modelling of charged point defects with density...
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Department of Computational Materials DesignDüsseldorf, Germany
Modelling of charged point defects with density-functional theory
Models and Data on Plasma-Material Interactions 2019June 18, 2019
Christoph Freysoldt
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 2
Point defects
How should we deal with defect-defect interactions?
Theoretical modelling is challenging• Supercell approach• Defect-defect interactions• Advanced electronic structure methods
(beyond standard DFT, i.e. hybrid functionals, GW, QMC)
• 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
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Charged defects with DFT
Outlook
Christoph Freysoldt – MPI Düsseldorf 3
Corrections for charged-cell artifacts Charged-defect
interactions at scale
bulk
Elastic interactionswithin thermodynamics
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 4
Supercell calculations
• Wave function overlap (decay e-αr) → k-integration• Strain (decay 1/r3) → often small• Coulomb interactions (decay 1/r) → ?
L
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 5
Slow supercell convergence
supercell calculations without corrections
DFT-LDA, norm-conserving PP, no ionic relaxation (=no strain)
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 6
Three-step process1. add N electrons to defect state bare charge 2. relax other electrons change in potential 3. Introduce periodicity periodic potential
+background
Exact formulation of artifacts
( )V∆ r( )V∆ r
2)()( rr ψNeq −=
1
23
CF, J. Neugebauer, C.G. Van de Walle, PRL 102, 016402 (2009).
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 7
Long-range treatment
Dsr lrFT (( ) ( )) CV V V− −= ∆r r r
lr sr( )( ) ( ) V V V= +r rrmodel
lr 3 ( ')( ) d ' qVε
=−∫
rr rr r'
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 8
• plateau indicates a successful modelling of long-range effects
Short-range effects
sr DFT lr( ) ( ) ( )V V V C= ∆ − −r r r
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 9
-3 Ga vacancy in GaAs (unrelaxed)
• Supercell-independent formation energies within 0.1 eV
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).
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Charged defects with DFT
Real interactions
Christoph Freysoldt – MPI Düsseldorf 10
• 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
200x200x200 cells (8 million)periodic boundary conditions
Simulation setup
• Fully compensated 𝑐𝑐+ = 𝑐𝑐−• Randomly distributed charges• Screening by mobile carriers
𝑉𝑉 𝑟𝑟 =14𝜋𝜋𝜋𝜋
𝑄𝑄𝑟𝑟𝑒𝑒−𝑟𝑟/𝜆𝜆
• Many realizations (104 -105 sites)• Collect statistics
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Charged defects with DFT
Interactions in random order
Christoph Freysoldt – MPI Düsseldorf 11
Coulomb interaction with randomly distributed defects yields Gaussian-like broadening of formation energy
𝑝𝑝 Δ𝐸𝐸 ∼ 𝑒𝑒−Δ𝐸𝐸
2
2𝛽𝛽2
−Δ𝐸𝐸2
2𝛽𝛽2
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Charged defects with DFT
How does broadening change with c?
Christoph Freysoldt – MPI Düsseldorf 12
𝛽𝛽2 ∼ 𝑐𝑐
𝛽𝛽
0.03 𝑒𝑒𝑉𝑉
0.01 𝑒𝑒𝑉𝑉
0.1 𝑒𝑒𝑉𝑉
1017𝑐𝑐𝑚𝑚−3 1019𝑐𝑐𝑚𝑚−31018𝑐𝑐𝑚𝑚−3
𝑎𝑎0 = 0.5 𝑛𝑛𝑚𝑚, 𝜋𝜋 = 10𝜋𝜋0
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Charged defects with DFT
How does broadening change with 𝜆𝜆?
