Wannier Function Based First Principles Method for Disordered Systems

Transcription

1 Wannier Function Based First Principles Method for Disordered Systems Tom Berlijn Stony Brook University & Brookhaven National Laboratory Excitations in Condensed Matter: From Basic Concepts to Real Materials November 23 2:00pm KITP Santa Barbara

2 Acknowledgements Collaborators: Dmitri Volja & Wei Ku Funding: DOE BNL + LDRD (John Hill & Wei Ku)

3 Outline Introduction: Super Cell Approximation Method 1: Unfolded Bandstructure (Wannier function) Method 2: Effective Hamiltonian (Wannier function) Results: Disordered Na x CoO 2 & Zn 1-x Cu x O 1-y

4 Introduction: Super Cell Approximation

5 What kind of disorder? Not like But like substitution interstitial vacancy

6 Goal: average Green function of disordered systems from first principles <G > = G i config i 1) Indirect Method 2) Direct Method

7 1) Indirect Methods 1 VCA: Virtual Crystal Approximation V virtual crystal = (1-x) V 0 + x V 1 no scattering CPA: Coherent Potential Approximation no cluster-scattering 1) A. Gonis, Green functions for ordered and disordered systems (1992)

8 Example 1: States at the band edge Energy Energy cluster-scattering important K K DOS disorder DOS

9 Example 2: Large-sized impurity states

10 2) Direct Method SCA: Super Cell Approximation <G > 1/N( G G N ) disordered system configuration 1 configuration N 1/N + +

11 Wannier Function Based Methods 1. Unfolded Bandstructure 2. Effective Hamiltonian

12 Method 1: Unfolded Bandstructure

13 Notation normal cell crystal momentum r lattice vector n Wannier index super cell K crystal momentum J band index R lattice vector KJ > Bloch state N Wannier index

14 Folded Bandstructure normal cell super cell r R E( ) normal cell crystal momentum r lattice vector E(K) K super cell K crystal momentum J band index R lattice vector KJ > Bloch state

15 Spectral Function A n (, ) = - Im < n G ( ) n> = - Im < n KJ > 2 < KJ G ( ) KJ > KJ Spectral Weight normal cell crystal momentum r lattice vector K crystal momentum R lattice vector state Green s function on Eigen basis super cell J band index KJ > Bloch

16 Green s function on Eigen basis < KJ G ( ) KJ >= ( - E KJ + i0 + ) -1 Eigenvalue normal cell crystal momentum r lattice vector super cell K crystal momentum J band index R lattice vector KJ > Bloch state

17 < n KJ >= Spectral Weight RN < n RN><RN KN><KN KJ > <KN KJ > = N (R/r) 3/2 e -i r nm K Eigenvector normal cell crystal momentum r lattice vector super cell K crystal momentum J band index R lattice vector KJ > Bloch state

18 Unfolded Bandstructure normal cell super cell r A n (, ) normal cell crystal momentum r lattice vector < n KJ > 0 K crystal momentum R lattice vector state super cell R A n (, ) J band index KJ > Bloch

19 Unfolded Bandstructure Example 1: Na 2 Co 6 O 12

20 Unfolded Bandstructure Example 1: Na x CoO 2 normal cell Co 2 O 4 Brillouin zone super cell Na 4 Co 6 O 12

21 Backfolded Bandstructure normal cell: Co 2 O 4 super cell: Na 4 Co 6 O 12 Energy (ev) Energy (ev) Co-e g Co-a g Co-e g O-p

22 Unfolded Bandstructure normal cell: Co 2 O 4 super cell: Na 4 Co 6 O 12 Energy (ev) Energy (ev) Co-e g Co-a g Co-e g O-p

23 Unfolded Bandstructure super cell: Na 4 Co 6 O 12 Q CDW Energy (ev) Q CDW Co-e g Co-a g Co-e g O-p

24 Unfolded Bandstructure Example 2: Zn 1-x Cu x O LDA+U spin

25 Zn 1-x Cu x O LDA+U spin X =0

26 Zn 1-x Cu x O LDA+U spin X =1/8

27 Zn 1-x Cu x O LDA+U spin X =1/4

28 Zn 1-x Cu x O LDA+U spin X =1

29 Method 2: Effective Hamiltonian

30 Concept: Linearity Drop impurity correlations in Hamiltonian (not in solutions) Influence impurity = linear

31 drop impurity correlations in Hamiltonian (not in solutions)

