periodic) thermodynamic limit. Problem: symmetry breaking Lemma (current work with Mathieu Lewin and Faizan Nazar) Let { I α (L) := 1 ˆ } L 3 inf Tr L 2 (Γ L ) ˆΓ ( γ) + V per ρ γ + D L (ρ γ, ρ γ ) α ρ 4/3 γ, γ P(Γ L ), L Γ L then, for α > α c, it holds that I α (2) < I α (1). Idea Enforce the 1-periodicity (for instance using convexity) = reduced Hartree-Fock (rhf). Non-interacting energy as a minimisation problem Set of one-body density matrices Recall the non-interacting case I per = Tr (H per γ per ) = inf { Tr (H per γ), γ P per, Tr (γ) = N }. P per = { γ S(L 2 (R 3 )), 0 γ 1, R R, τ R γ = γτ R }. Set of supercell one-body density matrices } P per = {γ S(L 2 per(γ L )), 0 γ 1, R R, τ R γ = γτ R. David Gontier Simulation of crystals 24 / 33

25 Charge density: µ per reduced Hartree-Fock I rhf per { } = inf Iper rhf (γ), γ P per, Tr (γ) = N. with (µ per is the charge density) Periodic Green s function G 1 = 4π R R I rhf per (γ) := 1 2 Tr ( γ) D 1(ρ γ µ per, ρ γ µ per ). ( δr Γ 1) so that G 1 (x) = 4π Γ k R \{0} Periodic Coulomb potential f, g L 2 per(γ), D 1 (f, g) := f(x)g 1 (x y)g(y) dxdy. (Γ) 2 e ik x k 2. Lemma ( Catto, Le Bris, Lions and Cancès, Deleurence, Lewin ) The problem Iper rhf admits a unique minimizer γ per. This minimizer satisfies the Euler-Lagrange equations γ per = 1(H per ε F ) H per = Vper acting on L2 (R 3 ) V per = (ρ γper µ per) Γ G 1. David Gontier Simulation of crystals 25 / 33

26 Supercell energy (rhf) ( EL {E rhf = inf rhf (γ L ), γ L P L, Tr L 2 per (Γ L ) γ L) = NL 3}, with E rhf (γ L ) := 1 2 Tr L 2 per (Γ L) ( L γ L) D L(ρ γ L µ per, ρ γ L µ per). Supercell Green s function G L = 4π R LR ( δr Γ L 1) so that G L (x) = 4π Γ L k L 1 R \{0} Supercell Coulomb potential f, g L 2 per(γ L ), D L (f, g) := f(x)g L (x y)g(y) dxdy. (Γ L ) 2 e ik x k 2 = 1 ( x ) L G 1. L Lemma The problem EL rhf admits a unique minimizer γl. This minimizer satisfies the Euler-Lagrange equations γ L = 1(H L ε L F ) H L = 1 2 L + V L acting on L 2 per(γ L ) V L = (ρ γ L µ per ) ΓL G L. David Gontier Simulation of crystals 26 / 33

27 Question: What is the speed of convergence of L 3 EL rhf IrHF per? Remark. Non-linear setting: The potential V depends on L: H per = V per and H L = 1 2 L + V L. Idea: Construct good test functions for the minimization problems. Problem: γ per P per and γ L P L cannot be compared. By convexity of the problems w.r.t. the density ρ, we deduce that ρ γper and ρ γ L are R-periodic. V per and V L are R-periodic. = Construct test functions from ρ γper and ρ γ L, and use the results for the linear case. David Gontier Simulation of crystals 27 / 33

28 Lemma (DG, Salma Lahbabi) There exists C R + and α > 0 such that, for all L L gap, L 3 EL rhf IrHF per Ce αl. (Convergence of the energy per unit cell) ρ γper ρ γ L Ce αl. (Convergence of the electronic density) L per Rates of convergence for the reduced Hartree-Fock model. David Gontier Simulation of crystals 28 / 33

29 Crystals with local defects David Gontier Simulation of crystals 29 / 33

30 General picture Perfect crystal H per = Vper acting on L2 per(γ). Local defect (charge) ν L 2 (R 3 ) with compact support in Γ. Crystal with local defect H ν := 1 ˆ 2 + V ν(y) per + R 3 y dy. Local defects How to define the energy of the defect? The energy per cell is no longer a meaningful quantity. The energy of the crystal (with and without defect) is infinite. Idea: Perform a supercell thermodynamic limit. David Gontier Simulation of crystals 30 / 33

31 Supercell thermodynamic limit Supercell local defect ν ν L L 2 per(γ L ) such that ν L = ν on Γ L. Local defect Supercell defect The defect interacts with its periodic images! Roadmap For all L N, the supercell energy of the crystal with and without defect is finite. Define the supercell energy of the defect as Jν L = Eν L E L. Does Jν L have limit as L? Yes in rhf 7 : J ν. Can we characterize the limit? Yes, as a minimization problem. What is the speed of convergence of Jν L J ν? 7 E. Cancès, A. Deleurence, and M. Lewin, Commun. Math. Phys. 281 (2008) David Gontier Simulation of crystals 31 / 33

32 Lemma (DG, Salma Lahbabi) There exists η > 0, L N, C R +, and α > 0 such that, for all ν L 2 per (Γ) with compact support in Γ and satisfying ν L 2 < η, and for all L L, it holds that J ν Jν L 1 mq 2 ( L 2ϵ C ν 3 L 2 + ν 2 L 2 L 3 + ν L 2 e αl where q = R 3 ν is the net charge of the defect, m is the Madelung constant of the crystal, defined by m := lim x 0 G 1 (x) 1 x, and ϵ is the macroscopic dielectric constant of the crystal. Comments Recover the term predicted by Leslie and Gillan 8, and by Makov and Payne 9. Slow convergence in the defect case (interaction of the defect with its images). The term 1 mq 2 can be computed with low computational time. L 2ϵ = We can improve the numerical convergence by substracting by hand this term. ), 8 M. Leslie and M.J. Gillan. J. Phys. C 18 (1985). 9 G. Makov and M.C. Payne. Phys. Rev. B, 51 (1995). David Gontier Simulation of crystals 32 / 33

33 Conclusion Convergence of supercell models Convergence of Riemann sums. Exponential rate of convergence for perfect crystals. Slow rate of convergence for crystal with local defects. Identification of the L 1 term allows to speed the rate of convergence. Work in progress (see Antoine Levitt s talk) Speed of convergence in the metallic case. (with E. Cancès, V. Ehrlacher, A. Levitt and D. Lombardi) References DG, S. Lahbabi, Convergence rates of supercell calculations in the reduced Hartree-Fock model (M2AN, 50,5 (2016), arxiv ). DG, S. Lahbabi, Supercell calculations in the reduced Hartree-Fock model for crystals with local defects (Appl. Math. Res. Express (2016), arxiv ). Thank you for your attention. David Gontier Simulation of crystals 33 / 33