Theoretical modelling for kesterite photovoltaics

1.1 About me

• 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

1.2 University of Bath

1.3 Kesterites at Bath

• Early experimental work
  • Laurie Peter (my co-supervisor)
    • Jonathan Scragg, Phillip Dale, Diego Columbara
• 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
• PVTEAM partners
  • SPECIFIC, Bristol, Loughborough, Northumbria

1.4 Aims of this talk

• 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

1.5 Not in this talk

• Nuts and bolts of running calculations
• Device modelling

3.1 Interatomic potentials

• 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)

3.2 Ab initio methods

• Focus on time-independent Schrödinger 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)

3.3 Ab initio codes

• 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)

3.4 Density Functional Theory

• 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

3.5 LDA and GGA

• 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
• Hugely underestimate bandgaps
  • DO NOT TRUST BANDGAPS FROM LDA/GGA

3.6 Tight-binding

• 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

3.7 DFT+U

• 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.

3.8 Hybrid functionals

• 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"

3.9 "Beyond-DFT" methods

• 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

4.1 Total energy 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)

4.2 Defect energies

• 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

• 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

4.3 Bandgaps

• 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 G₀W₀ quasiparticle calculation on HSE results, showing little change.
  • Expensive!

• Botti(2011) compares methods

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

4.4 Vibrations

• 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

4.5 Harmonic approximation

• 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

4.6 Beyond the harmonic approximation

• 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

5.1 Projected density of states

• 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

5.2 Surfaces and work functions

• 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

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

5.3 Molecular dynamics

• 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
• 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

6.1 Key work so far

• 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

6.2 Within reason, you can trust

• 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

6.3 Beware

• Optimisation with one method and electronic structure from another
• Conveniently fitted scissors operator or $U$

6.4 Some good areas for collaboration

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

6.5 It's hard to help with

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

6.6 Acknowledgements

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