FirstFirst--principles simulations of defects principles simulations of defects in oxides and nitridesin oxides and nitridesin oxides and nitridesin oxides and nitrides

Chris G. Van de WalleMaterials Department, UCSB

Acknowledgments:Acknowledgments:A. Janotti, J. Lyons, J. Varley, J. Weber (UCSB)P. Rinke (FHI), M. Scheffler (UCSB, FHI Berlin)G. Kresse (U. Vienna)C F ldt J N b (MPI Dü ld f)C. Freysoldt, J. Neugebauer (MPI Düsseldorf)NSF, DOE

School on Computational Modeling of MaterialsDecember 2-3, 2010Antwerp, Belgium

Why study defects?Why study defects?y yy y• “Defects”

Extended defectsVacancy

– Extended defects» dislocations

– Point defects: » Native defects» Impurities

• Defects often determine Interstitial

e ects o te dete ethe properties of materials– Doping and its limitations– Device degradation– Diffusion

» Mediated by point defects

Antisite

y p

Technological significanceTechnological significanceg gg g• Semiconductors

A hi hi h d i l l– Achieve higher doping levels» p-type doping of AlN would allow UV lasers

– Controllably dope materials p-type and n-typeControllably dope materials p type and n type» Oxides

• Photovoltaics– CuInxGa1-xSe2

• Hydrogen storage materials– Kinetics of hydrogen release in NaAlH4

• Embrittlement of structural metals

Links to experimentLinks to experimentpp• Secondary Ion Mass Spectrometry (SIMS)

– Impurity concentrations (to within factor of 2)P it ihil ti t• Positron annihilation spectroscopy– Most sensitive to negatively charged vacancy(-like) defects

• EXAFS (extended x-ray absorption fine structure)– Microscopic structure, atomic relaxations

• Electron paramagnetic resonance + ENDOR– Hyperfine parameters (wave functions, atomic positions)

• Vibrational spectroscopy (Raman, FTIR)– Local vibrational modes; sensitive to atomic positions

• Deep level transient spectroscopy– Electronic transition levels

» Total-energy differences, not Kohn-Sham levels• Photoluminescence (PL), PL excitation

– Optical transition levels

Defect calculations: geometryDefect calculations: geometryg yg y• Green’s functions

I l t ti diffi lt– Implementation difficult– Non-intuitive

• Clusters• Clusters– Surface effects– Not “bulk” (quantum confinement)Not bulk (quantum confinement)

• Supercells– “Molecular unit cell approach (MUCA)"Molecular unit cell approach (MUCA)

» R. P. Messmer and G. D. Watkins, in Radiation Damage and Defects in Semiconductors (Inst. of Phys. London, 1972), No. 16, p. 255.

Defect Formation EnergiesDefect Formation EnergiesggExample:  VO

1− +VO + 12

oxide: VO oxide ½ O2

Defect Formation EnergiesDefect Formation EnergiesggExample:  VO

+ 1 Fe‐− +VO ++ 12

oxide: VO oxide ½ O2 reservoir

e− @ F(electron reservoir))

FormalismFormalism• Eform: formation energy

Concentration of defects or impurities:C = Nsites exp [ Eform/kT]

• Example: oxygen vacancy in ZnOEform(VO+) = Etot(VO+) Etot(bulk) + O + EFEform(VO ) Etot(VO ) Etot(bulk) O EFO: energy of oxygen in reservoir, i.e., oxygen chemical potentialEF: energy of electron in its reservoir, i.e., the Fermi level

G l i• General expressionEform(Dq) = Etot(Dq) Etot(bulk) + ni i + qEF

b f t b i h d t f th d f tni: number of atoms being exchanged to form the defect

Formation energyFormation energygygy

• VO in ZnOOZn-rich conditions:Zn=Etot(bulk Zn)

O=Etot(O2)+Hf(ZnO)

Transition levelsTransition levels

• Charge-state gtransition levels

+/02+/+ CBEF2+/+

( / )

EF

VB(+/0)(2+/+)

2+/0

Issues...Issues...

