Lecture delivered by A. Walsh during CDT-PV training, University of Bath, March 2015

1. Sustainable
Centre for
Chemical Technologies
Prof. Aron Walsh
Department of Chemistry
University of Bath
a.walsh@bath.ac.uk
From theory to solar cells

2. Lecturers
Dr Keith Butler
Degree in Medicinal Chemistry (Dublin)
Postdoctoral Fellow (Hybrid Perovskites)
Dr Jarvist Frost
Degree in Physics (Imperial College London)
Postdoctoral Fellow (Energy Materials)
Prof. Aron Walsh
Degree in Computational Chemistry (Dublin)
Professor of Materials Theory

3. Overview
Background: Materials Modelling is widely used
as a tool for characterisation and prediction in
materials science. There is an expanding
literature on solar energy (e.g. active layers,
interfaces, transparent conducting oxides).
Aim: To have a basic understanding of the
terms and concepts, with the ability to critically
assess research papers in your field.

4. Mini-Module Outline
Class philosophy
Theory à Practice à Applications
Course structure
Three lectures with class literature review
1. Modelling (AW)
Electrons in a periodic potential
2. Interfaces (KTB)
Workfunctions, band bending and contacts
3. Multi-scale (JMF)
Bridging from atoms to solar cells

5. Literature Review
Small Group Activity
Task 1 (this afternoon): Find a relevant research
paper that uses materials modelling in the
context of photovoltaics.
Task 2 (tomorrow morning): 15 minute
presentation & discussion of the paper
(including possible limitations of the approach).

6. Recommended General Textbooks
Bonding in Solids
• Electronic Structure and Chemistry of Solids,
P. A. Cox, Oxford Publishing (1987)
• Principles of the Theory of Solids, J. M. Ziman,
Cambridge Press (1979)
Computational Chemistry
• Molecular Modelling, A. Leach, Prentice Hall (2001)
• Introduction to Computational Chemistry,
F. Jensen, Wiley (2006)
Density Functional Theory for Solids
• Electronic Structure, R. M. Martin, Cambridge (2008)
• Planewaves, Pseudopotentials and the LAPW Method,
D.J. Singh, Kluwer (1994)

7. Materials Modelling
1. Theory: What Equations to Solve
2. Practice: Codes & Supercomputers
3. Applications: From Kesterites to Hybrid
Halide Perovskites

8. The Scientific Method
*Robert Boyle (left); William Hamilton (right)
Theory
“Laws”
Experiment
“Evidence”
Models
“Chemical Intuition”
Computation
“in silico”

9. A Multi-Scale Simulation Toolbox

10. Quantum Mechanics
ˆHΨ = EΨ
Kinetic and Potential Energy Operators
ˆH = ˆT + ˆV
Non-relativistic
Relativistic
Schrödinger
(1887, Vienna)
Dirac
(1902, Bristol)

11. Electronic Structure Techniques
E[Ψ] → E[ρ]
Density based
quantum
mechanics
Wavefunction
based quantum
mechanics
Methods
Hatree-Fock
Møller–Plesset
Coupled Cluster
Configuration Interaction
Methods
Thomas–Fermi
Density Functional
Dynamical Mean Field
Optimised Effective Potential

12. Density Functional Theory (DFT)
Kohn-Sham DFT (Physical Review 1965)
Use one-electron Ψ that reproduce true interacting ρ
Core Electrons
all-electron
pseudopotential
frozen-core
Hamiltonian
non-relativistic
scalar-relativistic
spin-orbit coupling
Periodicity
0D (molecules)
1D (wires)
2D (surfaces)
3D (crystals)
Electron Spin
restricted
unrestricted
non-collinear
Basis Set
plane waves
numerical orbitals
analytical functions
Functional
beyond……..
hybrid-GGA
meta-GGA
GGA
LDA
QMC
GW
RPA
TD-DFT

