1. 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

2. 29/15/2015Hybrid functionals: DMS Overview GOAL: Good description of band 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

3. 39/15/2015Hybrid functionals: DMS 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

4. 49/15/2015Hybrid functionals: DMS Exact many electron Schrödinger Equation Complexity: basis set size Number 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 Ab initio modeling

5. 59/15/2015Hybrid functionals: DMS 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

6. 69/15/2015Hybrid functionals: DMS DFT Problems Precision of total energies Heats of formation of molecules are wrong by up to 0.5 eV/mol volume errors and errors in elastic constants Van der Waals bonding Self interaction error: no electron localization semiconductor modelling, magnetic properties One most go beyond a traditional one electron treatment Quantum Monte-Carlo Wave function based methods used in quantum chemistry CCSD(T), RPA

7. 79/15/2015Hybrid functionals: DMS One of the great lies: The band gap problem DFT is only accurate for ground state properties hence 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-discontinuity in the LDA/GGA/HF

8. 89/15/2015Hybrid functionals: DMS 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

9. 99/15/2015Hybrid functionals: DMS One-electron theories Density functional theory Hartree Fock theory GW

10. 109/15/2015Hybrid functionals: DMS Where is the correlation The electrons move in the exchange potential screened by all other electrons L. Hedin, Phys. Rev. 139, A796 (1965)

11. Hybrid functionals: two one-electron theories 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). 119/15/2015Hybrid functionals: DMS

12. PBEh and HSE functional The PBEh (PBE0) exchange-correlation functional 1 The HSE03 (HSE06) functional 2 129/15/2015Hybrid functionals: DMS 1.J. Perdew, M. Ernzerhof, and K. Burke, J. Chem. Phys. 105, 9982 (1996). 2.J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003).

13. HSE versus PBEh: convergence of exchange energy with respect to k points 1 139/15/2015Hybrid functionals: DMS 1 J. Paier, M. Marsman, K. Hummer, G. Kresse, I.C. Gerber, and J.G. Angyan, J. Chem. Phys. 124, 154709 (2006). Example: Aluminum - fcc PBEh HSE

14. PBE: Lattice constants and bulk moduli 149/15/2015Hybrid functionals: DMS Lattice constants Bulk moduli Paier, M. Marsmann, K. Hummer, G. Kresse,…, J. Chem. Phys. 122, 154709 (2006) PBE: MRE 0.8 %, MARE 1.0 % PBE: MRE -9.8 %, MARE 9.4 %

15. HSE: Lattice constants and bulk moduli 159/15/2015Hybrid functionals: DMS HSE: MRE 0.2 %, MARE 0.5 % HSE: MRE -3.2 %, MARE 6.4 % PBE: MRE 0.8 %, MARE 1.0 % PBE: MRE -9.8 %, MARE 9.4 % Paier, Marsmann, Hummer, Kresse,…, J. Chem. Phys. 122, 154709 (2006)

16. Vibrational properties: Phonons Kresse, Furthmüller, Hafner, EPL 32, 729 (1995). K. Hummer, G. Kresse, in preparation. 169/15/2015Hybrid functionals: DMS C Si Sn Ge

17. Vibrational Properties K. Hummer, G. Kresse, in preparation. 179/15/2015Hybrid functionals: DMS C Si Sn Ge

18. Hybrid functionals for solids: Band gaps 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 ! 189/15/2015Hybrid functionals: DMS  <4

19. Optical Absorptionspectra using PBE 199/15/2015Hybrid functionals: DMS

20. Two Problems 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 209/15/2015Hybrid functionals: DMS ε LDA RPA ε EXP GaAs12.811.1 Si12.011.9 SiC6.546.52 C5.555.70 ZnO5.123.74 LiF1.971.91

21. Better band gaps: HSE results 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 219/15/2015Hybrid functionals: DMS ε HSE RPA ε EXP GaAs9.511.1 Si10.2011.9 SiC5.656.52 C4.925.70 ZnO3.303.74 LiF1.801.91 SiC

22. Proper Absorption-spectra using HSE: Accurate band gaps and accurate absorption spectra [Dyson Equ. ] 229/15/2015Hybrid functionals: DMS Absorption spectrum  χ =iGG G from GW J.Paier, M. Marsman, G. Kresse, PRB 78, 121201(R) (2008)

23. Proper Absorption-spectra using HSE: Now spectra are very reasonable Distribution of intensities is about right Remarkable accurate static properties 239/15/2015Hybrid functionals: DMS ε HSE RPA ε EXP GaAs11.0211.1 Si11.3711.9 SiC6.446.52 C5.595.70 ZnO3.773.75 LiF1.911.9 SiC

24. Solve Cassidas equation Requires the diagonalisation of a large matrix with the dimension equal to number of electron-holes pairs Similar to usual BSE equation Includes an electrostatic interaction between electrons and holes from change of exchange potential Bethe Salpeter Equ. ε ab initio screening, hybrids ε=¼ 249/15/2015Hybrid functionals: DMS

25. Multivalent oxides: Ceria 259/15/2015Hybrid functionals: DMS CB VB f Usual from DFT to hybrid unsual J.L.F. Silva, …, G. Kresse, Phys. Rev. B 75, 045121 (2007).