• Band gap problemg p p– DFT (LDA, GGA)

• Affects formation energies and transition levels– Even for neutral charge

states, if defect-induced Kohn-Sham states areKohn Sham states are occupied with electrons

DFT/LDA band gap

BandBand--gap corrections: Empirical correctionsgap corrections: Empirical corrections• Ad hoc corrections

– “Scissors operator”Shift l l b d d ti l b d h t» Shift gap levels based on conduction- vs. valence-band character

– Delta-function(-like) term added to potential, shifts s states» N. E. Christensen, Phys. Rev. B 30, 5753 (1984).

– “Modified pseudopotentials”Modified pseudopotentials» D. Segev, A. Janotti, and C. G. VdW, Phys. Rev. B 75, 035201 (2007).

– Issues» Hard to control» May have unintended consequences (indirect vs. direct gaps, …)

• Extrapolations based on calculations that yield different gaps

Diff t l t ff– Different plane-wave cutoffs– Different exchange-correlation functionals– …

» S Zhang S Wei and A Zunger Phys Rev B 63 075 (2001)» S. Zhang, S. Wei, and A. Zunger, Phys. Rev. B 63, 075 (2001).– Issue: different choices of parameter not only produce different gap,

but also different levels of accuracy for description of defect

BandBand--gap corrections: SIC and LDA+gap corrections: SIC and LDA+UU• Physically meaningful improvements• Self-interaction corrections

» J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).– Difficult to implement in self-consistent calculations for solids– Incorporate in pseudopotentials

» D Vogel P Krueger and J Pollmann Z O(LDA) Z O(LDA+U)» D. Vogel, P. Krueger, and J. Pollmann, Phys. Rev. B 52, 14316 (1995).

ZnO(LDA) ZnO(LDA+U)

0 80

0.37Ecgap

• “LDA+U” approach» A Janotti and C G Van de Walle

0.34

0.80 1.51

5 17

Evgap» A. Janotti and C. G. Van de Walle,

Appl. Phys. Lett. 87, 122102 (2005).

» S. Lany and A. Zunger, Phys Rev B 72 035215 (2005)

7.16

2 33

Zn 3d-band

5.17Phys. Rev. B 72, 035215 (2005).• Issues:

– Determination of U– Band gap not fully corrected 2.33Band gap not fully corrected

» How to extrapolate to the experimental gap?

Oxygen vacancy in ZnOOxygen vacancy in ZnO

C t dLDA+ULDA CorrectedLDA+ULDA

BandBand--gap corrections: Hybrid functionalsgap corrections: Hybrid functionals• Mixing of Hartree-Fock (exact exchange) and DFT

– A. D. Becke, J. Chem. Phys. 98, 1372 (1993).F ti l i i 25% f t h• Functionals mix in ~25% of exact exchange

• PBE0» J. P. Perdew, K. Burke, and M. Ernzerhof,

Phys Rev Lett 77 3865 (1996)Phys. Rev. Lett. 77, 3865 (1996).» J. P. Perdew, M. Ernzerhof, and K. Burke,

J. Chem. Phys. 105, 9982 (1996).• HSE

– Exact exchange only for short-range interactions– J. Heyd, G. E. Scuseria and M. Enzernhof,

J. Chem. Phys. 118, 8207 (2003).Obt i d d i ti f ti• Obtain very good description of many properties– Band gaps close to experiment

• Details of the physics still to be explored/understood

BandBand--gap corrections: Beyond DFTgap corrections: Beyond DFT• Quasiparticle calculations

– Combine DFT and G0W0DFT d f t t l ti– DFT: good for structural properties

– G0W0: many-body perturbation theory for defects: accurate electron affinities in solids

» P. Rinke, A. Janotti, M. Scheffler, and C. G. Van de Walle, Phys. Rev. Lett. 102, 026402 (2009).