13. Materials Modelling with DFT
Input
Chemical Structure or Composition
Output
Total Energy + Electronic Structure
Structure
atomic forces
equilibrium coordinates
atomic vibrations
phonons
elastic constants
Thermodynamics
internal energy (U)
enthalpy (H)
free energy (G)
activation energies (ΔE)
Electron Energies
density of states
band structure
effective mass tensors
electron distribution
magnetism
Excitations
transition intensities
absorption spectra
dielectric functions
spectroscopy

14. Range of Applications
Materials Characterisation
Bulk physical and chemical properties.
Chemical Reactions
Catalysis; lattice defects; redox chemistry.
Materials Engineering
Beneficial dopants, alloys or morphology.
Substrate & Device Effects
Interfacial & strain phenomena.
Amorphisation
Conduction states in
InGaZnO4
Hybrid Network
Photochromic MIL-125
Understanding known compounds and
designing new materials

15. Exact Solution: Hydrogen
Building blocks for chemical bonding in all matter

16. From Atoms to Molecules (σ bonds)

17. Linear Combination of Atomic Orbitals
Ψ(r) = ciϕi (r)
i
∑
ϕ(r) =
1
N
re−αr2
ϕ(r) =
1
N
re−αr
ϕ(r) =
1
N
eikr
Gaussian functions (e.g. GAUSSIAN)
Slater functions (e.g. ADF)
Plane waves (e.g. VASP)
Numeric atom-centred functions (e.g. FHI-AIMS, SIESTA)
From simultaneous differential equations to linear algebra:

18. Basis Set Convergence
Source: Volker Blum – FHI-AIMS Summer School (2013)

19. From Molecules to Crystals
Lattice: an infinite array of points generated by translation
operations: R = n1a1+n2a2+n3a3
ñIntegerLattice vectorñ
Ψ(r) = u(r)eikr
Bloch Wave
Felix Bloch (1928)
Wavefunction of a particle in a
periodic potential (λ=2π/k)
1D
Bonding
Anti-Bonding
2D

20. k-point Sampling
All unique values of wave vector k are within the First
Brillouin Zone (primitive unit cell of the reciprocal lattice).
We just need to sample appropriately.
Dense k-point grids are used for converged total energy &
property calculations, but ‘band structures’ are
conventionally plotted along high symmetry lines.
Monkhorst & Pack, Physical Review B 13, 5188 (1976)

21. Lattice Settings: Diamond
1 0 0
0 1 0
0 0 1
!
!
0 0.5 0.5
0.5 0 0.5
0.5 0.5 0
!
!
Conventional cubic cell
8 atoms
Primitive fcc cell
2 atoms

22. Iterative Solutions: Electrons & Ions
Input
(ρtrial)
Electronic Minimisation
• Start from atomic or random density
• Apply variational principle
• Unique solution for closed-shell systems
Output
(ρ)
Input
(xyztrial)
Ionic Minimisation
• Start from X-ray or guess structures
• Calculate forces (-dE/dr)
• Usually exploring local structure
Output
(xyz)
(choice: diagonalisation method and mixing
between steps)
(choice: algorithm, e.g. conjugate gradient, quasi-
Newton, molecular dynamics)

23. Self-Consistent Cycle
(choice: algorithm, e.g. conjugate gradient, quasi-
Newton, molecular dynamics)
Source: Martijn Marsman – FHI-AIMS Summer School (2011)

24. Materials Modelling
1. Theory: What Equations to Solve
2. Practice: Codes & Supercomputers
3. Applications: From Kesterites to Hybrid
Halide Perovskites

25. Supercomputers (Top500.org – 11.14)

26. Next Step: Exascale Computing
1,000,000,000,000,000,000
floating point operations per second
1000
times faster calculations than current supercomputers
100
Megawatt power consumption (1 million 100W lightbulbs)
5
years before we have access

27. Tiered Computing Resources
Local:
Desktops
(4 – 8 cores)
Departmental:
Servers
(10s cores)
University:
Clusters
(1000s cores)
National:
Supercomputers
(100,000s cores)
BALENA Modest production runs and
project students.
ARCHER Large-scale production
runs (limited by wall-time).
NEON Interactive jobs; testing; non-
standard implementations.

