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Sustainable
Centre for
Chemical Technologies
Prof. Aron Walsh
Department of Chemistry
University of Bath
a.walsh@bath.ac.uk
From theory to solar cells
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
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.
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
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).
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)
Materials Modelling
1.  Theory: What Equations to Solve
2.  Practice: Codes & Supercomputers
3.  Applications: From Kesterites to Hybrid
Halide Perovskites
The Scientific Method
*Robert Boyle (left); William Hamilton (right)
Theory
“Laws”
Experiment
“Evidence”
Models
“Chemical Intuition”
Computation
“in silico”
A Multi-Scale Simulation Toolbox
Quantum Mechanics
ˆHΨ = EΨ
Kinetic and Potential Energy Operators
ˆH = ˆT + ˆV
Non-relativistic
 Relativistic
Schrödinger
(1887, Vienna)
Dirac
(1902, Bristol)
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
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
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
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
Exact Solution: Hydrogen
Building blocks for chemical bonding in all matter
From Atoms to Molecules (σ bonds)
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:
Basis Set Convergence
Source: Volker Blum – FHI-AIMS Summer School (2013)
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
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)
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
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)
Self-Consistent Cycle
(choice: algorithm, e.g. conjugate gradient, quasi-
Newton, molecular dynamics)
Source: Martijn Marsman – FHI-AIMS Summer School (2011)
Materials Modelling
1.  Theory: What Equations to Solve
2.  Practice: Codes & Supercomputers
3.  Applications: From Kesterites to Hybrid
Halide Perovskites
Supercomputers (Top500.org – 11.14)
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
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.
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.
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).
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
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]
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.
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).
Step 1: Structure
Generate crystal structure by hand, from supplementary
information, or from a database (e.g. ICSD).
Step 1: Structure
Check POSCAR
Step 2: Create other Input Files
cat ./C/POTCAR ./N/POTCAR ./H/POTCAR ./Pb_d/POTCAR ./I/POTCAR > POTCAR
INCAR (Partial)
KPOINTS
Step 3: Run VASP
Let’s see…
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!
Dependence on Exc
Journal of Chemical Physics 123, 174101 (2005)
Recommend: PBEsol (GGA for solids) & HSE06 (Screened hybrid GGA)
Electronic Spectroscopy
Source: Patrick Rinks (FHI-AIMS Workshop 2011)
Approximate: Kohn-Sham eigenvalues
Accurate: Quasi-particle energies (GW) TD-DFT/BSE
Photoemission (DFT vs XPS): HgO
Chemical Physics Letters 399, 98 (2004) [1st Publication!]
XPS
(weighted DOS)
O K XES
(O 2p DOS)
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]
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….
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
Materials Modelling
1.  Theory: What Equations to Solve
2.  Practice: Codes & Supercomputers
3.  Applications: From Kesterites to Hybrid
Halide Perovskites
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
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)
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)
Beyond Periodic Solids: Point Defects
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.
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
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”
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]
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)
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)
Mixed Ionic-Electronic Conductors
Angewandte Chemie 54, 1791 (2015); Under Review (2015)
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.

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Materials Modelling: From theory to solar cells (Lecture 1)

  • 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”
  • 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
  • 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).
  • 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.