Tutorial: Understanding and Modelling Defects in Semiconductors (with VASP)

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Defects Tutorial I: Fundamentals Seán R. Kavanagh & Joe Willis 21/01/2021

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Overview This time… § Motivation § Types of defect § Formation energies § Transition level diagrams § Understanding results § Experimental defects Next time… § Calculations § INCAR settings § Dos and don’ts

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Motivation § Defects control device behaviour

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Point Defect Classification Intrinsic: Defect arising from imperfect arrangement of the bulk atoms. Always present; concentrations can be controlled just with temperature. Extrinsic: Defect arising from an external impurity. (e.g. doping, intentional or otherwise). Vacancy: Missing atom at lattice site. Interstitial: Atom located at interstitial site (i.e. void in the crystal structure). Substitution: Atom located at the lattice site of a different element. (Often “Antisite” if intrinsic). Complex: Two or more point defects located nearby, with elastic/electronic/magnetic/etc interactions.

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Kröger–Vink Point Defect Notation X is the species: - Element (Si, Cd…) - Vacancy “V” or “v” S is the lattice site: - Elemental Symbol of original site - “i” for an interstitial site q is the electronic charge of the species relative to the site that it occupies - q = Current Charge – Original Charge Kröger- Vink (Chemistry) Materials Science × 0 ′′ -2 •• +2 Charge Notation:

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Kröger–Vink Point Defect Notation

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Defect Formation Energy § Overview Goyal, A. et al, Comp. Mat. Sci., 2017, 130, 1-9

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Supercells § D = defect § q = charge state § H = host

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Chemical Potential § Accounting for the change in Gibbs free energy when adding or removing an atom § ni = number of atoms added to (negative) or removed from (positive) the host § µi = chemical potential § i = atomic species … but what is it?

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Chemical Potential: Definition § Rate of change of free energy of a system with respect to the change in number of atoms in the system, i.e.

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Chemical Potential: Limits § Thermodynamic stability of host material, e.g. ZnO

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Chemical Potential: Limits Zn-rich: Zn-poor:

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Chemical Potential: Limits § For multiple stable competing phases, solve simultaneous equations:

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Chemical Potential: Limits https://github.com/SMTG-UCL/wiki/wiki/Defects:-Chemical-Potential-Limits § For TiO2 :

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Chemical Potential: Dopant Limits § For a VTi defect: § The same process can be applied for dopant competing phases, but within chemical potential limits of the host material

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Chemical Potential: Dopant Limits Species Ti rich Ti poor Nb2 O5 -0.44 -9.24 NbO2 -0.42 -7.46 NbO -0.58 -4.10 µO = -3.52 µO = 0 µ Ti rich Ti poor Ti -2.10 -9.14 O -3.52 0.00 Nb -0.58 -9.24

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Chemical Potential: Doping Limits § For a substitution, you now must account for the energy needed to remove a host atom and to add a dopant atom § e.g. for a NbTi defect:

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Defect Formation Energy Goyal A, Gorai P, Peng H, Lany S and Stevanović V 2017 A computational framework for automation of point defect calculations Comput. Mater. Sci. 130 1–9 Fermi level = the thermodynamic work required to add one electron to the system

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Defect Formation Energy Goyal A, Gorai P, Peng H, Lany S and Stevanović V 2017 A computational framework for automation of point defect calculations Comput. Mater. Sci. 130 1–9 Defect-Defect Interactions:

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Defect Formation Energy § Overview Goyal, A. et al, Comp. Mat. Sci., 2017, 130, 1-9

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Transition Level Diagram

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Fermi Level EF / eV Formation energy / eV 𝑦 = 𝑚𝑥 + 𝑐 Fermi Level EF / eV Formation energy / eV +1 0 -1

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Transition Level Diagram Fermi Level EF / eV Formation energy / eV +1 0 -1

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Deep Defect Fermi Level EF / eV Formation energy / eV +1 0 0 3 CBM

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Deep Defect Huang, Y.-T. et al, Nano., 2021, 32, 132004

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Shallow Defect Fermi Level EF / eV Formation energy / eV +1 0 0 3 CBM