28. Popular DFT Packages
• CASTEP (Plane wave – pseudopotential)
• CP2K (Mixed Gaussian/plane wave)
• FHI-AIMS (Numeric orbitals – all electron)
• GPAW (Numeric orbitals – pseudopotential)
• QUANTUM-ESPRESSO (Plane wave – pseudopotential)
• SIESTA (Numeric orbitals - pseudopotential)
• VASP (Plane wave – pseudopotential)
• WIEN2K (Augmented plane wave – all electron)
With the same exchange-correlation functional, all codes
should produce the same equilibrium properties.

29. Vienna Ab Initio Simulation Package
A widely used code from Austria (Prof. Georg Kresse):
• License fee ~€5000 (small academic group)
• Site: http://www.vasp.at
• Forum: http://cms.mpi.univie.ac.at/vasp-forum
• Wiki: http://cms.mpi.univie.ac.at/wiki
• Many pre- and post-processing tools.
• Visualisation: http://jp-minerals.org/vesta
A popular package because of reliable pseudopotentials for
periodic table (benchmarked against all-electron methods).

30. Compiling VASP (and other codes)
General Requirements:
Program source code (e.g. x.f, x.f90, x.c); Makefile or
configure script; Math libraries; Fortran or C compiler
Common Compilers:
Intel Fortran (ifort); Portland Group (pgf90); Gnu-Fortran
(gfort); Pathscale (pathf90); Generic links (f77 or f90)
Common Libraries:
LAPACK (Linear algebra - diagonalisation)
- ScaLAPACK (Distributed memory version)
BLAS (Linear algebra – vector / matrix multiplication)
BLACS (Linear algebra communication subprograms)
Examples: MKL (Intel); ACML (AMD); GotoBLAS

31. Example Makefile
(customised section only)
FC = ifort
FFLAGS = -O3
LAPACKBLAS = -L/$(MKL) -lmkl_intel_lp64
-lmkl_intel_thread -lmkl_core -lmkl_lapack
USE_MPI = yes
MPIFC = mpif90
…type “make”, the code will compile and a binary file is
created. Test and benchmark!
[Tip: intel-mkl-link-line-advisor for optimal MKL flags]

32. VASP Input Files
• POSCAR (“Position Card”)
• POTCAR (“Potential Card”)
• INCAR (“Input Card”)
• KPOINTS (k-point Sampling)
All four files should be in the same directory for VASP
to run successfully.
Caution: The order of the elements in POTCAR must be
the same as POSCAR.

33. VASP Output Files
• OUTCAR (“Output Card”)
• CONTCAR (“Continue [Positions] Card”)
• DOSCAR (“Density of States Card”)
• CHGCAR (“Charge Density Card”)
• vasprun.xml (Auxiliary output as xml)
A number of additional files that are generated
depending on flags set in INCAR.
Caution: If NSW > 0, a number of the properties are
averaged over past structures (rerun with NSW=0 at end).

34. Step 1: Structure
Generate crystal structure by hand, from supplementary
information, or from a database (e.g. ICSD).

35. Step 1: Structure
Check POSCAR

36. Step 2: Create other Input Files
cat ./C/POTCAR ./N/POTCAR ./H/POTCAR ./Pb_d/POTCAR ./I/POTCAR > POTCAR
INCAR (Partial)
KPOINTS

37. Step 3: Run VASP
Let’s see…

38. Step 4: Investigate Output Files
• OUTCAR – all basic output (including energy and forces)
• CONTCAR – the final structure
• DOSCAR – the electronic density of states
• PROCAR – the detailed band structure
• CHGCAR – the total electron density
See group guide for more details:
http://people.bath.ac.uk/aw558/presentations/
Many scripts and tools available online!

