Hybrid functionals: Dilute Magnetic semiconductors

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

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

4
Ab initio modeling

• Exact many electron Schrödinger Equation
• Complexity basis set sizeNumber of electrons
• Wavefunctions based methods (HFMP2, CCSD(T))
• QMC
• Central idea map onto best one-electron
  theory
• Complexity basis set size Number of electrons

5
Kohn Sham Density functional theory

• Density and kinetic energy are the sum of one
  electron wave functions
• KS functional has its minimum at the electronic
  ground state

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
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

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

9
One-electron theories

• Density functional theory
• Hartree Fock theory
• GW

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
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).

12
PBEh and HSE functional

• The PBEh (PBE0) exchange-correlation functional1
• The HSE03 (HSE06) functional 2

• J. Perdew, M. Ernzerhof, and K. Burke, J. Chem.
  Phys. 105, 9982 (1996).
• 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 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).

14
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
15
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
16
Vibrational properties Phonons

• Kresse, Furthmüller, Hafner, EPL 32, 729 (1995).
  K. Hummer, G. Kresse, in preparation.

Ge
C
Si
Sn
17
Vibrational Properties

• K. Hummer, G. Kresse, in preparation.

Ge
C
Si
Sn
18
Hybrid functionals for solids Band gaps

• Band gaps improved
• But fairly larger errors prevail for materials
  with weak screening(elt4)
• for these materials half-half functionals are
  quite accurate but these will be worse for the
  rest !

?lt4
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Optical Absorptionspectra using PBE
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

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 ?/ (???)Required by
  sum rule

Si
C
22
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.
  

Absorption spectrum
? ?iGG G from GW
23
Proper Absorption-spectra using HSE

• Now spectra are very reasonable
• Distribution of intensities is about right
• Remarkable accurate static properties

Si
C
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. e ab initio screening,
  hybrids e¼

25
Multivalent oxides Ceria
J.L.F. Silva, , G. Kresse, Phys. Rev. B 75,
045121 (2007).
VB
CB
f
Usual from DFT to hybrid
unsual
26
3d transition metal oxides 1

• Hybrids substantially improve upon PBE
• HSE latt. const. and local spin mag. moments are
  excellent

• M. Marsman et al., J. Phys. Condens. Matter 20,
  64201 (2008).

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3d metals When hybrids fail
28
RPA correlation

• The electrons move in the exchange potential
  screened by all other electrons

L. Hedin, Phys. Rev. 139, A796 (1965)
-1
29
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
• ?-1(G)
• No screening for large G
• Strong screening for small G(static screening
  properties)
• Hybrids ¼ is a compromise

30
GW0 approximation
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 Greens function and self-energy (W fixed
  to W0)

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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).

32
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).

33
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)

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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).

35
Cu2ZnSnS4 or CZTS
DFT
hybrid

• In this case HSE hybrid functional and GW give
  identical answers

GW
J. Paier, R. Asahi, A. Nagoya, and Georg Kresse,
PRB 79, 115126 (2009).
36
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.

37
PBE results

• Ga3
• Mn3 4 electrons in majority component
• 1 hole in t orbitals
• DFT predicts almost degenerate t2 orbitals
• Metallic behavior

3 t2-orbitals
2 e-orbitals
A. Stroppa and G. Kresse, PRB RC in print.
38
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

39
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

40
Charge density
PBE
HSE

• PBE predicts symmetric solution
• HSE predicts D2d symmetry (no trigonal axis)

A. Stroppa and G. Kresse, PRB RC in print.
41
Mn_at_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

GaN
GaAs
42
Summary

• 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

43
Acknowledgement

• FWF for financial support
• And the group for their great work...
• You
• for listening