Modelling of charged point defects
with density-functional theory
Christoph Freysoldt

Department of Computational Materials Design
Düsseldorf, Germany
Models and Data on Plasma-Material Interactions 2019
June 18, 2019

Point defects• Point defects (dopants, impurities, vacancies,interstitials, …) critical for semiconductor &insulator properties• Low concentrations (1016-1020 cm-3
≈10-6 – 10-2 relative)• Important insights from theory & calculations

Theoretical modelling is challenging• Supercell approach• Defect-defect interactions• Advanced electronic structure methods(beyond standard DFT, i.e. hybrid
functionals, GW, QMC)

How should we deal with defect-defect interactions?
Christoph Freysoldt – MPI Düsseldorf

2Charged defects with DFT

Outlook

Corrections forcharged-cell artifacts

Charged-defectinteractions at scale

Elastic interactions
within thermodynamics

bulk
Christoph Freysoldt – MPI Düsseldorf

3

Charged defects with DFTSupercell calculations

L• Wave function overlap (decay e-αr) → k-integration• Strain (decay 1/r3)→ often small• Coulomb interactions (decay 1/r) → ?
Christoph Freysoldt – MPI Düsseldorf

4Charged defects with DFTSlow supercell convergencesupercell calculations
without corrections

DFT-LDA, norm-conserving PP, no ionic relaxation (=no strain)
Christoph Freysoldt – MPI Düsseldorf

5Charged defects with DFTExact formulation of artifacts

12
3

Three-step process
1. add N electrons to defect state
2. relax other electrons
3. Introduce periodicity+backgroundbare charge q (r ) = − Neψ (r )2change in potential ∆V (r )periodic potential ∆V (r )CF, J. Neugebauer, C.G. Van de Walle, PRL 102, 016402 (2009).
Christoph Freysoldt – MPI Düsseldorf
6
Charged defects with DFTLong-range treatment

V (r )=V lr (r ) + V sr (r )model(r ')V (r ) = ∫ d r 'ε r − r'lrq

3DFTlrV (r ) = ∆V (r ) − V (r ) − C
srChristoph Freysoldt – MPI Düsseldorf

7

Charged defects with DFT

Short-range effects

V sr (r ) =∆V DFT (r ) − V lr (r ) − C• plateau indicates a successful modelling of long-range effects
Christoph Freysoldt – MPI Düsseldorf

8Charged defects with DFT

-3 Ga vacancy in GaAs (unrelaxed)bulk: CF, J. Neugebauer, C.G. van de Walle, PRL 102, 016402 (2009).
sxdefectalign available at https://sxrepo.mpie.de
Surfaces, interfaces, 2D materials:
CF, J. Neugebauer, Phys. Rev. B 97, 205425 (2018).

• Supercell-independent formation energies within 0.1 eV
Christoph Freysoldt – MPI Düsseldorf

9Charged defects with DFTReal interactions• Real defects occur at finite concentrations (10-6..10-3 relative)• Random distribution (at high T, low concentration)• Formation energy will vary from site to site due to interactions

Simulation setup• Fully compensated 𝑐𝑐 + = 𝑐𝑐 −• Randomly distributed charges• Screening by mobile carriers
1 𝑄𝑄 −𝑟𝑟/𝜆𝜆𝑉𝑉 𝑟𝑟 =𝑒𝑒
4𝜋𝜋𝜋𝜋 𝑟𝑟
• Many realizations (104 -105 sites)• Collect statistics
200x200x200 cells (8 million)
periodic boundary conditions
Christoph Freysoldt – MPI Düsseldorf

10Charged defects with DFTInteractions in random order

𝑝𝑝 Δ𝐸𝐸Δ𝐸𝐸 2− 2∼ 𝑒𝑒 2𝛽𝛽

Δ𝐸𝐸 2− 22𝛽𝛽Coulomb interaction with randomly distributed defectsyields Gaussian-like broadening of formation energyChristoph Freysoldt – MPI Düsseldorf

11Charged defects with DFTHow does broadening change with c?
1017 𝑐𝑐𝑚𝑚−3 1018 𝑐𝑐𝑚𝑚−31019 𝑐𝑐𝑚𝑚−3𝛽𝛽0.1 𝑒𝑒𝑒𝑒0.03 𝑒𝑒𝑒𝑒0.01 𝑒𝑒𝑒𝑒
𝑎𝑎0 = 0.5 𝑛𝑛𝑛𝑛, 𝜖𝜖 = 10𝜖𝜖0

Christoph Freysoldt – MPI Düsseldorf

𝛽𝛽 2 ∼ 𝑐𝑐
12Charged defects with DFTHow does broadening change with 𝜆𝜆?