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Resonant Defect Fermi Level EF / eV Formation energy / eV +1 0 0 3 CBM

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Comparing Chemical Potential Limits

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Thermodynamic Driving Forces of Disorder Thermodynamic equilibrium: ΔF is minimised Gibbs Free Energy ΔF = ΔH – TΔS Configurational entropy S = kB lnW kB. = Boltzmann constant W = Multiplicity (of defect formation) Energy Free Energy Minimum (Thermodynamic Equilibrium) nd Nexp −∆H kB T Image Credits: Prof. Graeme Watson

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Defect Formation Energy Diagrams Fermi Level Pinning Charge Neutrality: p + ND + = n + NA – EF

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Defect Formation Energy Diagrams Fermi Level Pinning Charge Neutrality: p + ND + = n + NA – EF

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Defect Formation Energy Diagrams Fermi Level Pinning Charge Neutrality: p + ND + = n + NA – EF

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Defect Formation Energy Diagrams Fermi Level Pinning Charge Neutrality: p + ND + = n + NA – EF

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Defect Properties: Optical Transition Levels Td C3v Configuration Coordinate (Q) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Relative Energy (eV) 0.47 eV 0.22 eV 0.36 eV 0.58 eV 2.61 Å 2.59 Å 3.05 Å

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Optical Transition Levels

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Defect Formation Energy Diagrams Deep, shallow or resonant?

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Defect Formation Energy Diagrams Deep Levels -> Efficiency Reduction in Solar Cells

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Experiment - HAXPES HAXPES Credit: T. J. Featherstone

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Experiment - DLTS Lang, D. V., Jour. Appl. Phys., 1974, 45, 3023-32

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Experiment – DLTS Scanlon, D. O. et al, PRL, 2009, 103, 096405

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Next Time § More complicated chemical potential spaces § How to set up calculations § Finite size corrections in practice § Polarons § Optical transitions

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Further Reading § Lany, S. and Zunger, A., Phys. Rev. B, 2008, 78, 235104 § Freysoldt, C. et al, Rev. Mod. Phys., 2014, 86, 253 § Williamson, B. A. D. et al, Chem. Mater., 2020, 32, 1964-73 § Swallow, J. E. N. et al, Mater. Horiz., 2020, 7, 236-43 § Wickramaratne, D. et al, Appl. Phys. Lett., 2018, 113, 192106

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Chemical Potential: Ternary + § The problem clearly gets larger with more competing phases and more complex materials. § Use CPLAP! § Input: https://github.com/jbuckeridge/cplap

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Chemical Potential: Ternary + § Output:

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Chemical Potential: Ternary + Ternary: § 2D x-y plot with colour chart as 3rd dimension § 3D x-y-z plot For quaternary: § 3D x-y-z- plot with colour chart as 4th dimension § 2D x-y plots with colour chart as 3rd dimension, keep one chemical potential constant § 2D x-y plots keep two chemical potentials constant

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Chemical Potentials: Quaternary Example § Overview Xiao, Z. et al, Adv. Func. Mater., 2020, 1909906

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Defects Tutorial II: Implementation Seán R. Kavanagh & Joe Willis 04/02/2020

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Overview § Primitive – supercell relationship § Constructing defective supercells § INCAR tags § Calculating defects efficiently § Finite correction schemes in practice § Polarons § CPLAP for complex systems § Summary

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From primitive to supercell 3 x 3 x 2 $ super –f POSCAR 3 3 2 Initial structure has 4 atoms Final structure has 72 atoms import pymatgen.core.structure make_supercell(scaling_matrix)

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From primitive to supercell $ kgrid-series POSCAR_prim Length cutoff KSPACING Samples ------------- -------- ------------ 10.419 0.3015 8 8 4 $ kgrid-series POSCAR_super_3_3_2 Length cutoff KSPACING Samples ------------- -------- ------------ 10.419 0.3015 3 3 2 IMPORTANT Remember the relationship between real and reciprocal space Change your k-point grid accordingly

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Creating Point Defects • Manual • Python 🐍

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Creating Point Defects: Manual

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Creating Point Defects: Manual POSCAR: Beware POSCAR elemental ordering!

