(This presentation is online at https://wmd-bath.github.io/theory4kest)

- Adam J. Jackson
- Undergraduate MEng Chemical Engineering (Bath, UK)
- Doctoral Training Centre in Sustainable Chemical Technologies (Bath, UK)
- Currently writing up a PhD thesis
- "Ab initio thermodynamics for practical kesterite photovoltaics"

- Based at University of Bath
- Near Bristol, < 2 hours from London

- Early experimental work
- Laurie Peter (my co-supervisor)
- Jonathan Scragg, Phillip Dale, Diego Columbara

- Laurie Peter (my co-supervisor)
- Current/recent
- Aron Walsh [computational]
- Adam Jackson, Jarvist Frost, Suzy Wallace

- Mark Weller [crystallography, solid-state synthesis]
- Mako Ng

- Laurie Peter, Kieran Molloy [solution processing]
- Anna Sudlow, Gabriela Kissling

- Aron Walsh [computational]
- PVTEAM partners
- SPECIFIC, Bristol, Loughborough, Northumbria

- Introduce the main theoretical methods that are relevant to kesterites research
- Highlight key existing theoretical work
- Identify problematic areas and opportunities for collaboration
- Help with terminology and critical reading of literature

- Nuts and bolts of running calculations
- Device modelling

- Simple interaction functions such as Buckingham potentials \((V = Ae^{-Br} - \frac{C}{r^6})\)
- Polarisation can be included approximately through shell models

- Simple to implement, inexpensive calculations
- Main challenge is in fitting the potentials
- No information about electronic structure
- We have access to a parameterisation for CZTS (Chris Eames, University of Bath)

- Focus on time-independent SchrÃ¶dinger equation \(H\Psi = E\Psi\)
- (Time-dependent methods exist, are more complicated…)
- Solution for a system locates the
*ground state*configuration and energy - Ground state energy \(E \approx U(T=0)\)
- \(H\) is an operator, must be defined for a particular
*basis*

- Bloch's theorem:
- The single-electron eigenvectors in a periodic system are the products of a single function with the periodicity of the lattice and all the points of the reciprocal lattice

- Crystalline systems can be modelled with a periodic boundary condition
- but the sum over the reciprocal lattice must be converged
- (integration over the Brillouin zone)

- but the sum over the reciprocal lattice must be converged

- A wide range of packages are available
- Wikipedia maintains a handy table
- Most popular codes include
- VASP (very efficient, if archaic, code for periodic systems)
- Gaussian (fast and quite user-friendly, non-periodic)

- Kohn-Sham DFT replaces \(E_{XC}(\Psi)\) with \(E_{XC}(\mathbf{\rho})\)
- Accuracy of DFT calculation depends on
- Numerical convergence of iterations
- Convergence of basis set, k-points
- Quality of XC-functional
- Does not "converge" systematically

- "Jacob's ladder" of methods 10.1063/1.1390175

- These methods are fairly affordable for large (100+ atoms) systems
**Local (spin)-Density Approximation**(LDA)- Consider only the local electron density, \(E_{XC} = f(\mathbf{\rho})\)
- Surprisingly good for metals

**Generalised Gradient Approximation**(GGA)- Include local density gradient, \(E_{XC} = f(\rho, \nabla rho)\)
- Better performance for semiconductors and insulators
- Popular GGAs are
**PW91**and**PBE**- We like
**PBEsol**

- We like

- Hugely underestimate bandgaps
**DO NOT TRUST BANDGAPS FROM LDA/GGA**

- Tight-binding models solve a simplified form of \(H\Psi = E\Psi\), where \(H\) contains a minimal set of interactions.
- Calculations are inexpensive and scalable
- The interactions must be parameterised
- "DFTB" refers to automated fitting of a tight-binding model to data from DFT

- Unlike forcefield methods, this is an electronic structure method

- DFT tends to underestimate on-site Coulomb interactions
- Leads to significant errors for localised d- and f-electrons
- A simple energy correction, \(U\), is included in SCF cycle

- The parameter \(U\) must be sourced responsibly!
- If \(U\) has been fitted to reproduce the experimental bandgap,
then good agreement can be obtained between calculated and measured results.
- This does not mean that good science is being done…

- If a paper uses DFT+U, check where the \(U\) comes from and bear this in mind.