• Quantum Monte Carlo– Richard Hennig

“Deep” “Deep” vs. vs. “shallow” defects“shallow” defects

CBCBCB

VBVB

• Note: dispersion– Due to finite supercell sizep– Energetics taken care of by special-point sampling– Make sure correct occupation of states

“Deep” “Deep” vs. vs. “shallow” defects“shallow” defects

CBCB CB

VB VB

DeepLocalized wave function

ShallowDefect-induced state is

Level (usually) far from band edges

resonance in VB or CBNear VB or CB: only small

perturbation“Effective mass” state

Alignment of Fermi levelAlignment of Fermi level

• Charges are exchanged with EF, referenced to EvEf (Dq) = Et t(Dq) Et t(bulk) + ni i + qEFEform(Dq) = Etot(Dq) Etot(bulk) + ni i + qEF

– Presence of defect in supercell shifts average electrostatic potential with respect to bulk

defect bulk

V

Supercell size effectsSupercell size effectspp• Neutral defects:

Supercell should be large enough to ensure– Supercell should be large enough to ensure atomic relaxations are included, and overlap of wave functions is small enough

• Charged defects:– Inbalance in electronic and

ionic charge would lead toionic charge would lead to Coulomb divergence

– Neutralizing backgroundNo need to explicitly include;» No need to explicitly include; G=0 term calculated for neutral system

Supercell size effectsSupercell size effectspp

• Neutralization leads to unintended terms in total energytotal energy– Interactions with neutralizing background and

between periodic images of defectsbetween periodic images of defects • “Makov-Payne” correction

E(L) = E() q2/L C3q2/L3 + O(L-5)( ) ( ) q 3q ( )» G. Makov and M.C. Payne, Phys. Rev. B 51, 4014 (1995).

– Correct in vacuum– Effect of solid: dielectric constant

Supercell size effectsSupercell size effectspp• Explicit studies as a function of supercell size have

shown complex behaviorshown complex behavior– Fits to 1/L and 1/L3 terms– In some cases, Makov-Payne correction satisfactory– In other cases: Makov-Payne

correction makes things significantly worse

2» Example: V2+ in diamond» Shim, Lee, Lee, and Nieminen,

Phys. Rev. B 71, 035206 (2005).

With MP

MP

Supercell size effectsSupercell size effectspp• Need rigorous analysis of electrostatic interactions

in dielectric mediain dielectric media» C. Freysoldt, J. Neugebauer, and C. G. Van de Walle,

Phys. Rev. Lett. 102, 106402 (2009). Produces rigorous definitions for– Produces rigorous definitions for

» Coefficient of 1/L3 (“quadrupole”) term» Alignment term

All t ti h th• Allows testing whether “point charge” correction (Makov-Payne) suffices or not– Fails if defect state decays slowly; point-charge model

overcorrects– Prescription for addressing this problemPrescription for addressing this problem

Native point defects in ZnONative point defects in ZnOpp• VO, VZn dominate

– A. Janotti and C. G. Van de Walle,Zn-rich

A. Janotti and C. G. Van de Walle, Appl. Phys. Lett. 87, 122102 (2005).

– S. B. Zhang et al., Phys. Rev. B 63, 075205 (2001).

– F. Oba et al., Phys. Rev. B 77,F. Oba et al., Phys. Rev. B 77, 245202 (2008).

• VO: deep donor– Also high formation energy in g gy

n-type ZnO• VZn: deep acceptor

– Cause of green luminescenceCause of green luminescence – A. F. Kohan, G. Ceder,

D. Morgan, C. G. Van de Walle, Phys. Rev. B 61, 15019 (2000)

Native defects Native defects vsvs. impurities. impuritiespp

• Native defects cannot explain n-type doping• Impurities: donors?

Interstitial Hydrogen in ZnOInterstitial Hydrogen in ZnOy gy g

2

3y

(eV)

0

1

+H

mat

ion

ener

gy

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-1For

m

EF (eV)

H+ is the only stable charge state hydrogen acts as shallow donorUnexpected! In other semiconductors hydrogen reduces the conductivity

C. G. Van de Walle, Phys. Rev. Lett. 85, 1012 (2000).H drogen is a likel candidate for nintentional incorporationHydrogen is a likely candidate for unintentional incorporation• But: highly mobile

M. G. Wardle, J. P. Goss and P. R. Briddon, Phys. Rev. Lett. 96, 205504 (2006). unstable at temperatures where n-type conductivity is known to

persist (>500oC)Also cannot explain dependence of conductivity on oxygen partial pressure…