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Creating Point Defects: Python

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Creating Point Defects: Python github.com/kavanase/doped

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INCAR tags Bread and butter tags: ISIF = 2 ISYM = 0 IBRION = 1 POTIM = 0.2 ISPIN = 2 Play around with as usual if relaxation is getting stuck

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INCAR tags Trade-off between accuracy and supercell size/number of electrons in system: LREAL = False or LREAL = Auto ROPT = 1E-03 ! For each species in POSCAR Fast! More accurate, may be required for non- convergent relaxations

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INCAR tags If using AIDE: ICORELEVEL = 0 or 1 LVHAR = .TRUE. 0 prints average electrostatic potential at core 1 prints core state eigenenergies AIDE will calculate potential alignment using either. Be consistent across your project!

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INCAR tags If using DOPED: LVHAR = .TRUE. or ICORELEVEL = 0 Freysoldt Kumagai More on this from Seán to come…

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INCAR tags Setting charge states: 1. DOPED – reads POSCAR and POTCAR and assigns charge states based on pymatgen.core module. Can then manually pick/delete charge states provided. 2. Manually for AIDE – use your chemical intuition to consider each defect site and change NELECT and NUPDOWN accordingly.

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Chemical intuition…

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INCAR tags VO in In2 O3 : … or just use DOPED! VO x VO • VO •• NUPDOWN sets the difference between the number of electrons in the spin up and spin down channels. NELECT sets total number of electrons. NELECT = 378 NUPDOWN = 0 NELECT = 377 NUPDOWN = 1 NELECT = 376 NUPDOWN = 0

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Calculation Optimisation: Converge Test! davidbowler.github.io/AtomisticSimulations/blog/dft-scaling DFT Computation Time ∝ N3 Hybrid DFT ∝ N3 – N4 github.com/kavanase/vaspup2.0

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Calculation Optimisation: Test your System ‘Spot the difference’

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Calculation Optimisation: Test your System • AlGO • Supercell Size and k- grid combinations • KPAR

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Calculation Optimisation: Relaxation Pre-Convergence 𝚪-point only vasp_g am • For each inequivale nt defect site and likely magnetic configurat ion. NKRED = 2 vasp_s td • Only the lowest energy vasp_gam - predicted configurat ion, vasp_ std • Continua tion from NKRED run (often only 1 or 2 steps). vasp_ ncl • Spin-orbit single- shot energy calculatio n (possibly with (Can’t use WAVECAR from vasp_gam) (Can’t use WAVECAR from vasp_std)

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Calculation Optimisation: Relaxation Pre-Convergence 𝚪-point only vasp_gam •For each inequivalent defect site and likely magnetic configuration. NKRED = 2 vasp_std •Only the lowest energy vasp_gam- predicted configuration, unless tiny 𝛥E. vasp_std •Continuation from NKRED run (often only 1 or 2 steps). vasp_ncl •Spin-orbit single-shot energy calculation (possibly with ISMEAR=-5) (Can’t use WAVECAR from vasp_gam) (Can’t use WAVECAR from vasp_std)

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Post-Processing: Finite-Size Corrections AIDE (Lany-Zunger Correction Scheme) defects.dat: chem_pots.dat -> Chemical Potentials limits.dat -> Chemical Potential Limits diel.dat -> Dielectric Tensor

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Post-Processing: Finite-Size Corrections doped (FNV, Kumagai & Oba or Lany-Zunger Correction Scheme)

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Calculation Optimisation: Chemical Potentials: Smearing 5 10 15 20 25 30 k-point Mesh (i x i x i) −60 −50 −40 −30 −20 −10 0 10 Ground-State Energy wrt Converged Value [meV] Energy Convergence wrt k-point Mesh ISMEAR = -5 ISMEAR = 2 ISMEAR = 0 Good k-point convergence particularly important for calculations of metallic phases. (Metallic Bismuth)

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Calculation Optimisation: Chemical Potentials: Provided the k-point mesh is well-converged, NKRED{X,Y,Z} = 2 does not significantly affect the accuracy of the energy (𝛥E < 1 meV/atom), and reduces calculation cost by approx. an order of magnitude. Use of tetrahedron smearing (ISMEAR = -5) permits a reduced k-point mesh – but note is not suitable for metal relaxations.