- GGA functionals tend to under-estimate the "exchange" portion of E
_{XC}. - The Hartree-Fock (HF) method calculates
*exact*exchange, but*no*correlation. - Calculated energies can be improved by mixing in some HF exchange
- The main implementations are
**PBE0**(below),**B3LYP**(older, very succesful) and**HSE06**(PBE0 with range-screening)

- Unfortunately, the scaling of HF is worse than GGA, so calculations become slower
- Generally hybrid functionals give "reasonable" results, approaching "chemical accuracy"

- Quasi-particle methods
- \(G_{0}W_{0}\), sc\(GW\)

- Time-dependent DFT (TD-DFT)
- Useful for studying transition states, optical properties

- Post-Hartree Fock methods
- coupled cluster [CCSD, CCSD(T)]
- quantum monte carlo
- configuration interaction (CI)

- Some of these methods are very expensive, poor scaling
- Used for "gold standard" reference calculations

- Geometry optimisation required
- DFT calcs were key for confirming kesterite ground structure
- Conventional unit cells with LDA, GGA sufficient
- Several papers in 2009

(Paier et al. 2009)

- By studying larger supercells, simple defects and defect clusters can be studied
- V
_{Cu}, V_{Cu}+ Cu_{Zn}, Cu_{Zn}+ Zn_{Cu}clusters shown to have low formation energy

- V

- Defect formation energies depend on energies of other phases; and, ultimately temperature
- Kosyak et al. (2013) combines lattice vibrations with defect formation; finds S vacancies are also likely under annealing conditions

- LDA and GGA fail badly
- It is possible to apply an empirical correction or "scissors operator" but
there is no guarantee of accuracy
- This was used by Chen et al. (2009) when looking at CZTS analogues containing Ge

- LDA+U or GGA+U can improve matters, and was used by Persson (2010) for high-quality electronic structure calculations.
- Optimised with LDA for GGA+U calcs
- Est. \(E_g\) 1.5 eV for CZTS, 1.0 eV CZTSe
- From band structure obtain effective masses, dielectric constant, optical absorption coefficient

- Hybrid DFT calculations by Paier et al. (2009) found \(E_g\) 1.49 for kest CZTS, 1.30 for stannite
- Followed up with G
_{0}W_{0}quasiparticle calculation on HSE results, showing little change. - Expensive!

- Followed up with G

- Botti(2011) compares methods

- Same paper shows how sensitive bandgap is to anion displacement
- Shaded regions are experimental values

- Many theoretical papers say "enthalpy" and mean "ground-state energy"
- Thermodynamic potentials (H, S, \(\mu\) …) are temperature-dependent
- In the solid state this is primarily driven by lattice vibrations

- Some lattice vibrations can be directly observed by IR and/or Raman spectroscopy

- Assume simple harmonic motion of ground-state lattice
- Two main methods:
*Density functional perturbation theory*(DFPT)- Often quicker, built into many DFT codes. Only gives information at Γ-point.

*frozen phonon method*(AKA "direct method" and "supercell method")- Obtain whole phonon band structure

- Diagonalize matrix of force constants to obtain normal modes and frequencies.
- Results tend to agree qualitatively with experimental frequencies, with errors ~ 10 cm
^{-1}

- The first step is the quasi-harmonic approximation
- carry out harmonic approximation at several lattice expansions to form Equation of State
- Accounting for thermal expansion tends to "soften" frequencies

- Phonon-phonon interactions
- Very demanding, requires many displacement calculations and slow post-processing
- Allows peaks to be broadened using phonon lifetimes
- For a recent test case we used… CZTS!
- Skelton et al. (2015) improves agreement with IR and Raman data.
- Also shows that CZTS has very low thermal conductivity

- Energy is mapped to atom-centred functions
- Choice of procedures, no "correct" method

- Application: ZnS/Se resonant Raman spectroscopy
- HSE06 hybrid functional used for "good enough" bandgap
- Main challenge was converging k-point mesh

- Alignment of valence and conduction bands is important for device design
- Ideal reference point is the vacuum level
- Typical approach is look at electrostatic potential of "slab" models
- Slab cannot be a dipole
- dipole + periodic boundary = infinite energy

- Slab cannot be a dipole

- Band alignment has been done by modelling junctions with CdS and using CdS as the reference. Chen et al. (2011)

- Calculate forces and follow Newtonian mechanics
- Various methods to introduce temperature
- Many time steps needed for good quality data
- No problem for forcefield models, expensive for
*ab initio*

- No problem for forcefield models, expensive for
- We've done some simple tests
- https://www.youtube.com/watch?v=BuvtoyDF4vU
- Activation barrier to disorder is low, but still requires long MD runtimes

- Ab initio calculations have provided qualitative and quantitative information about the electronic structure of CZT(S,Se)
- Due of scaling challenges we know a lot more about highly-crystalline phases than disordered ones

- Lattice parameters (~1%) from LDA or better
- Elastic properties from GGA or better
- Optical properties from GGA+U or better
- Bandgaps from hybrid DFT or better

- Optimisation with one method and electronic structure from another
- Conveniently fitted scissors operator or \(U\)

- Γ-point vibrations
- Values for effective mass, absorption
- Properties of well-defined hypothetical materials

- "Fractional" structures
- Bandgaps of disordered, heavily-defective systems
- Surfaces and interfaces - but we must try!

- Centre for Sustainable Chemical Technologies
- PVTEAM
- Computer time from EPSRC, STFC and University of Bath
- Walsh Materials Design group