32 Construction 1. DFT doped & undoped 2. Wannier-transformation 3. Linear superposition

33 1) Density Functional Theory two DFT Calculations undoped (normal cell) 1 impurity (per super cell)

34 k 2) Wannier transfomation DFT undoped 2 Wannier transformations Energy Energy k DFT 1 impurity 2 Tight Binding Hamiltonians undoped 1 impurity

35 3) Linear Superposition Influence 1 impurity: Hamiltonian N impurities:

36 Testing DFT v.s. effective Hamiltonian 1. Linearity: x 1 dft > x eff 2. Partition: H eff break symmetry H 1 dft

37 Test Linearity & Partition: Na x CoO 2 x=0 x=2/3 x=1/8

38 Test Linearity & Partition: Na x CoO 2 DFT Effective Hamiltonian Energy (ev) Co-e g Co-a g Co-e g O-p 2019 LAPW s LO s self consistency 66 Wanier Functions 1 diagonalization

39 Test Linearity Zn 1-x Cu x O (rock salt) x=0 x=1 x=1/4

40 ZnO Reminder CuO

41 Test Linearity : Zn1-xCuxO (rock salt) 8 DFT spin 8 Effective Hamiltonian spin Energy (ev) spin spin

42 Test Partition Zn 1-x Cu x O (rock salt) x=0 x=1/8 x=1/4

43 Test Partition : Zn1-xCuxO (rock salt) 8 DFT spin 8 Effective Hamiltonian spin Energy (ev) spin spin

44 Disordered Results : Na x CoO 2 & Zn 1-x Cu x O 1-y

45 Disordered Results big super cells (100 s of atoms) disordered system configuration 1 configuration N 1/N + + <G > 1/N( G G N )

46 Intercalation: NaxCoO2 a g LDA 1 ARPES 2 e g Q) Does Na disorder destroy e g pockets 3? 1) D.J. Singh, PRB 20, (2000) 2) D. Qian et al, PRL (2006) 3) David J. Singh et al, PRL 97, (2006)

47 Na x CO 2 : x configurations of ~200 atoms configuration 1 configuration

48 Energy (ev) Co-e g Co-a g O-p A) Na disorder does not destroy e g 1 1) T. B., D. Volja, W. Ku, to be published

49 Substitution & Vacancies: Zn1-xCuO1-y SQUID 1 300K (O-poor) 300K (O-rich) 5K (O-poor) 300K (O-poor) Q) What is the role of oxygen vacancy? 1) T. S. Herng, D.-C. Qi, T. B, J. B. Yi, K. S. Yang, Y. Dai, Y. P. Feng, I. Santoso, C. H. Sanchez, X. Y. Gao, A. T. S. Wee, W. Ku, J. Ding, A. Rusydi, to be published

50 Zn 1-x Cu x O 1-y : x 0.02, y 0.01, M 0.8 B /Cu 10 configurations of ~200 atoms configuration 1 configuration 10 layer 1 layer 2 layer 1 layer 2 layer 3 layer layer 5 layer 6 = ZnO = Cu-up = Cu-dn = Vo layer 3 layer 4

51 spin minority spin majority Energy (ev) Cu-d V O Cu-d V O A) To mediate Cu moments no vacancies with vacancies V O Cu d 9 Cu d 10

52 Next New materials: CaC 6, Fe 1+y Te 1-x Se x, (solid solution GaN/ZnO with Phil Allen & Li Li?) Short range order, 2-particle Green function Extensions of the method: - Total energy - Higher-order impurity correlations - Self-consistency - Relaxed atoms

53 Summary Introduction: Super Cell Approximation Method 1: Unfolded Bandstructure (Wannier function) Method 2: Effective Hamiltonian (Wannier function) Results: Disordered Na x CoO 2 & Zn 1-x Cu x O 1-y