39. Dependence on Exc
Journal of Chemical Physics 123, 174101 (2005)
Recommend: PBEsol (GGA for solids) & HSE06 (Screened hybrid GGA)

40. Electronic Spectroscopy
Source: Patrick Rinks (FHI-AIMS Workshop 2011)
Approximate: Kohn-Sham eigenvalues
Accurate: Quasi-particle energies (GW) TD-DFT/BSE

41. Photoemission (DFT vs XPS): HgO
Chemical Physics Letters 399, 98 (2004) [1st Publication!]
XPS
(weighted DOS)
O K XES
(O 2p DOS)

42. The DFT Band Gap
There is much debate (and literature) on whether
the electronic band gap is a ground state
property and whether the exact exchange-
correlation functional would reproduce it, e.g.
Sham and Schluter, PRL 51, 1888 (1983)
Eg = IP – EA
For finite systems: the ionisation potentials can
be far from the Kohn-Sham eigenvalues.
For solids: Eg = IP – EA = -εKS
VB + εKS
CB
[Dilute limit: a one-electron change in an extended system]

43. DFT Caution!
While crystal structures, band widths and density
of states can be well described, many (LDA and
GGA) functionals predict band gaps too small.
This results in an exaggerated dielectric response
(too polarisable) and an incorrect onset of optical
absorption.
Common solutions:
• Scissors operator (shift conduction band
eigenvalues to match experimental gap).
• Use a hybrid exchange-correlation functional,
which reproduces the band gap.
• Go beyond DFT….

44. Beyond DFT
Many-body GW theory
L. Hedin, Phys. Rev. 139, A796 (1965)
From Kohn-Sham eigenvalues to quasi-particle
electron addition (N+1) and removal (N-1) energies.
Limitations:
• Self-consistency
• No total energy
• Excitons à GW+BSE Source: Patrick Rinke
Time-dependent DFT
E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984)
Inclusion of time-dependent potentials (electric, etc).
Limitations:
• Unknown functional (kernel) / different approximations
• Few full implementations for extended solids

45. Materials Modelling
1. Theory: What Equations to Solve
2. Practice: Codes & Supercomputers
3. Applications: From Kesterites to Hybrid
Halide Perovskites

46. Multi-component Semiconductors
Multernary Materials Screening
• Build database of plausible (stoichiometric) materials.
• Assess structural, electronic and thermodynamic properties.
• Screen & tailor for specific applications.
2
4
2

47. Quaternary Semiconductors
Predicted (and Confirmed) Photovoltaic Absorbers
Cu2ZnSnS4, Cu2ZnSnSe4 and Cu2ZnGeS4
Applied Physics Letters 94 041903 (2009) [> 340 citations]
Predicted Spin-transport Materials
ZnSiAl2As4, CdGeAl2As4 and CuAlCd2Se4
Applied Physics Letters 95 052102 (2010)
Predicted Topological Insulators
Cu2HgPbSe4, Cu2CdPbSe4 and Ag2HgPbSe4
Physical Review B 83 245202 (2011)

48. Cu2ZnSnS4 (13% Record Efficiency)
Advanced Energy Materials 2, 400 (2012)
• Crystal structure (kesterite vs stannite vs disordered)
• Band gaps (as a function of composition)
• Phase stability (disproportionation into secondary phases)
• Lattice defects (origin of electrons and holes)

49. Beyond Periodic Solids: Point Defects

50. Defects: Theory & Experiment
Calculable Observable
Total Energy Differences
• Heats of formation and
reaction: relative stabilities
and concentrations.
• Diffusion barriers.
Defect Ionisation Energy
(Vertical)
Optical absorption;
photoluminscence;
photoconductivity.
Defect Ionisation Energy
(Adiabatic)
Deep-level transient
spectroscopy; thermally
stimulated conductivity.
Defect Vibrational Modes
• IR / Raman spectra.
• Diffusion rates; free energy.

