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Hybrid functionals: Dilute Magnetic semiconductors

Hybrid functionals: Dilute Magnetic semiconductors . Georg Kresse J. Paier , K. Hummer, M. Marsman , A. Stroppa Faculty of Physics, University of Vienna and Center for Computational Materials Science Funded by the Austrian FWF. Overview.

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Hybrid functionals: Dilute Magnetic semiconductors

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  1. Hybrid functionals: Dilute Magnetic semiconductors Georg KresseJ. Paier, K. Hummer, M. Marsman, A. Stroppa Faculty of Physics, University of Viennaand Center for Computational Materials Science Funded by the Austrian FWF

  2. Overview • GOAL: Good description ofband structures, magnetic properties and magnetic defects at reasonable cost • DFT and Hybrid functionals • When hybrid functionals are better than DFT • Prototypical solids: lattice constants and bulk moduli • Band gaps • Vibrational properties • Static and dynamic dielectric function • Magnetic properties: TM, TMO, ceria, DMS • Why hybrid functionals are (not) good enough Hybrid functionals: DMS

  3. Take home messages • Hybrid functionals are a step forward compared to local functionals except for itinerant systems • But not a universal improvement • ¼ exact exchange is a good compromise for semiconductors and some insulators • Band gaps • Optical properties • Structural properties • Going further is difficult • Test results using GW Hybrid functionals: DMS

  4. Ab initio modeling • Exact many electron Schrödinger Equation • Complexity:basis set sizeNumber of electrons • Wavefunctions based methods (HF+MP2, CCSD(T)) • QMC • Central idea: map onto “best” one-electron theory • Complexity:basis set size • Number of electrons Hybrid functionals: DMS

  5. Density and kinetic energy are the sum of one electron wave functions KS functional has its minimum at the electronic ground state Kohn Sham Density functional theory Hybrid functionals: DMS

  6. DFT Problems • Precision of total energies • Heats of formation of molecules are wrong by up to 0.5 eV/molvolume errors and errors in elastic constants • Van der Waals bonding • Self interaction error: no electron localizationsemiconductor modelling, magnetic properties • One most go beyond a traditional one electron treatment Wave function based methodsused in quantum chemistryCCSD(T), RPA Quantum Monte-Carlo Hybrid functionals: DMS

  7. One of the great lies: The band gap problem • DFT is only accurate for ground state propertieshence the error in the band gap does not matter • The band gap is a well defined ground state property wrong using local and semi-local DFT • Fundamental gap Large errors in LDA/GGA/HF • Lack of Integer-discontinuityin the LDA/GGA/HF Hybrid functionals: DMS

  8. Hartree-Fock theory • Effective one electron equation • Lacks correlation, unoccupied states only Hartree pot. • Exchange potential (anti-symmetry of wave functions in Slater determinant) • Hartree potential Hybrid functionals: DMS

  9. One-electron theories • Density functional theory • HartreeFock theory • GW Hybrid functionals: DMS

  10. Where is the correlation The electrons move in the exchange potential screened by all other electrons L. Hedin, Phys. Rev. 139, A796 (1965) -1 Hybrid functionals: DMS

  11. Hybrid functionals: two one-electron theories Hybrid functionals: DMS • Hartree-Fock • Much too large band gaps • Density-functional theory • Too small band gaps • Generalized Kohn-Sham schemes Seidl, Görling, Vogl, Majewski, Levy, Phys. Rev. B 53, 3764 (1996).

  12. HSE versus PBEh: convergence of exchange energy with respect to k points1 Example: Aluminum - fcc HSE PBEh 1 J. Paier, M. Marsman, K. Hummer, G. Kresse, I.C. Gerber, and J.G. Angyan, J. Chem. Phys. 124, 154709 (2006). Hybrid functionals: DMS

  13. PBE: Lattice constants and bulk moduli Paier, M. Marsmann, K. Hummer, G. Kresse,…, J. Chem. Phys. 122, 154709 (2006) PBE: MRE 0.8 %, MARE 1.0 % Lattice constants PBE: MRE -9.8 %, MARE 9.4 % Bulk moduli Hybrid functionals: DMS

  14. HSE: Lattice constants and bulk moduli Paier, Marsmann, Hummer, Kresse,…, J. Chem. Phys. 122, 154709 (2006) PBE: MRE 0.8 %, MARE 1.0 % HSE: MRE 0.2 %, MARE 0.5 % PBE: MRE -9.8 %, MARE 9.4 % HSE: MRE -3.2 %, MARE 6.4 % Hybrid functionals: DMS

  15. Vibrational properties: Phonons Ge C Si Sn Hybrid functionals: DMS Kresse, Furthmüller, Hafner, EPL 32, 729 (1995). K. Hummer, G. Kresse, in preparation.

  16. Vibrational Properties Ge C Si Sn Hybrid functionals: DMS K. Hummer, G. Kresse, in preparation.

  17. Hybrid functionals for solids: Band gaps <4 Hybrid functionals: DMS Band gaps improved But fairly larger errors prevail for materials with weak screening(ε<4) for these materials half-half functionals are quite accurate but these will be worse for the rest !

