Hybrid functionals: Dilute Magnetic semiconductors Georg Kresse J. Paier, K. Hummer, M. Marsman, A. Stroppa Faculty of Physics, University of Vienna and

Hybrid functionals: Hybrid functionals: Dilute Magnetic semiconductors Dilute Magnetic semiconductors

Georg KresseGeorg KresseJ. Paier, K. Hummer, M. Marsman, J. Paier, K. Hummer, M. Marsman,

A. StroppaA. Stroppa

Faculty of Physics, University of ViennaFaculty of Physics, University of Viennaand Center for Computational Materials Scienceand Center for Computational Materials Science

Funded by the Austrian FWF

2204/19/2304/19/23 Hybrid functionals: DMSHybrid functionals: DMS

OverviewOverview

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


Take home messagesTake 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


Exact many electron Schrödinger EquationExact many electron Schrödinger Equation

Complexity: basis set sizeNumber 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 modelingAb initio modeling

)r,...,Ψ(r)r,...,Ψ(r)r,...,V(rΔ2m n1n1n1

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Density and kinetic energy are the sum of one Density and kinetic energy are the sum of one electron wave functionselectron wave functions

KS functional has its minimum at the electronic ground stateKS functional has its minimum at the electronic ground state

Kohn Sham Density functional theoryKohn Sham Density functional theory

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DFT ProblemsDFT Problems

Precision of total energiesPrecision 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 treatmentOne most go beyond a traditional one electron treatment

Quantum Monte-Carlo Wave function based methodsused in quantum chemistry

CCSD(T), RPA


One of the great lies: The band One of the great lies: The band gap problemgap problem

DFT is only accurate for ground state propertieshence the error in the band gap does not matterThe band gap is a well defined ground state property The band gap is a well defined ground state property wrong using local and semi-local DFTwrong using local and semi-local DFT

Fundamental gap

Large errors in LDA/GGA/HF

Lack of Integer-discontinuityLack of Integer-discontinuityin the LDA/GGA/HFin the LDA/GGA/HF

LDA/GGAinVBMAXCBMIN ][][

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Hartree-Fock theoryHartree-Fock theory

Effective one electron equationEffective one electron equation

Lacks correlation, unoccupied states only Hartree pot.Lacks correlation, unoccupied states only Hartree pot.

Exchange potential Exchange potential (anti-symmetry of wave functions in Slater determinant)(anti-symmetry of wave functions in Slater determinant)

Hartree potentialHartree potential

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One-electron theoriesOne-electron theories

Density functional theoryDensity functional theory

Hartree Fock theoryHartree Fock theory

GWGW

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Where is the correlationWhere is the correlation

-1

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

L. Hedin, Phys. Rev. 139, A796 (1965)

Hybrid functionals: two one-electron theoriesHybrid functionals: two one-electron theories

Hartree-FockHartree-Fock

Much too large band gaps

Density-functional theoryDensity-functional theory

Too small band gaps

Generalized Kohn-Sham schemesGeneralized Kohn-Sham schemesSeidl, Görling, Vogl, Majewski, Levy, Phys. Rev. B 53, 3764 (1996).


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HSE versus PBEh: convergence of HSE versus PBEh: convergence of exchange energy with respect to k pointsexchange energy with respect to k points11


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

PBE: Lattice constants and bulk moduliPBE: Lattice constants and bulk moduli


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 %

HSE: Lattice constants and bulk moduliHSE: Lattice constants and bulk moduli


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)

Vibrational properties: PhononsVibrational properties: PhononsKresse, Furthmüller, Hafner, EPL 32, 729 (1995). K. Hummer, G. Kresse, in preparation.


C

SiSn

Ge

Vibrational PropertiesVibrational PropertiesK. Hummer, G. Kresse, in preparation.


C

SiSn

Ge

Hybrid functionals for solids: Band gapsHybrid 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 !


<4

Optical Absorptionspectra using PBEOptical Absorptionspectra using PBE


Two ProblemsTwo 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

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

Better band gaps: HSE resultsBetter 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

RPA

εEXP

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

Si C

Proper Absorption-spectra using HSE:Proper Absorption-spectra using HSE:

Accurate band gaps and accurate absorption spectra [Dyson Equ. ]


Absorption spectrum

χ=iGG G from GW

)( xcipip fv

J.Paier, M. Marsman, G. Kresse, PRB 78, 121201(R) (2008)

Proper Absorption-spectra using HSE:Proper Absorption-spectra using HSE:

Now spectra are very reasonable

Distribution of intensities is about right

Remarkable accurate static properties


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

Si C

Multivalent oxides: CeriaMultivalent oxides: Ceria


CB VB

f

Usual from DFT to hybrid

unsual

J.L.F. Silva, …, G. Kresse,Phys. Rev. B 75, 045121 (2007).