Christoph Freysoldt – MPI Düsseldorf
13
𝛽𝛽2 ∼ 𝜆𝜆
𝑝𝑝 Δ𝐸𝐸 ∼ 𝑒𝑒−Δ𝐸𝐸
2
2𝛽𝛽2 𝛽𝛽 ∼𝑄𝑄2
4𝜋𝜋𝜋𝜋𝑎𝑎0𝑐𝑐𝜆𝜆
𝑎𝑎0 = 0.5 𝑛𝑛𝑚𝑚, 𝜋𝜋 = 10𝜋𝜋0, 𝑄𝑄 = ±1𝑒𝑒, 𝜆𝜆 = 10𝑛𝑛𝑚𝑚, 𝑐𝑐 = 1018𝑐𝑐𝑚𝑚−3
≈ 0.03𝑒𝑒𝑉𝑉
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Charged defects with DFT
Outlook
Christoph Freysoldt – MPI Düsseldorf 14
Corrections for charged-cell artifacts Charged-defect
interactions at scale
bulk
Elastic interactionswithin thermodynamics
![Page 15: Modelling of charged point defects with density …dpc.nifs.ac.jp/dkato/MoD-PMI2019/presentation/I-6_MoDPMI...charged-cell artifacts Charged-defect interactions at scale bulk Elastic](https://reader033.vdocument.in/reader033/viewer/2022053017/5f1cd5baad943b7ff42c07bf/html5/thumbnails/15.jpg)
Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf
Elastic interactions between defects
One approach: Lattice Green’s function [Tewary, Phys. Rev. B 094109 (2004)].
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 16
Thermodynamic view
bulk
defect Ub+d= Ub + UdVb+d = Vb + Vd
Hb+d(p)= Hb(p) + Hd(p)
Enthalpy: H(p) = U(V) + pV
p = −𝜕𝜕𝜕𝜕𝜕𝜕
ΔE = Ub+d(Vb+d) - Ub(Vb+d)
Vb+d(p) = Vb(p) + Vd(p)Ub+d(Vb+d(p))= Ub(Vb(p)) + Ud(Vd(p))
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Charged defects with DFT
ΔE at constant volume
64 atom8 atom
216 atom
substitutional Ge in Si
ΔE at constant volume converges to formation enthalpy ΔHf
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 18
Get ΔUf, ΔHf, ΔVrel from DFT
Ub+d(Vb+d) = Ub(Vb+d) + ΔE
p = pb + Δp−𝜕𝜕𝜕𝜕𝜕
ΔVrel (p) = Vb+d(p) - Vb(p)
ΔUf = Δ E +Ub(Vb+d) - Ub(Vb)
Fitted analytic form, e.g. Murnaghan
DFT error cancellation
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 19
Ge in bulk Si: energy and enthalpy
ΔUf(ε)
64 atom
8 atom216 atom
ΔHf(p)
64 atom8 atom
216 atom
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Charged defects with DFT
Conclusions
Christoph Freysoldt – MPI Düsseldorf 20
Corrections for charged-cell artifacts Charged-defect
interactions at scale
bulk
Elastic interactionswithin thermodynamics
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Charged defects with DFT
Thermodynamics
Christoph Freysoldt – MPI Düsseldorf 22
𝐺𝐺system = 𝐺𝐺bulk + 𝑁𝑁𝑋𝑋 Δ𝐺𝐺𝑋𝑋𝑓𝑓 − 𝑇𝑇 𝑆𝑆conf 𝑐𝑐𝑋𝑋 + Δ𝐺𝐺interact[𝑐𝑐𝑋𝑋]
Gibb’s free energy of formationof isolated(!) defect
𝐺𝐺 𝑝𝑝,𝑇𝑇, 𝜇𝜇𝑠𝑠 = 𝑈𝑈 + 𝑝𝑝𝑉𝑉 − 𝑇𝑇𝑆𝑆 − 𝜇𝜇𝑠𝑠Δ𝑁𝑁𝑠𝑠
𝑈𝑈 𝑉𝑉,𝑇𝑇 = 0,𝑁𝑁 → 𝐸𝐸𝐷𝐷𝐷𝐷𝐷𝐷
System containing defect “X” at concentration cX (in total NX)
0..4 eV 0..±0.2 eV1..10 meV
Glensk et al., PRX 4, 011018 (2014)
interactions
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Charged defects with DFT
Coulomb interactions
Christoph Freysoldt – MPI Düsseldorf 23
𝐸𝐸~𝑄𝑄2