Substitutional hydrogen in ZnOSubstitutional hydrogen in ZnOy gy g• Forced to reconsider the role of

hydrogen...and in the process some interesting– … and in the process some interesting

new physics/chemistry emerged!• Substitutional hydrogen

Hydrogen on a substitutional oxygen site

VO HOH– Hydrogen on a substitutional oxygen site

– Formation energy: low– Ionization energy: small; shallow donor

C i t tl l i d d f

Hi

• Consistently explains dependence of n-type conductivity on oxygen partial pressure Zn

g[X] [n]

g[X] [n][n]

HO [Hi]

log

[VO]

[HO][Hi][Hi]

log

[VO][VO]

[HO][HO]

pO21/2pO21/2

pp--type doping of ZnOtype doping of ZnO• Nitrogen is often regarded as most suitable

hole dopant– Shallow acceptor in ZnSe

• Numerous reports of p-type ZnO crystalsUsing N and other group V impurities (Sb As P)– Using N and other group-V impurities (Sb, As, P)

• Reliability? Reproducibility?

C.H. Park, S.B. Zhang, and S.-H. Wei, PRB 66, 073202 (2002).

BandBand--gap corrections: Hybrid gap corrections: Hybrid functionalsfunctionals• Mixing of Hartree-Fock (exact exchange) and DFT

– A. D. Becke, J. Chem. Phys. 98, 1372 (1993).F ti l i i 25% f t h• Functionals mix in ~25% of exact exchange

• PBE0» J. P. Perdew, K. Burke, and M. Ernzerhof,

Phys Rev Lett 77 3865 (1996)Phys. Rev. Lett. 77, 3865 (1996).» J. P. Perdew, M. Ernzerhof, and K. Burke,

J. Chem. Phys. 105, 9982 (1996).• HSE

– Exact exchange only for short-range interactions– J. Heyd, G. E. Scuseria and M. Enzernhof,

J. Chem. Phys. 118, 8207 (2003).Obt i d d i ti f ti• Obtain very good description of many properties– Band gaps close to experiment

HSE calculations for HSE calculations for ZnOZnO• Hybrid functionals

include a portion of pexact exchange, correct band gap– J Heyd G E ScuseriaJ. Heyd, G. E. Scuseria,

and M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003).

– =0.36

• VASP code (ver. 5.1)

Nitrogen acceptor in Nitrogen acceptor in ZnOZnOg pg p• Deep acceptor• Large ionization energy: 1.3 eVLarge ionization energy: 1.3 eV• Low formation energy under n-type

conditions• Localized wavefunction

ZnZn

NOJ. L. Lyons, A. Janotti, and C. G. Van de

A i l b d 2 12 Åy

Walle, Appl. Phys. Lett. 95, 252105 (2009).Axial bond = 2.12 ÅPlanar bonds = 1.94 Å

Test case: Test case: NNSeSe in in ZnSeZnSe

• Similar semiconductor• NSe known shallow

Zn-rich(0/-)

Seacceptor in ZnSe

• Effective p-type dopant NSe• Results agrees well with

experimental values– Theory: EA = 150 meVTheory: EA 150 meV– Exp.: EA = 110-130 meV

NNOO acceptor: Experimentacceptor: Experiment

• Configuration• Configuration coordinate diagram

• Absorption: exciting electron from

2.4 eV 1.7 eVexciting electron from NO0 to CB– 2.4 eV = 520 nm– Zero phonon line at p

2.1 eV• Emission:

electron falling from CB t N l lCB to NO level– 1.7 eV = 730 nm

NNOO acceptor: experimentacceptor: experiment

• Localization on N atom• Directed towards axial zinc

neighbor• Agrees with EPR signals• Agrees with EPR signals

• W.E. Carlos, E.R. Glaser, and D.C. Look Physica B 308 976 (2001)Look, Physica B 308, 976 (2001).

• N. Y. Garces, N. C. Giles, L. E. Halliburton, G. Cantwell, D. B. Eason, D. C. Reynolds, and D. C. Look, Appl Ph s Lett 80 1334 (2002)

Spin density of nitrogen state(Isosurface is 5% of maximum

density)

Appl. Phys. Lett. 80, 1334 (2002).