𝛽𝛽 2 ∼ 𝜆𝜆
13𝑎𝑎0 = 0.5 𝑛𝑛𝑛𝑛, 𝜖𝜖 = 10𝜖𝜖0 , 𝑄𝑄 = ±1𝑒𝑒, 𝜆𝜆 = 10𝑛𝑛𝑛𝑛, 𝑐𝑐 = 1018 𝑐𝑐𝑚𝑚−3𝑝𝑝 Δ𝐸𝐸

Δ𝐸𝐸 2− 2∼ 𝑒𝑒 2𝛽𝛽Christoph Freysoldt – MPI Düsseldorf

𝑄𝑄 2𝛽𝛽 ∼𝑐𝑐𝑐𝑐 ≈ 0.03𝑒𝑒𝑒𝑒
4𝜋𝜋𝜋𝜋𝑎𝑎0
Charged defects with DFT

Outlook

Corrections forcharged-cell artifacts

Charged-defectinteractions at scale

Elastic interactions
within thermodynamics

bulk
Christoph Freysoldt – MPI Düsseldorf

14

Charged defects with DFTElastic interactions between defects

One approach: Lattice Green’s function [Tewary, Phys. Rev. B 094109 (2004)].
Christoph Freysoldt – MPI Düsseldorf

Charged defects with DFTThermodynamic view𝜕𝜕𝜕p=−𝜕𝜕𝜕defect Ub+d= Ub + Ud
Vb+d = Vb + Vdbulk

Vb+d(p) = Vb(p) + Vd(p)Ub+d(Vb+d(p))= Ub(Vb(p)) + Ud(Vd(p))
Enthalpy: H(p) = U(V) + pV
Hb+d(p)= Hb(p) + Hd(p)ΔE = Ub+d(Vb+d) - Ub(Vb+d)
Christoph Freysoldt – MPI Düsseldorf

16

Charged defects with DFT

ΔE at constant volumesubstitutional
Ge in Si

64 atom
8 atom
216 atom

ΔE at constant volume converges to formation enthalpy ΔHf
Charged defects with DFTGet ΔUf, ΔHf, ΔVrel from DFT
𝜕𝜕−𝜕𝜕𝜕

Ub+d(Vb+d) = Ub(Vb+d) + ΔEpDFT error
cancellation+ Δp= pb

Fitted analytic form, e.g. Murnaghan
ΔVrel (p) = Vb+d(p) - Vb(p)ΔUf = Δ E +Ub(Vb+d) - Ub(Vb)

Christoph Freysoldt – MPI Düsseldorf

18

Charged defects with DFT

Ge in bulk Si: energy and enthalpy

ΔUf(ε)
64 atom

ΔHf(p)

8 atom
216 atomChristoph Freysoldt – MPI Düsseldorf

8 atom64 atom
216 atom19

Charged defects with DFTConclusions

Corrections forcharged-cell artifacts

Charged-defectinteractions at scale

Elastic interactions
within thermodynamics

bulk
Christoph Freysoldt – MPI Düsseldorf

20Charged defects with DFTThermodynamics
System containing defect “X” at concentration cX (in total NX)
𝑓𝑓

𝐺𝐺 system = 𝐺𝐺 bulk + 𝑁𝑁𝑋𝑋 Δ𝐺𝐺𝑋𝑋 − 𝑇𝑇 𝑆𝑆 conf 𝑐𝑐𝑋𝑋 + Δ𝐺𝐺 interact [𝑐𝑐𝑋𝑋 ]interactions