4𝜋𝜋𝜋𝜋𝑎𝑎0
𝑉𝑉 𝑟𝑟 =14𝜋𝜋𝜋𝜋
1𝑟𝑟𝑒𝑒−𝑟𝑟/𝜆𝜆
𝜆𝜆2 =𝜋𝜋𝜖𝜖𝑇𝑇𝑛𝑛𝑒𝑒2
𝑎𝑎0 = 0.5 𝑛𝑛𝑚𝑚, 𝜋𝜋 = 10𝜋𝜋0,𝑛𝑛 = 1017𝑐𝑐𝑚𝑚−3,𝑇𝑇 = 300𝐾𝐾 → 𝜆𝜆 = 12𝑛𝑛𝑚𝑚 ≈ 20𝑎𝑎0
≈ 0.3 𝑒𝑒𝑉𝑉
Most calculations done on periodic 200x200x200 lattice (8 million sites)
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 24
Si(111) surface: convergence
c
• Role of the vacuum (for lateral: 2x2)• Need corrections for charge states• Lateral convergence (size of surface unit cell) for c=40 bohr
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 25
• Corrections work perfectly for vacuum convergence• and also improve NxN lateral convergence
DFT results + sxdefectalign-2D corrections
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Charged defects with DFT
Host material
coherentlystrained
Christoph Freysoldt – MPI Düsseldorf 26
Real materials may be strained
Goal: understand point defect formation energetics fromDFT (here: LDA or PBE) with “reasonable” settings
InxGa1-xNcompressivestrain (x∙11%)
GaN
H C
O
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 27
Beyond isotropic pressure/volume
Generalized enthalpy:H(σ) = U(V0 ε) + pV - V0σdevεdev
𝜎𝜎𝑖𝑖𝑖𝑖 =1𝜕0
𝜕𝜕𝜕𝜕𝜕𝜋𝜋𝑖𝑖𝑖𝑖
General energy-strain curve:U(ε) = P(ln(1+ ε)ij)
Example: In in wurtzite GaN (32 atoms)• free parameters: a and c lattice constant• 4th-order P
nth –order polynomial
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 28
In in GaN
ΔUf
ΔHf
[bohr]a c
0
1
ΔUfΔHf
0.49
0.46
[eV]
[eV]
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 29
In in GaN
ΔVaΔVc
[bohr]
ac
8
24
[bohr3]
Relaxation volume varies with strain
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 30
Conclusions
• Robust, efficient scheme for thermodynamic properties of defects in strained material
• General-purpose non-linear energy-strain model
• Relaxation volume tensor depends on strain
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 31
Elastic interactions between defects
electrostaticcharge
potential
𝑫𝑫𝑬𝑬
dipole
scalarvector
rank 2 tensorrank 3 tensor
elasticityforce
displacement
stress 𝜎𝜎strain 𝜋𝜋
elastic dipole tensorrelaxation volume tensor
vectorrank 2 tensorrank 4 tensorrank 6 tensornonlinear effects?
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 32
What is DFT’s ΔE in thermodynamics?
bulk bulk bulk
Vb+d(pb) Vb(p)
ΔEΔUf
12𝑝𝑝 + 𝑝𝑝𝑏𝑏 𝑉𝑉𝑑𝑑
p -Δp
Δ𝐸𝐸 = Δ𝑈𝑈𝑓𝑓 + 𝑝𝑝𝑉𝑉𝑑𝑑 −12Δ𝑝𝑝𝑉𝑉𝑑𝑑 = Δ𝐻𝐻𝑓𝑓 −
12Δ𝑝𝑝𝑉𝑉𝑑𝑑
V
U
-pb
-p
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 33
Ge in bulk Si: relaxation volume
Relaxation volume varies with strain
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Charged defects with DFTChristoph Freysoldt – MPI Düsseldorf 34
Ge in bulk Si: improvement over ΔE