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Polarons

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Polarons When a charge carrier interacts with ions in a system resulting in a bound state. This will lower the overall energy of a system. Need to account for this in highly ionic/polar systems, i.e. oxides. https://github.com/SMTG-UCL/wiki/wiki/Defect-Calculation-Workflow#accounting-for-polarons

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Polarons VIn in In2 O3 : VIn x VIn ’ VIn ’’ VIn ’’’

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Polarons VIn x 3 holes NELECT = 381 MAGMOM sets the initial magnetic moment for each atom NUPDOWN = 1 MAGMOM = 31*0 45*0 2*1 1*-1 NUPDOWN = 3 MAGMOM = 31*0 45*0 3*1 POSCAR In O 31 48

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Polarons VIn ’ 2 holes NUPDOWN = 2 MAGMOM = 31*0 46*0 2*1 POSCAR In O 31 48 NELECT = 382 NUPDOWN = 0 MAGMOM = 31*0 46*0 1*1 1*-1

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Polarons VIn ’’ 1 hole NUPDOWN = 1 MAGMOM = 31*0 47*0 1*1 POSCAR In O 31 48 NELECT = 383

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Polarons VIn ’’’ 0 holes NUPDOWN = 0 MAGMOM = 31*0 48*0 POSCAR In O 31 48 NELECT = 384

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Polarons VIn x VIn ’ VIn ’’ VIn ’’’

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Polawrong VIn ’’ VIn ’’ Higher energy structure

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Polarons Consider the ionicity of your material. Less likely in materials with disperse VBMs. Test all combinations with vasp_gam. Visualise hole charge density using PARCHG file. (Apply reverse logic if expecting localized electrons) https://github.com/SMTG-UCL/wiki/wiki/VASP-Partial-Charges

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CPLAP Chemical Potential Limits Analysis Program. Calculates values of µ for all elements in complex chemical potential spaces. Example run through with ternary material BaSnO3 .

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CPLAP 1. Calculate elemental reference energies. 2. Calculate formation energies of all stable competing phases (NOTE energy per formula unit) 3. Run CPLAP.

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CPLAP Sample input file: 3 !Number of species in host compound. 1 Ba 1 Sn 3 O -11.464243 !Stoichiometry + formation energy. none !Dependent variable. 3 !Number of competing phases. 2 !Number of species in comp. phase. 1 Ba 1 O -5.1386635 !Stoichiometry + formation energy. 2 !Number of species in comp. phase. 1 Sn 1 O -2.539629 !Stoichiometry + formation energy. 2 !Number of species in comp. phase. 1 Sn 2 O -5.2876295 !Stoichiometry + formation energy.

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CPLAP Sample output file Intersection points in chemical potential space: mu_Ba mu_Sn | mu_O = ----------------------|----------- -1.9759 0.0000 | -3.1628 -5.1387 -6.3256 | 0.0000 -3.5328 0.0000 | -2.6438 -6.1766 -5.2876 | 0.0000 -2.4948 0.0000 | -2.9898 -5.1387 -5.2876 | -0.3460 O-poor O-rich O-rich

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CPLAP Plotting: • 3D xyz plot • 2D xy plot with colour bar (set a dependent variable) • 3D xyz plot with colour bar (set a dependent variable) • 3D xyz plot with one fixed variable • 2D xy plots with multiple fixed variables

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CPLAP Two elements Three elements

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CPLAP § Overview Xiao, Z. et al, Adv. Func. Mater., 2020, 1909906

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The Main Event: Defect Thermodynamics

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Defect Analysis • Structure Visualisation (VESTA, CrystalMaker…) • Electronic Density of States (via sumo-dosplot and/or EIGENVAL & PROCAR files) • Transition Level Position (Deep, Shallow, Resonant?) • Charge/Magnetization Density Isosurfaces (from CHGCAR or PARCHG files) • Structural and Bond Length Analysis (via VESTA, doped, pymatgen…) • COHP (Bonding Analysis), via LOBSTER − 1 0 1 2 Energy(eV) Density of States Total DOS Bi (s) Bi (p) Br (p) Sn (s)

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Conclusions