“Why nitrogen cannot lead to p-type conductivity in ZnO”, J. L. Lyons, A. Janotti, and C. G. Van de Walle, Appl. Phys. Lett. 95, 252105 (2009).

TiOTiO22 in GGAin GGA--PBEPBE

Rutile

VBM O t tVBM - O p statesCBM - Ti d states

Hybrid functional calculations for TiOHybrid functional calculations for TiO22• HSE functional

» Heyd, Scuseria, & Ernzerhof, J. Chem. Phys. 118, 8207 (2003); » erratum: J. Chem. Phys. 124, 219906 (2006)

• H-F mixing parameter 0.20; screening parameter μ = 0.2 Å-1g p μ

• PAW, VASP 5.1• 72 atom-supercell

2 k i t (0 0 0) (0 5 0 5 0 5) t t d f (2 2 2)• 2 k-points: (0,0,0) (0.5,0.5,0.5); tested for (2x2x2)• cutoff: tested for up to 400 eV

A. Janotti, J. B. Varley, P. Rinke, N. Umezawa, G. Kresse, and C. G. Van de Walle, Phys. Rev. B 81, 085212 (2010).

TiOTiO22 -- structural propertiesstructural propertiesPBE versus HSEPBE versus HSEPBE versus HSEPBE versus HSE

Rutile

Band structure of TiOBand structure of TiO22GGA versus hybrid functional (HSE)GGA versus hybrid functional (HSE)GGA versus hybrid functional (HSE)GGA versus hybrid functional (HSE)

HSE can be tuned to reproduce the exp. band gap value

Effects of HSE on valence band Effects of HSE on valence band versus conduction bandversus conduction band

(determined from surface calculations)(determined from surface calculations)

HSE lowers the VBM by 0.6 eV and raises the CBM by 0.7 eV

SingleSingle--particle states of particle states of VVOO in TiOin TiO22GGA results for unrelaxed vacancyGGA results for unrelaxed vacancyGGA results for unrelaxed vacancyGGA results for unrelaxed vacancy

unrelaxed vacancy induced single-particle states are in the gap

SingleSingle--particle states of particle states of VVOO in TiOin TiO22l d l dl d l drelaxed versus unrelaxed vacancyrelaxed versus unrelaxed vacancy

GGA cannot describe relaxed vacancy in 0 and +1 charge statesoccupied states above the CBM

SingleSingle--particle states of particle states of VVOO in TiOin TiO22GGA h b id f ti l (HSE)GGA h b id f ti l (HSE)GGA versus hybrid functional (HSE)GGA versus hybrid functional (HSE)

relaxed VO0 and VO+ can be described in HSE

Formation energies of Formation energies of VVOO in TiOin TiO22OOunrelaxed

all transition levels in the gapall transition levels in the gapformation energy of +2 and +

are lowered, consistent with the PBE/HSE b d li tPBE/HSE band alignment

Fermi level (eV)

Formation energies of Formation energies of VVOO in TiOin TiO22unrelaxed

all transition levels in the gap

OO

all transition levels in the gapformation energy of +2 and +

are lowered, consistent with the PBE/HSE b d li t

relaxed

PBE/HSE band alignment

relaxation energies for +2 are similar in PBE and HSE

transition levels (+2/0) and (+/0) are near the CBM

Fermi level (eV)

Oxygen vacancy in TiOOxygen vacancy in TiO22 --relaxations and charge relaxations and charge

densitiesdensities+2 (PBE and HSE)

+1 (HSE)

neutral (HSE)

t t l li d i lstate localized on axial Ti neighbors

Oxygen vacancy in TiOOxygen vacancy in TiO22

shallow donor - can cause conductivitylow formation energy in O-poor conditions

Summary and OutlookSummary and Outlookyy• The field of defects in materials is active with

direct applications for many technologically pp y g yimportant systems– Much broader than merely “point defects in semiconductors”

• Significant progress has recentlySignificant progress has recently been made on crucial issues:– DFT band-gap errors– Supercell size corrections– Supercell size corrections

• Still need for deeper physical understandingR f /O i• References/Overviews

» J. Neugebauer and C. G. Van de Walle, J. Appl. Phys. 95, 3851 (2004).