3d transition metal oxides 3d transition metal oxides [1][1]

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).

PBE HSE EXPT.

MnO aoEg

4.440.93

4.442.8

4.453.9

FeO aoEg

4.30metal

4.332.2

4.332.4

CoO aoEg

4.22metal

4.263.4

4.252.5

NiO aoEg

4.190.81

4.184.2

4.174.0

3d metals: When hybrids fail3d metals: When hybrids fail


Fe Hund‘s ruleferromagnet using HSE

Spin up

Spin down


RPA correlationRPA correlation

-1

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

L. Hedin, Phys. Rev. 139, A796 (1965)


The right physics: screened exchangeThe right physics: screened exchange

Screened exchange:Screened exchange: Screening system dependent

For bulk materials dielectricmatrix is diagonal in reciprocalspace

Ɛ-1(G) No screening for large G Strong screening for small G

(static screening properties)

Hybrids: ¼ is a compromiseHybrids: ¼ is a compromise

M. S. Hybertsen, S. G. Louie, Phys. Rev. B 34, 5390 (1986)

Vacuum noscreening Insulators

weak screening

Semiconductors/ metalsstrong screening

hybrids


GWGW00 approximationapproximation

Calculate DFT/hybrid functional wavefunctionsCalculate DFT/hybrid functional wavefunctions

Determine Green function and W using DFT wavefunctionsDetermine Green function and W using DFT wavefunctions

Determine first order change of energiesDetermine first order change of energies

Update Green’s function and self-energy (W fixed to WUpdate Green’s function and self-energy (W fixed to W00))

M. S. Hybertsen, S. G. Louie, Phys. Rev. B 34, 5390 (1986)

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PBE: GWPBE: GW00 band gaps band gaps11

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).


HSE: GHSE: G00WW00 band gaps band gaps11

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).


Self-consistent QPGWSelf-consistent QPGWTC-TCTC-TC band gaps band gaps11

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)

Strategy for true ab-initio modellingStrategy for true ab-initio modelling

Apply HSE functional as zero order descriptionApply HSE functional as zero order description

Perform Perform GWGW on top of the HSE functional 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 efficientFairly 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).


CuCu22ZnSnSZnSnS4 4 or CZTS or CZTS

In this case HSE hybrid functional and GW give identical answers


GW

hybrid

DFT

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

GaNGaN

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.


PBE resultsPBE results


GaGa3+3+

MnMn3+3+ 4 electrons in 4 electrons in majority majority componentcomponent

1 hole in t orbitals1 hole in t orbitals

DFT predicts almost DFT predicts almost degenerate degenerate tt22 orbitals orbitals

Metallic behavior Metallic behavior

2 e-orbitals

3 t2-orbitals

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

HSE resultsHSE results


GaGa3+3+

MnMn3+3+ 4 electrons in 4 electrons in majority componentmajority component


HSE predicts a HSE predicts a splitting within in splitting within in tt2 2 manifoldmanifold

Localized hole on MnLocalized hole on Mn

GW resultsGW results


GaGa3+3+



HSE predicts a HSE predicts a splitting within in splitting within in tt2 2 manifoldmanifold

Localized hole on MnLocalized hole on Mn

GW confirms resultsGW confirms results

Charge densityCharge density

PBE predicts symmetric solutionPBE predicts symmetric solution

HSE predicts DHSE predicts D2d2d symmetry (no trigonal axis) symmetry (no trigonal axis)


PBE HSE

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

Mn@GaAsMn@GaAs


GaGa3+3+



HSE predicts no HSE predicts no splitting within in splitting within in tt2 2 manifoldmanifold

Strong hybridization Strong hybridization with valence bandwith valence band

Delocalized holeDelocalized hole

GaN GaAs

SummarySummary

HSE is better compromise than classical local DFT HSE is better compromise than classical local DFT functionalsfunctionals But a compromise it is

Metals !! GW is more universal

although not necessarily more accurate

Why HSE works so wellWhy HSE works so wellis not quite understoodis not quite understood

¼ seems to be very good¼ seems to be very goodfor states close to the for states close to the Fermi levelFermi level


Vacuum noscreening Insulators

weak screening

Semiconductors/ metalsstrong screening

hybrids

AcknowledgementAcknowledgement

FWF for financial supportFWF for financial support

And the group for theirAnd the group for theirgreat work...great work...

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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