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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
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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

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Test results using GW
Hybrid functionals: DMS
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Ab initio modeling
Exact many electron Schrödinger Equation
 2

 
Δ  V(r1,..., rn ) Ψ(r 1,..., rn )  EΨ(r 1,..., rn )
 2me

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 )]

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Complexity: basis set size • Number of electrons
Hybrid functionals: DMS
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Kohn Sham Density functional theory
Density and kinetic energy are the sum of one
electron wave functions
 (r ) 2 | n (r ) |2
occ
2
2
e
 (r )  (r ' ) 3 3
*
3
ion
3
xc


(
r
)


(
r
)
d
r

V
(
r
)

(
r
)
d
r

d
r
d
r
'

E
[  (r )]

n
n



2me occ
2
| r  r' |
KS functional has its minimum at the electronic ground state
 2

ion
el
xc
 
  V (r )  V (r )  V (r ) n (r )  Enn (r )
 2me

Possion : V el (r )  e2 
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 (r ' )
d 3r'
| r  r' |
LDA : V xc (r )  V xc  (r ) 
Hybrid functionals: DMS
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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
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Wave function based methods
used in quantum chemistry
CCSD(T), RPA
Hybrid functionals: DMS
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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
ECBMIN [ N ]
I
 EVBMAX [ N ] in LDA/GGA
Large errors in LDA/GGA/HF
Lack of Integer-discontinuity
in the LDA/GGA/HF
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Hartree-Fock theory
Effective one electron equation
DFT : V xc (r )n (r )
 2

ion
el
 
  V (r )  V (r ) n (r )   V x (r, r' )n (r' )d 3r'  Enn (r )
 2me

Lacks correlation, unoccupied states only Hartree pot.
Exchange potential
(anti-symmetry of wave functions in Slater determinant)
2

e
V x (r, r)  m (r )m* (r) 
| r  r |
occ
Hartree potential
2
e
V H (r )   d 3r m (r)m* (r) 
| r  r |
occ
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One-electron theories
Density functional theory
 2

ion
el
xc
 
  V (r )  V (r )  V (r ) n (r )  Enn (r )
 2me

Hartree Fock theory
 2

ion
el
 
  V (r )  V (r ) n (r )   V x (r, r' )n (r' )d 3r'  Enn (r )
 2me

GW
 2

ion
el
 
  V (r)  V (r) n (r)    xc (r, r' ,  )n (r' )d 3r' d  Enn (r)
 2me

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Where is the correlation
The electrons move in the exchange potential screened by
all other electrons
L. Hedin, Phys. Rev. 139, A796 (1965)
-1
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Hybrid functionals: two one-electron theories
Hartree-Fock
1
4
V (r, r) 
x

e
2

m
m (r ) (r)
*
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).
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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).
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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
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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 %
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Vibrational properties: Phonons
Kresse, Furthmüller, Hafner, EPL 32, 729 (1995).
K. Hummer, G. Kresse, in preparation.
C
Ge
Si
Sn
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Vibrational Properties
K. Hummer, G. Kresse, in preparation.
C
Ge
Si
Sn
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Hybrid functionals for solids: Band gaps
Band gaps improved
But fairly larger errors
prevail for materials
with weak screening
(ε<4)
<4
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Hybrid functionals: DMS
for these materials
half-half functionals
are quite accurate but
these will be worse for
the rest !
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Optical Absorptionspectra using PBE
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Two Problems
Red shift of spectrum compared to experiment
Too weak cross scattering cross section at low energies



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In many cases these effects compensate each other
Dominant peak in C in pretty much spot on
Static properties are pretty good in DFT
Hybrid functionals: DMS
εLDA
RPA
εEXP
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
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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
Si
εHSE
RPA
εEXP
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
C
GaAs
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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
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Proper Absorption-spectra using HSE:
Now spectra are very reasonable
Distribution of intensities is about right
Remarkable accurate static properties
Si
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εHSE
RPA
εEXP
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
C
Hybrid functionals: DMS
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Multivalent oxides: Ceria
J.L.F. Silva, …, G. Kresse,
Phys. Rev. B 75, 045121 (2007).
VB
CB
f
Usual from
DFT to hybrid
unsual
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3d transition metal oxides [1]
PBE
HSE
EXPT.
MnO
ao
Eg
4.44
0.93
4.44
2.8
4.45
3.9
FeO
ao
Eg
4.30
metal
4.33
2.2
4.33
2.4
CoO
ao
Eg
4.22
metal
4.26
3.4
4.25
2.5
NiO
ao
Eg
4.19
0.81
4.18
4.2
4.17
4.0
Hybrids
substantially
improve upon
PBE
HSE latt. const.
and local spin
mag. moments
are excellent
1. M. Marsman et al., J. Phys.: Condens. Matter 20, 64201 (2008).
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3d metals: When hybrids fail
Spin up
Spin down
Fe Hund‘s rule
ferromagnet using HSE
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RPA correlation
The electrons move in the exchange potential screened by
all other electrons
L. Hedin, Phys. Rev. 139, A796 (1965)
-1
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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 dielectric
matrix is diagonal in reciprocal
space

Insulators
weak screening
Semiconductors/ metals
strong screening
Ɛ-1(G)

Vacuum no
screening
No screening for large G
Strong screening for small G
(static screening properties)
hybrids
Hybrids: ¼ is a compromise
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GW0 approximation
M. S. Hybertsen, S. G. Louie, Phys. Rev. B 34, 5390 (1986)
Calculate DFT/hybrid functional wavefunctions
 2

ion
el
xc
 
  V (r )  V (r )  V (r ) n (r )  Enn (r )
 2me

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 ]
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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
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Hybrid functionals: DMS
M. Shishkin, G. Kresse, Phys
Rev. B 75, 235102 (2007).
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HSE: G0W0 band gaps1
About same quality as
using PBE wave
functions and
screening properties
Overall slightly too
large
1
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Hybrid functionals: DMS
F. Fuchs, J. Furthmüller,
F. Bechstedt, M. Shishkin,
G. Kresse, PRB 76, 115109
(2007).
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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
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Hybrid functionals: DMS
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).
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Hybrid functionals: DMS
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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).
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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.
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PBE results
3 t2-orbitals
2 e-orbitals
Ga3+
Mn3+
4 electrons in
majority
component
1 hole in t orbitals
DFT predicts almost
degenerate
t2 orbitals
Metallic behavior
A. Stroppa and G. Kresse, PRB RC in print.
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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
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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
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Charge density
PBE
HSE
PBE predicts symmetric solution
HSE predicts D2d symmetry (no trigonal axis)
A. Stroppa and G. Kresse, PRB RC in print.
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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
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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
Vacuum no
screening
Insulators
weak screening
Semiconductors/ metals
strong screening
hybrids
¼ seems to be very good
for states close to the
Fermi level
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Acknowledgement
FWF for financial support
And the group for their
great work...
You
for listening
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