Wavefunction and electronic struture in solids: Bloch functions, Fermi level and other concepts.

Transcription

1 Wavefunction and electronic struture in solids: Bloch functions, Fermi level and other concepts. Silvia Casassa Università degli Studi di Torino July 12, 2017 Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems July 12, / 24

2 Outline 1 Symmetry exploitation in Reciprocal Space 2 SCF iterative procedure in Solids Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

3 3 - Fock matrix in Reciprocal Space Fock matrix diagonalization is still a dreadful task... Exploiting the periodic boundary condition! Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

4 Born-Von Karman boundary conditions Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

5 Born-Von Karman boundary conditions Periodic (or cyclic) Boundary Conditions the translational symmetry group becomes an abelian cyclic group; N N IRREP N Class N SymmOperators N cells Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

6 Born-Von Karman boundary conditions Periodic (or cyclic) Boundary Conditions the translational symmetry group becomes an abelian cyclic group; N N IRREP N Class N SymmOperators N cells Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

7 Born-Von Karman boundary conditions the characters of the operators are the N-rows of the unity for the k-irreducible representation the character of the identity is A(Ê) k = 1 exp[2πik] for the ˆT 1 operator: ( ˆT 1 ) N = ˆT x ˆT x... = Ê (A( ˆT 1 ) k ) N = exp[2πik)] A( ˆT 1 ) k = exp[2πi( k N )] for any operators ˆT T : ( ˆT T ) = ( ˆT 1 ) T A( ˆT T ) k = (A( ˆT 1 ) k ) T = exp[2πi( k N T )] Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

8 Born-Von Karman boundary conditions the characters of the operators are the N-rows of the unity for the k-irreducible representation the character of the identity is A(Ê) k = 1 exp[2πik] for the ˆT 1 operator: ( ˆT 1 ) N = ˆT x ˆT x... = Ê (A( ˆT 1 ) k ) N = exp[2πik)] A( ˆT 1 ) k = exp[2πi( k N )] for any operators ˆT T : ( ˆT T ) = ( ˆT 1 ) T A( ˆT T ) k = (A( ˆT 1 ) k ) T = exp[2πi( k N T )] Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

9 Born-Von Karman boundary conditions the characters of the operators are the N-rows of the unity for the k-irreducible representation the character of the identity is A(Ê) k = 1 exp[2πik] for the ˆT 1 operator: ( ˆT 1 ) N = ˆT x ˆT x... = Ê (A( ˆT 1 ) k ) N = exp[2πik)] A( ˆT 1 ) k = exp[2πi( k N )] for any operators ˆT T : ( ˆT T ) = ( ˆT 1 ) T A( ˆT T ) k = (A( ˆT 1 ) k ) T = exp[2πi( k N T )] Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

10 Given the Character Table of a group the Wigner-Seitz projector operators can be derived ˆ P k = T exp[2πi( k T)] ˆT N..and applied to get functions of proper Symmetry i.e. basis of each Irreducible Representation of the translational group ˆ P k χ 0 µ(r) = T exp[2πi( k N T)] ˆT χ 0 µ(r) Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

11 From Atomic Orbitals to Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

12 Bloch functions Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

13 Questions What is k? What can we do with Bloch Functions? Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

14 What is k? Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

15 What is k? Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

16 What is k? Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

17 What is k? Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

18 What is k? Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

19 What is k? Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

20 The momentum Space from the real to reciprocal space metric Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

21 The Reciprocal lattice A lattice with basis vectors b 1, b 2, b 3 corresponds to every real (or direct) lattice with lattice basis vectors a 1, a 2, a 3. Any lattice point can be written as (K 1, K 2, K 3 integer): K = K 1 b 1 + K 2 b 2 + K 3 b 3 Basis vectors obey the following orthogonality rules: b i a j = 2πδ ij the scalar product between any direct and reciprocal lattice vectors is: < K T > = 2π(K 1 t 1 + K 2 t 2 + K 3 t 3 ) Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

22 The first Brillouin zone Wigner-Seitz cell in reciprocal space The portion of space closer to the Gamma (Γ) lattice point than to anyone else is the Brillouin zone Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