26. 3d transition metal oxides [1] Hybrids substantially improve upon PBE HSE latt. const. and local spin mag. moments are excellent 269/15/2015Hybrid functionals: DMS 1.M. Marsman et al., J. Phys.: Condens. Matter 20, 64201 (2008). PBEHSEEXPT. MnO aoEgaoEg 4.44 0.93 4.44 2.8 4.45 3.9 FeO aoEgaoEg 4.30 metal 4.33 2.2 4.33 2.4 CoO aoEgaoEg 4.22 metal 4.26 3.4 4.25 2.5 NiO aoEgaoEg 4.19 0.81 4.18 4.2 4.17 4.0

27. 3d metals: When hybrids fail 279/15/2015Hybrid functionals: DMS Fe Hund‘s rule ferromagnet using HSE Spin up Spin down

28. 289/15/2015Hybrid functionals: DMS RPA correlation The electrons move in the exchange potential screened by all other electrons L. Hedin, Phys. Rev. 139, A796 (1965)

29. 299/15/2015Hybrid functionals: DMS The right physics: screened exchange Screened exchange: Screening system dependent For bulk materials dielectric matrix is diagonal in reciprocal space Ɛ -1 (G) No screening for large G Strong screening for small G (static screening properties) Hybrids: ¼ is a compromise M. S. Hybertsen, S. G. Louie, Phys. Rev. B 34, 5390 (1986) Vacuum no screening Insulators weak screening Semiconductors/ metals strong screening hybrids

30. 309/15/2015Hybrid functionals: DMS GW 0 approximation 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 W 0 ) M. S. Hybertsen, S. G. Louie, Phys. Rev. B 34, 5390 (1986)

31. 319/15/2015Hybrid functionals: DMS PBE: GW 0 band gaps 1 Improvement over G 0 W 0 G 0 W 0 : MARE 8.5 % GW 0 : 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).

32. 329/15/2015Hybrid functionals: DMS HSE: G 0 W 0 band gaps 1 About same quality as using PBE wave functions and screening properties Overall slightly too large 1 F. Fuchs, J. Furthmüller, F. Bechstedt, M. Shishkin, G. Kresse, PRB 76, 115109 (2007).

33. 339/15/2015Hybrid functionals: DMS Self-consistent QPGW TC-TC band gaps 1 Excellent results across all materials MARE: 3.5 % Further slight improvement over GW 0 (PBE) Too expensive for large scale applications but fundamentally important 1 M. Shishkin, M. Marsman, PRL 95, 246403 (2007)

34. Strategy for true ab-initio modelling 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). 349/15/2015Hybrid functionals: DMS

35. Cu 2 ZnSnS 4 or CZTS In this case HSE hybrid functional and GW give identical answers 359/15/2015Hybrid functionals: DMS GW hybrid DFT J. Paier, R. Asahi, A. Nagoya, and Georg Kresse, PRB 79, 115126 (2009).

36. GaN Lattice constant a, bulk-modulus B 0, 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. 369/15/2015Hybrid functionals: DMS

37. PBE results 379/15/2015Hybrid functionals: DMS Ga 3+ Mn 3+ 4 electrons in majority component 1 hole in t orbitals DFT predicts almost degenerate t 2 orbitals Metallic behavior 2 e-orbitals 3 t 2 -orbitals A. Stroppa and G. Kresse, PRB RC in print.

38. HSE results 389/15/2015Hybrid functionals: DMS Ga 3+ Mn 3+ 4 electrons in majority component 1 hole in t orbitals HSE predicts a splitting within in t 2 manifold Localized hole on Mn

39. GW results 399/15/2015Hybrid functionals: DMS Ga 3+ Mn 3+ 4 electrons in majority component 1 hole in t orbitals HSE predicts a splitting within in t 2 manifold Localized hole on Mn GW confirms results

40. Charge density PBE predicts symmetric solution HSE predicts D 2d symmetry (no trigonal axis) 409/15/2015Hybrid functionals: DMS PBE HSE A. Stroppa and G. Kresse, PRB RC in print.

41. Mn@GaAs 419/15/2015Hybrid functionals: DMS Ga 3+ Mn 3+ 4 electrons in majority component 1 hole in t orbitals HSE predicts no splitting within in t 2 manifold Strong hybridization with valence band Delocalized hole GaNGaAs

42. Summary HSE is better compromise than classical local DFT functionals But a compromise it is Metals !! GW is more universal although not necessarily more accurate Why HSE works so well is not quite understood ¼ seems to be very good for states close to the Fermi level 429/15/2015Hybrid functionals: DMS Vacuum no screening Insulators weak screening Semiconductors/ metals strong screening hybrids

43. Acknowledgement FWF for financial support And the group for their great work... You for listening for listening 439/15/2015Hybrid functionals: DMS