A New First-Principles Computational Method for Disordered Materials

1. A New First-Principles Computational Method for Disordered Materials Wei Ku CM-Theory, CMPMSD, Brookhaven National Lab Department of Physics, SUNY Stony Brook Tom Berlijn, Dmitri Volja CM-Theory, CMPMSD, Brookhaven National Lab Department of Physics, SUNY Stony Brook

2. Acknowledgment • DOE/BES + BNL-LDRD (John Hill & Wei Ku)

3. Relevance Inter-atomic bonding entirely controlled by the electronic structure Defects, vacancy, chemical substitution, doping & intercalation Thermal/quantum fluctuation induced spatial inhomogeneity Paramagnetic phase with disordered local moment (no intrinsic difficulty with high pressure) Different approaches Mean-field potential: virtual crystal approximation Mean-field scattering: coherent potential approximation Perturbation: diagrammatic summation Non-perturbed direct method: ensemble average over super cell First-Principles Methods for Disordered Materials

4. Localization  Inter-atomic interaction depends on correlation between disorders, not just a local property! Difficulty with Large-Sized Impurity States

5. Ensemble Average over Super Cell disordered system configuration N configuration 1 + … + ≈ 1/N <G> ≈ 1/N( G1 + … + GN )

6. New Methods First-principles Wannier functions Unfolding band structure Effective Hamiltonian Case Studies Na-doped Cobaltates (thermoelectric material) Cu-doped ZnO with oxygen vacancy (photovoltaic material & room temperature dilute magnetic semiconductor) Recent Developments

7. Energy (eV) truncated First-Principles Wannier Functions Wannier functions DFT CoO2 Wei Ku et al, PRL 2002 see also W. Yin & Wei Ku, PRL 2006 B. Larson et al, PRL 2007 P. Abbamonte et al, PNAS 2008 I. Zaliznyak et al, Nature Physics 2009

8. Folded Band Structure of Super Cell normal cell: CoO2 super cell: NaCo3O6

9. L L H H A A Γ Γ K K M M Γ Γ A A Unfolded Band Structure normal cell: CoO2 super cell: NaCo3O6 Energy (eV)

10. Unfolded Band Structure Zn1-xCuxO LDA+U spin ↓ X = 0

11. Unfolded Band Structure Zn1-xCuxO LDA+U spin ↓ X = 1/8

12. Unfolded Band Structure Zn1-xCuxO LDA+U spin ↓ X = 1/4

13. Unfolded Band Structure Zn1-xCuxO LDA+U spin ↓ X = 1

14. Effective Hamiltonian via Wannier Functions Keep only linear influence of disorder for now

15. Step 1. Density Functional Theory two DFT Calculations 1 impurity (per super cell) undoped (normal cell)

16. Step 2. Wannier Transformation DFT undoped 2 Wannier transformations Energy Reduced Hilbert space k DFT 1 impurity 2 Tight Binding Hamiltonians undoped 1 impurity Energy k

17. Step 3. Effective Hamiltonian Influence 1 impurity: Hamiltonian N impurities:

18. Test Linearity & Partition: NaxCoO2 Effective Hamiltonian DFT Energy (eV) Co eg Co ag Co eg’ 2019 LAPW’s + 164 LO’s self consistency 66 Wanier Functions 1 diagonalization

19. spin spin spin spin 8 8 8 8 4 4 4 4 0 0 0 0 -4 -4 -4 -4 -8 -8 -8 -8 Test Linearity : Zn1-xCuxO (rock salt) Effective Hamiltonian DFT Energy (eV)

20. Test Partition : Zn1-xCuxO (rock salt) spin spin spin spin 8 8 8 8 4 4 4 4 0 0 0 0 -4 -4 -4 -4 -8 -8 -8 -8 Effective Hamiltonian DFT Energy (eV)

21. Ensemble Average over Super Cell disordered system configuration N configuration 1 + … + ≈ 1/N <G> ≈ 1/N( G1 + … + GN )

22. Intercalation: NaxCoO2 ARPES2 LDA1 ag eg’ Q) Does Na disorder destroy eg’ pockets3 ? • D.J. Singh, PRB 20, 13397 (2000) • D. Qian et al, PRL 97 186405 (2006) • David J. Singh et al, PRL 97, 016404-1 (2006)

23. -0.2 0.0 0.2 -0.2 0.0 0.2 -3.8 -4.0 -4.2 0.1 0.2 Intercalation: NaxCoO2 Co-eg’ Co-ag Energy (eV) O-p A) Na disorder does not destroy eg’ ! T. Berlijn, D. Volja, and Wei Ku, to be published

24. Substitution & Vacancies: Zn1-xCuxO1-y SQUID ZnO @ 300K (O-poor) ZnO:Cu @ 300K (O-rich) ZnO:Cu @ 5K (O-poor) ZnO:Cu @ 300K (O-poor) Q1) Where do the doped electrons reside? Q2) What is the role of oxygen vacancy? T. S. Herng, D.-C. Qi,…., T. Berlijn, W. Ku, to be published

25. VO Cu d 9 Cu d 10 spin majority spin minority VO VO Cu-d Cu-d Energy (eV) A1) Doped electrons reside in Cu orbitals A2) VO mediates Cu moments T. S. Herng, D.-C. Qi, T. Berlijn, 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

26. Conclusion • Electronic structure of disordered materials • Defects, vacancy, chemical substitution, doping & intercalation • Thermal/quantum fluctuation induced spatial inhomogeneity • Paramagnetic phase with disordered local moment • Non-perturbative approach • Hilbert space reduction via Wannier functions • unfolding band structure • linear construction of effective Hamiltonian • ensemble average over configurations • Test cases • Na-doped cobaltates • Cu-doped ZnO with oxygen vacancy • Next: including consideration of total energy

27. QCDW QCDW Energy (eV) L H A Γ K M Γ A Unfolded Band Structure super cell: NaCo3O6

28. Mean-Field Approaches VCA: Virtual Crystal Approximation Vvirtual crystal = (1-x) V0 + x V1 no scattering CPA: Coherent Potential Approximation lack cluster-scattering A. Gonis, “Green functions for ordered and disordered systems” (1992)

29. Difficulty with Band Edge States Energy K DOS disorder cluster-scattering important Energy DOS K