23 Answers What is k? k is a point in the Reciprocal Space belonging to the first Brillouin zone < k T > = 2π N (k 1t 1 + k 2 t 2 + k 3 t 3 ) A( ˆT ) k = exp[ 2πi (k T)] = exp(ikt ) N What can we do with Bloch Functions? Crystalline Orbitals can be found as linear combination of BF Ψ k i (r) = µ = T c(k) µi Φ k µ(r) = µ exp(ikt) ϕ T i (r) c(k) µi exp(ikt) χ T µ(r) T Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

24 Consequences of Bloch Theorem 1- due to the orthogonality between functions belonging to different IR, elements of the Fock matrix in the BF basis set are orthogonal F µν (k 1, k 2 ) = < Φ k 1 µ (r) ˆF Φ k 2 ν (r) >= δ k1, k 2 Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

25 Consequences of Bloch Theorem 2- continuity of the energy in k-space, E(k): the BAND structure interpolation Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

26 SCF iterative procedure in Solids Outline 1 Symmetry exploitation in Reciprocal Space 2 SCF iterative procedure in Solids Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

27 SCF iterative procedure in Solids The procedure from direct lattice, DL to reciprocal lattice, RL 1 Construction of the (M 2 xn) elements of Fock matrix, F T µν, in the AOs basis set 2 Fourier tranformation of F T µν at every point k of the Pack-Monkhorst net (k BZ) F µν (k) = N T exp (ikt ) F T µν 3 Diagonalization of every M-dimension Fock matrix F(k)C(k) = S(k)C(k)E(k) 4 In each k, M crystalline orbitals (CO) Ψ j (k, r) and eigenvalues ɛ j (k) are obtained: Ψ j (k, r) = M µ c µj(k)φ k µ (r) Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

28 SCF iterative procedure in Solids F(k)C(k) = S(k)C(k)E(k) {c(k)} 5 Calculation of the RL-density matrix P k µν = j c µj(k)c νj (k) 6 back-fourier transformation of the density matrix P T µν = k exp( ik T ) P k µν 7 from P T µν to a new F T µν Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

29 SCF iterative procedure in Solids geometry initial information basis set, {χ µ (r)} guess for the density matrix, P µν calculation of F µν (P) P µν = 2 occ j cjµ c jν update the new P matrix diagonalize F FC = SCE no E n E n 1 < TOL? yes stop Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

30 SCF iterative procedure in Solids geometry initial information basis set, {χ 0 µ(r)} k-grid in the first BZ guess for the density matrix, P g µν calculation of F g µν(p) F (k) µν = g exp(ikg ) F g µν P(k) µν = 2 occ j c (k) jµ c(k) jν P g µν = k exp(ikg ) P(k) µν diagonalize F (k) matrices F (k)c(k) = S(k)C(k)E(k) no E n E n 1 < TOL? yes stop Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

31 SCF iterative procedure in Solids Fermi energy insulator system n states = 2 M N θ[ɛ F ɛ j (k)] N j k 2 M = θ[ɛ F ɛ j (k)]dk Ω BZ j Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

32 SCF iterative procedure in Solids Fermi energy insulator system n states = 2 M N θ[ɛ F ɛ j (k)] N j k 2 M = θ[ɛ F ɛ j (k)]dk Ω BZ j metallic systems: definition of a new function and iterative evaluation of it n states (ɛ) = 2 Ω BZ M j θ[ɛ F ɛ j (k)]dk Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

33 SCF iterative procedure in Solids Acknowledgments I would like to thank: the whole CRYSTAL group Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

34 SCF iterative procedure in Solids Acknowledgments I would like to thank: the whole CRYSTAL group and invite all of you at Platform 9 and 3/4 where some strange boundary conditions are waiting for us... Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24

35 SCF iterative procedure in Solids Acknowledgments I would like to thank: the whole CRYSTAL group and invite all of you at Platform 9 and 3/4 where some strange boundary conditions are waiting for us... Minnesota Workshop on ab initio MSC Symmetry and Periodic Systems Minneapolis - July / 24