51. Defect Concentrations in Cu2ZnSnS4
Advanced Materials 25, 1522 (2013)
0.5
1
1.5
2
2.5
ElementRatio
1e+14
1e+16
1e+18
1e+20
DefectDensity
0.1
0.2
0.3
FermiEnergy
-0.5 -0.4 -0.3 -0.2 -0.1 0
Cu
(eV)
1e+14
1e+16
1e+18
HoleDensity
VCu
+ZnCu
CuZn
CuZn
-
VCu
-
Cu
Zn
Sn
ZnSn
+2ZnCu
2CuZn
+SnZn
2.5
0.5
1
1.5
2
2.5
ElementRatio
1e+14
1e+16
1e+18
1e+20
1e+22
DefectDensity
0
0.05
0.1
0.15
FermiEnergy
-0.4 -0.3
1e+16
1e+17
1e+18
HoleDensity
VCu
+ZnCu
VCu
-
Cu
Zn
Sn
ZnSn
+2ZnCu
VCu
2.5
(b) Cu2ZnSnS4
(a) Cu2ZnSnS4 (d) Cu2ZnSnSe4
(e) Cu2ZnSnSe4
“Cu-rich”
“Cu-poor”
Cu:Zn:Sn
Defects
Carriers
μelectron

52. Hybrid Halide Perovskites
Snaith (Oxford)
Grätzel (EPFL)
Park (SKKU)
Il Seok (KRICT)
APL Mater. 1, 042111 (2013); Nano Letters 14, 2484 (2014)
A B X3 a (Å) Eg (eV)
NH4
+ Pb I 6.21 1.38
CH3NH3
+ Pb I 6.29 1.67
CH(NH2)2
+ Pb I 6.34 1.55
(1991) Dye cell à (2015) Perovskite cell [20.1% efficiency]
See Mendeley Group “Hybrid Perovskite Solar Cells”

53. CH3NH3PbI3 (or MAPI for short)
Configuration: PbII [5d106s26p0]; I-I [5p6]
F. Brivio et al, Physical Review B 89, 155204 (2014)
Relativistic QSGW theory with Mark van Schilfgaarde (KCL)
Conduction
Band
Valence
Band
Dresselhaus
Splitting (SOC)
[Molecule breaks
centrosymmetry]

54. First-principles Dynamics (300 K)
“MAPI is as soft as jelly”
25 fs per frame
J. M. Frost et al, APL Materials 2, 081506 (2014)
Jarvist
http://dx.doi.org/10.6084/m9.figshare.1061490
ß
Focus on one
CH3NH3 ion3D periodic
boundary
(80 - 640 atoms)

55. Domains of Molecular Dipoles
Ferroelectric Hamiltonian (Monte Carlo solver)
Regions of high (red) and low (blue) electrostatic potential
J. M. Frost et al, APL Materials 2, 081506 (2014)

56. Mixed Ionic-Electronic Conductors
Angewandte Chemie 54, 1791 (2015); Under Review (2015)

57. Lecture 1 Conclusions
For reliable materials modelling, follow :
:
• Basis sets & k-points
• Forces & cell pressure
:
• Exchange-correlation functional
:
• Measured values and properties
• Previous calculations
ng
ve
s
ed
f
n
n
t
simulation at the forefront of the search
for new materials2
. Using quantum
mechanical techniques, quantitative
information on the structure and properties
of a material can be provided at relatively
modest computational and economic cost.
Efforts such as the Materials Project have
succeeded in tabulating the properties of
many known inorganic systems, with more
shows one such process), and validated
by ‘searching’ known compounds — the
method did correctly predict their stability
and structures. A crystal structure search
was carried out to ensure a global minimum
configuration was identified, and the
vibrational spectrum of each candidate
material was investigated to confirm its
dynamic stability. Finally thermodynamic
cted structures and properties.
Structural
prediction
Property
simulation
Targeted
synthesis
Chemical
input
Figure 1 | A modular materials design procedure, where an initial selection of chemical elements is
subject to a series of optimization and screening steps. Each step may involve prediction of the crystal
structure, assessment of the chemical stability or properties of the candidate materials, before being
followed by experimental synthesis and characterization. A material may be targeted based on any
combination of properties, for example, a large Seebeck coefficient and low lattice thermal conductivity
for application to heat-to-electricity conversion in a thermoelectric device.