  18. Optical Absorptionspectra using PBE Hybrid functionals: DMS

  19. Two Problems Hybrid functionals: DMS • Red shift of spectrum compared to experiment • Too weak cross scattering cross section at low energies • In many cases these effects compensate each other • Dominant peak in C in pretty much spot on • Static properties are pretty good in DFT

  20. Better band gaps: HSE results Si C Hybrid functionals: DMS • Now onset of optical absorption is quite reasonable • But too weak cross section at low energies • Error compensation is gone • Reduction of intensity by ω/ (ω+Δω)Required by sum rule

  21. Proper Absorption-spectra using HSE: J.Paier, M. Marsman, G. Kresse, PRB 78, 121201(R) (2008) Absorption spectrum χ=iGG G from GW Hybrid functionals: DMS Accurate band gaps and accurate absorption spectra [Dyson Equ. ]

  22. Proper Absorption-spectra using HSE: Si C Hybrid functionals: DMS Now spectra are very reasonable Distribution of intensities is about right Remarkable accurate static properties

  23. Multivalent oxides: Ceria J.L.F. Silva, …, G. Kresse, Phys. Rev. B 75, 045121 (2007). VB CB f Usual from DFT to hybrid unsual Hybrid functionals: DMS

  24. 3d transition metal oxides [1] • Hybrids substantially improve upon PBE • HSE latt. const. and local spin mag. moments are excellent M. Marsmanet al., J. Phys.: Condens. Matter 20, 64201 (2008). Hybrid functionals: DMS

  25. 3d metals: When hybrids fail Spin up Spin down Fe Hund‘s ruleferromagnet using HSE Hybrid functionals: DMS

  26. RPA correlation The electrons move in the exchange potential screened by all other electrons L. Hedin, Phys. Rev. 139, A796 (1965) -1 Hybrid functionals: DMS

  27. The right physics: screened exchange M. S. Hybertsen, S. G. Louie, Phys. Rev. B 34, 5390 (1986) • Screened exchange: • Screening system dependent • For bulk materials dielectricmatrix is diagonal in reciprocalspace • Ɛ-1(G) • No screening for large G • Strong screening for small G(static screening properties) • Hybrids: ¼ is a compromise Vacuum no screening Insulators weakscreening Semiconductors/ metals strong screening hybrids Hybrid functionals: DMS

  28. GW0approximation M. S. Hybertsen, S. G. Louie, Phys. Rev. B 34, 5390 (1986) • Calculate DFT/hybrid functional wavefunctions • Determine Green function and W using DFT wavefunctions • Determine first order change of energies • Update Green’s function and self-energy (W fixed to W0) Hybrid functionals: DMS

  29. PBE: GW0 band gaps1 • Improvement over G0W0 • G0W0: MARE 8.5 % • GW0 : MARE 4.5 % • Overall still slightly too small, in particular for materials with shallow d-electrons 1 M. Shishkin, G. Kresse, Phys Rev. B 75, 235102 (2007). Hybrid functionals: DMS

  30. HSE: G0W0band gaps1 • About same quality as using PBE wave functions and screening properties • Overall slightly too large 1F. Fuchs, J. Furthmüller, F. Bechstedt, M. Shishkin, G. Kresse, PRB 76, 115109 (2007). Hybrid functionals: DMS

  31. Self-consistent QPGWTC-TC band gaps1 • Excellent results across all materials • MARE: 3.5 % • Further slight improvement over GW0 (PBE) • Too expensive for large scale applications but fundamentally important 1 M. Shishkin, M. Marsman, PRL 95, 246403 (2007) Hybrid functionals: DMS

  32. Strategy for true ab-initio modelling Hybrid functionals: DMS • Apply HSE functional as zero order description • Perform GW on top of the HSE functional • Screening properties are determined either using PBE or HSE • A little bit of pragmatism is used to select on which level the screening properties are calculated • For most materials PBE screening properties are very good • If band the PBE gap is inverted or much too small, HSE screening properties are preferable • Initial wave functions are from HSE, since they are usually closer to GW wave functions • Fairly efficient F. Fuchs, J. Furthmüller, F. Bechstedt, M. Shishkin, G. Kresse, PRB 76, 115109 (2007). J. Paier, M. Marsman, G. Kresse, PRB 78, 121301(R) (2008).

  33. Cu2ZnSnS4 or CZTS DFT hybrid GW J. Paier, R. Asahi, A. Nagoya, and Georg Kresse, PRB 79, 115126 (2009). Hybrid functionals: DMS In this case HSE hybrid functional and GW give identical answers

  34. GaN Hybrid functionals: DMS Lattice constant a, bulk-modulus B0, energy gap at , L, X, dielectric constant , valence band-width W, and the energy position of Ga d states determined using PBE, HSE and GW0.

  35. PBE results 3 t2-orbitals 2 e-orbitals A. Stroppa and G. Kresse, PRB RC in print. Hybrid functionals: DMS Ga3+ Mn3+4 electrons in majority component 1 hole in t orbitals DFT predicts almost degenerate t2 orbitals Metallic behavior

  36. HSE results Hybrid functionals: DMS Ga3+ Mn3+4 electrons in majority component 1 hole in t orbitals HSE predicts a splitting within in t2 manifold Localized hole on Mn

  37. GW results Hybrid functionals: DMS Ga3+ Mn3+4 electrons in majority component 1 hole in t orbitals HSE predicts a splitting within in t2 manifold Localized hole on Mn GW confirms results

  38. Charge density PBE HSE A. Stroppa and G. Kresse, PRB RC in print. Hybrid functionals: DMS PBE predicts symmetric solution HSE predicts D2d symmetry (no trigonal axis)

  39. Mn@GaAs GaN GaAs Hybrid functionals: DMS Ga3+ Mn3+4 electrons in majority component 1 hole in t orbitals HSE predicts no splitting within in t2 manifold Strong hybridization with valence band Delocalized hole

  40. Summary Vacuum no screening Insulators weakscreening Semiconductors/ metals strong screening hybrids Hybrid functionals: DMS • HSE is better compromise than classical local DFT functionals • But a compromise it isMetals !! • GW is more universalalthough not necessarily more accurate • Why HSE works so wellis not quite understood¼ seems to be very goodfor states close to the Fermi level

  41. Acknowledgement Hybrid functionals: DMS FWF for financial support And the group for their great work... You for listening

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