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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 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 6/25/2011 Hybrid functionals: DMS 2 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 6/25/2011 Hybrid functionals: DMS 3 Ab initio modeling Exact many electron Schrödinger Equation 2 2m Δ V(r1,..., rn ) Ψ(r 1,..., rn ) EΨ(r 1,..., rn ) e Complexity: basis set sizeNumber of electrons Wavefunctions based methods (HF+MP2, CCSD(T)) QMC Central idea: map onto “best” one-electron theory r1 , r2 ,..., rN S[1 (r1 ) , 2 (r2 ),..., N el (rN el )] Complexity: basis set size • Number of electrons 6/25/2011 Hybrid functionals: DMS 4 Kohn Sham Density functional theory Density and kinetic energy are the sum of one electron wave functions (r ) 2 | n (r ) |2 occ 2 e 2 (r ) (r ' ) 3 3 2me occ n (r )* n (r )d 3r V ion (r ) (r )d 3r 2 | r r' | d rd r' E xc [ (r )] KS functional has its minimum at the electronic ground state 2 2m V (r ) V (r ) V (r ) n (r ) Enn (r ) ion el xc e (r ' ) Possion : V el (r ) e2 d 3r' LDA : V xc (r ) V xc (r ) | r r' | 6/25/2011 Hybrid functionals: DMS 5 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 6/25/2011 Hybrid functionals: DMS 6 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 Eg ( E[ N 1] E[ N ]) ( E[ N ] E[ N 1]) A I ECBMIN [ N ] EVBMAX [ N ] in LDA/GGA Large errors in LDA/GGA/HF Lack of Integer-discontinuity in the LDA/GGA/HF 6/25/2011 Hybrid functionals: DMS 7 Hartree-Fock theory DFT : V xc (r )n (r ) Effective one electron equation 2 2m V (r ) V (r ) n (r ) V x (r, r' )n (r' )d 3r' Enn (r ) ion el e Lacks correlation, unoccupied states only Hartree pot. Exchange potential (anti-symmetry of wave functions in Slater determinant) e2 V x (r, r) m (r )m (r) * occ | r r | Hartree potential e2 V H (r ) d 3r m (r)m (r) * occ | r r | 6/25/2011 Hybrid functionals: DMS 8 One-electron theories Density functional theory 2 2m V (r ) V (r ) V (r ) n (r ) Enn (r ) ion el xc e Hartree Fock theory 2 2m V (r ) V (r ) n (r ) V x (r, r' )n (r' )d 3r' Enn (r ) ion el e GW 2 2m V (r ) V (r) n (r) xc (r, r' , )n (r' )d 3r' d Enn (r) ion el e 6/25/2011 Hybrid functionals: DMS 9 Where is the correlation The electrons move in the exchange potential screened by all other electrons L. Hedin, Phys. Rev. 139, A796 (1965) -1 6/25/2011 Hybrid functionals: DMS 10 Hybrid functionals: two one-electron theories Hartree-Fock 1 m (r ) (r) * V (r, r) x 4 e 2 m m | r r | Much too large band gaps Density-functional theory 3 Vxc (n(r )) 4 Too small band gaps Generalized Kohn-Sham schemes Seidl, Görling, Vogl, Majewski, Levy, Phys. Rev. B 53, 3764 (1996). 6/25/2011 Hybrid functionals: DMS 11 HSE versus PBEh: convergence of exchange energy with respect to k points1 Example: Aluminum - fcc PBEh HSE 1 J. Paier, M. Marsman, K. Hummer, G. Kresse, I.C. Gerber, and J.G. Angyan, J. Chem. Phys. 124, 154709 (2006). 6/25/2011 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 6/25/2011 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 % 6/25/2011 Hybrid functionals: DMS 15 Vibrational properties: Phonons Kresse, Furthmüller, Hafner, EPL 32, 729 (1995). K. Hummer, G. Kresse, in preparation. C Ge Si Sn 6/25/2011 Hybrid functionals: DMS 16 Vibrational Properties K. Hummer, G. Kresse, in preparation. C Ge Si Sn 6/25/2011 Hybrid functionals: DMS 17 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 <4 are quite accurate but these will be worse for the rest ! 6/25/2011 Hybrid functionals: DMS 18 Optical Absorptionspectra using PBE 6/25/2011 Hybrid functionals: DMS 19 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 εLDA εEXP RPA GaAs 12.8 11.1 Si 12.0 11.9 SiC 6.54 6.52 C 5.55 5.70 ZnO 5.12 3.74 LiF 1.97 1.91 6/25/2011 Hybrid functionals: DMS 20 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 εHSE εEXP Si C RPA GaAs 9.5 11.1 Si 10.20 11.9 SiC 5.65 6.52 C 4.92 5.70 ZnO 3.30 3.74 LiF 1.80 1.91 6/25/2011 Hybrid functionals: DMS 21 Proper Absorption-spectra using HSE: J.Paier, M. Marsman, G. Kresse, PRB 78, 121201(R) (2008) Accurate band gaps and accurate absorption spectra [Dyson Equ. ip ip ( v f xc ) ] Absorption spectrum χ=iGG G from GW 