Gibb’s free energy of formationof isolated(!) defect

𝐺𝐺 𝑝𝑝, 𝑇𝑇, 𝜇𝜇𝑠𝑠 = 𝑈𝑈+ 𝑝𝑝𝑝𝑝

0..4 eV 1..10 meV− 𝑇𝑇𝑇𝑇

0..±0.2 eV

𝑈𝑈 𝑉𝑉, 𝑇𝑇 = 0, 𝑁𝑁 → 𝐸𝐸 𝐷𝐷𝐷𝐷𝐷𝐷− 𝜇𝜇𝑠𝑠 Δ𝑁𝑁𝑠𝑠

Glensk et al., PRX 4, 011018 (2014)
Christoph Freysoldt – MPI Düsseldorf

22

Charged defects with DFTCoulomb interactions
1 1 −𝑟𝑟/𝜆𝜆𝑒𝑒
𝑉𝑉 𝑟𝑟 =4𝜋𝜋𝜋𝜋 𝑟𝑟

𝜖𝜖𝜖𝜖𝜖𝜖𝜆𝜆 = 2𝑛𝑛𝑒𝑒
2𝑄𝑄2≈ 0.3 𝑒𝑒𝑒𝑒
𝐸𝐸~4𝜋𝜋𝜋𝜋𝑎𝑎0𝑎𝑎0 = 0.5 𝑛𝑛𝑛𝑛, 𝜖𝜖 = 10𝜖𝜖0 ,𝑛𝑛 = 1017 𝑐𝑐𝑚𝑚−3 , 𝑇𝑇 = 300𝐾𝐾 → 𝜆𝜆 = 12𝑛𝑛𝑛𝑛 ≈ 20𝑎𝑎0

Most calculations done on periodic 200x200x200 lattice (8 million sites)

Christoph Freysoldt – MPI Düsseldorf

23Charged defects with DFT

Si(111) surface: convergencec• Role of the vacuum (for lateral: 2x2)
• Need corrections for charge states• Lateral convergence (size of surface unit cell) for c=40 bohr
Christoph Freysoldt – MPI Düsseldorf

24

Charged defects with DFTDFT results + sxdefectalign-2D corrections• Corrections work perfectly for vacuum convergence• and also improve NxN lateral convergence
Christoph Freysoldt – MPI Düsseldorf

25

Charged defects with DFTReal materials may be strained

InxGa1-xN

compressivestrain (x∙11%)

GaN

H

CO

coherently
strained

Host material
Goal: understand point defect formation energetics fromDFT (here: LDA or PBE) with “reasonable” settings
Christoph Freysoldt – MPI Düsseldorf

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Charged defects with DFTBeyond isotropic pressure/volume
Generalized enthalpy:
H(σ) = U(V0 ε) + pV - V0σdevεdev
1 𝜕𝜕𝜕𝜎𝜎𝑖𝑖𝑖𝑖 =V0 𝜕𝜕𝜖𝜖𝑖𝑖𝑖𝑖

General energy-strain curve:U(ε) = P(ln(1+ ε)ij)
nth –order polynomial
Example: In in wurtzite GaN (32 atoms)• free parameters: a and c lattice constant• 4th-order P
Christoph Freysoldt – MPI Düsseldorf

27

Charged defects with DFT

In in GaN
1ΔHf

[eV]

ΔUf

0a0.49ΔHf[bohr]cΔUf

[eV]
0.46

Christoph Freysoldt – MPI Düsseldorf

28

Charged defects with DFT

In in GaN
24

ΔVaΔVc[bohr3]8ac[bohr]

Relaxation volume varies with strain
Christoph Freysoldt – MPI Düsseldorf

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Charged defects with DFTConclusions• Robust, efficient scheme forthermodynamic properties ofdefects in strained material• General-purpose non-linear energy-strain model• Relaxation volume tensor
depends on strain
Christoph Freysoldt – MPI Düsseldorf

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Charged defects with DFTElastic interactions between defects
electrostatic elasticity
chargeforce
potential
displacement
stress 𝜎𝜎
𝑫𝑫strain 𝜖𝜖𝑬𝑬dipole elastic dipole tensor

relaxation volume tensor

nonlinear effects?
Christoph Freysoldt – MPI Düsseldorf

vectorscalarrank 2 tensor
vectorrank 2 tensor rank 4 tensor
rank 3 tensor rank 6 tensor
31

Charged defects with DFTWhat is DFT’s ΔE in thermodynamics?

ΔEΔUf

U

bulk

Vb+d(pb)

bulk

bulk

Vb(p)

-pp -Δp1
𝑝𝑝 + 𝑝𝑝𝑏𝑏 𝑉𝑉𝑑𝑑
2
V-pb

1
1
𝑓𝑓
Δ𝐸𝐸 = Δ𝑈𝑈 + 𝑝𝑝𝑉𝑉𝑑𝑑 − Δ𝑝𝑝𝑉𝑉𝑑𝑑 = Δ𝐻𝐻 − Δ𝑝𝑝𝑉𝑉𝑑𝑑
22𝑓𝑓

Christoph Freysoldt – MPI Düsseldorf

32

Charged defects with DFTGe in bulk Si: relaxation volumeRelaxation volume varies with strain
Christoph Freysoldt – MPI Düsseldorf

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Charged defects with DFTGe in bulk Si: improvement over ΔE

Christoph Freysoldt – MPI Düsseldorf

34

Charged defects with DFT