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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 )  Enn (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'  Enn (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 )  Enn (r )
                      ion      el       xc
                                                
                  e                            
      Hartree Fock theory

 2                       

 2m     V (r )  V (r ) n (r )   V x (r, r' )n (r' )d 3r'  Enn (r )
            ion      el
                           
    e                     
      GW
 2                      

 2m     V (r )  V (r) n (r)    xc (r, r' ,  )n (r' )d 3r' d  Enn (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 )  Enn (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

				
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