6/25/2011 Hybrid functionals: DMS 22 Proper Absorption-spectra using HSE: Now spectra are very reasonable Distribution of intensities is about right Remarkable accurate static properties Si C εHSE εEXP RPA GaAs 11.02 11.1 Si 11.37 11.9 SiC 6.44 6.52 C 5.59 5.70 ZnO 3.77 3.75 LiF 1.91 1.9 6/25/2011 Hybrid functionals: DMS 23 Multivalent oxides: Ceria J.L.F. Silva, …, G. Kresse, Phys. Rev. B 75, 045121 (2007). CB VB f Usual from DFT to hybrid unsual 6/25/2011 Hybrid functionals: DMS 25 3d transition metal oxides [1] Hybrids PBE HSE EXPT. substantially MnO ao 4.44 4.44 4.45 improve upon Eg 0.93 2.8 3.9 PBE FeO ao 4.30 4.33 4.33 HSE latt. const. Eg metal 2.2 2.4 and local spin CoO ao 4.22 4.26 4.25 mag. moments Eg metal 3.4 2.5 are excellent NiO ao 4.19 4.18 4.17 Eg 0.81 4.2 4.0 1. M. Marsman et al., J. Phys.: Condens. Matter 20, 64201 (2008). 6/25/2011 Hybrid functionals: DMS 26 3d metals: When hybrids fail Spin up Spin down Fe Hund„s rule ferromagnet using HSE 6/25/2011 Hybrid functionals: DMS 27 RPA correlation The electrons move in the exchange potential screened by all other electrons L. Hedin, Phys. Rev. 139, A796 (1965) -1 6/25/2011 Hybrid functionals: DMS 28 The right physics: screened exchange M. S. Hybertsen, S. G. Louie, Phys. Rev. B 34, 5390 (1986) Screened exchange: Screening system dependent Vacuum no screening Insulators For bulk materials dielectric weak screening matrix is diagonal in reciprocal space Semiconductors/ metals Ɛ-1(G) strong screening No screening for large G hybrids Strong screening for small G (static screening properties) Hybrids: ¼ is a compromise 6/25/2011 Hybrid functionals: DMS 29 GW0 approximation M. S. Hybertsen, S. G. Louie, Phys. Rev. B 34, 5390 (1986) Calculate DFT/hybrid functional wavefunctions 2 2m V (r ) V (r ) V (r ) n (r ) Enn (r ) ion el xc e Determine Green function and W using DFT wavefunctions Determine first order change of energies 2 n V ion V el (GW0 ) n En 2me Update Green‟s function and self-energy (W fixed to W0) m (r )m (r) G (r, r) m E Em i sgn[ Em E Fermi ] 6/25/2011 Hybrid functionals: DMS 30 PBE: GW0 band gaps1 Improvement over G 0W 0 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). 6/25/2011 Hybrid functionals: DMS 31 HSE: G0W0 band gaps1 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). 6/25/2011 Hybrid functionals: DMS 32 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) 6/25/2011 Hybrid functionals: DMS 33 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). 6/25/2011 Hybrid functionals: DMS 34 Cu2ZnSnS4 or CZTS DFT hybrid In this case HSE hybrid functional GW and GW give identical answers J. Paier, R. Asahi, A. Nagoya, and Georg Kresse, PRB 79, 115126 (2009). 6/25/2011 Hybrid functionals: DMS 35 GaN 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. 6/25/2011 Hybrid functionals: DMS 36 PBE results Ga3+ Mn3+ 4 electrons in 3 t2-orbitals majority 2 e-orbitals component 1 hole in t orbitals DFT predicts almost degenerate t2 orbitals Metallic behavior A. Stroppa and G. Kresse, PRB RC in print. 6/25/2011 Hybrid functionals: DMS 37 HSE results Ga3+ Mn3+ 4 electrons in majority component 1 hole in t orbitals HSE predicts a splitting within in t2 manifold Localized hole on Mn 6/25/2011 Hybrid functionals: DMS 38 GW results 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 6/25/2011 Hybrid functionals: DMS 39 Charge density PBE HSE PBE predicts symmetric solution HSE predicts D2d symmetry (no trigonal axis) A. Stroppa and G. Kresse, PRB RC in print. 6/25/2011 Hybrid functionals: DMS 40 Mn@GaAs GaN GaAs 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 6/25/2011 Hybrid functionals: DMS 41 Summary HSE is better compromise than classical local DFT functionals But a compromise it is Vacuum no Metals !! screening Insulators weak screening GW is more universal although not necessarily more accurate Semiconductors/ metals strong screening Why HSE works so well is not quite understood hybrids ¼ seems to be very good for states close to the Fermi level 6/25/2011 Hybrid functionals: DMS 42 Acknowledgement FWF for financial support And the group for their great work... You for listening 6/25/2011 Hybrid functionals: DMS 43