Download Computational study of structural and electrical properties of

The University of Toledo
The University of Toledo Digital Repository
Theses and Dissertations
2015
Computational study of structural and electrical
properties of methylammonium lead iodide
perovskite
Vamshidhar Rao Boinapally
Follow this and additional works at: http://utdr.utoledo.edu/theses-dissertations
Recommended Citation
Boinapally, Vamshidhar Rao, "Computational study of structural and electrical properties of methylammonium lead iodide perovskite"
(2015). Theses and Dissertations. 1815.
http://utdr.utoledo.edu/theses-dissertations/1815
This Thesis is brought to you for free and open access by The University of Toledo Digital Repository. It has been accepted for inclusion in Theses and
Dissertations by an authorized administrator of The University of Toledo Digital Repository. For more information, please see the repository's About
page.
A Thesis
entitled
Computational Study of Structural and Electrical Properties of Methylammonium
Lead Iodide Perovskite
by
Vamshidhar Rao Boinapally
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Masters of Science Degree in Electrical Engineering
Dr. Sanjay V. Khare, Committee Chair
Dr. Junghwan Kim, Committee Member
Dr. Rashmi Jha, Committee Member
Dr. Patricia R. Komuniecki, Dean
College of Graduate Studies
The University of Toledo
May 2015
Copyright 2015, Vamshidhar Rao Boinapally
This document is copyrighted material. Under copyright law, no parts of this
document may be reproduced without the expressed permission of the author.
An Abstract of
Computational Study of Structural and Electrical Properties of Methylammonium
Lead Iodide Perovskite
by
Vamshidhar Rao Boinapally
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Masters of Science Degree in Electrical Engineering
The University of Toledo
May 2015
Methyl ammonium lead iodide perovskite (CH3 NH3 PbI3 ) plays an important role
in light absorption in perovskite solar cells. The main aim of this thesis is to investigate the structural and electrical properties of cubic and tetragonal phases of
CH3 NH3 PbI3 . The optimized structure and minimum energy lattice constants of relaxed cubic unit cells were initially computed. The most stable orientation of methylammonium cation was found to be in the [1 1 -1] direction. This directional preference
is described by bonding analysis of the atomic cage of PbI3 with the C≡N dimer. The
variation of c/a ratio with the distortion angle of PbI6 underlies the understanding
of the transition from the cubic to the tetragonal phase.
For the equilibrium structures band structures and effective masses were computed. The computed effective masses of both holes and electrons of CH3 NH3 PbI3
are comparable to the widely used silicon in commercial inorganic solar cells. These results describe the light absorption nature of methylammonium lead iodide perovskite
and its importance in future solar cell technology.
iii
Dedicated to my parents.
Thank you.
Acknowledgments
I would like to start by thanking my research advisor, Prof. Sanjay V. Khare. Not
only for his guidance throughout my research but also for the knowledge and tips he
has shared with me. I thank him for playing major role in my career development.
Next, I would like to thank my committee members Prof. Junghwan Kim and
Prof. Rashmi Jha for their support and guidance.
Next, I would like to thank all the current and former members of Dr. Khare’s
research group for their help and friendship.
Next, I would like to thank all the faculty and staff of both Engineering and
Physics departments.
I want to mention a special thanks to Dr. Richard Irving for his computer and
technical support throughout my research.
I would like to thank my parents, my brother, friends and my loved ones for their
love and support.
And finally I would like to thank God for his eternal love and blessings.
v
Contents
Abstract
iii
Acknowledgments
v
Contents
vi
List of Tables
ix
List of Figures
x
List of Abbreviations
xii
1 Introduction
1
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2 Solar Energy and Solar Cells
4
2.1
Solar Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2
Photovoltaic System and Need of PV . . . . . . . . . . . . . . . . . .
4
2.3
Basic Semiconductor Physics
. . . . . . . . . . . . . . . . . . . . . .
6
2.4
Types of Solar Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.4.1
Crystalline Solar Cells . . . . . . . . . . . . . . . . . . . . . .
6
2.4.2
Concentrating Photovoltaic Solar Cells (CPV) . . . . . . . . .
7
2.4.3
Thin Film Solar cells (TFSC) . . . . . . . . . . . . . . . . . .
7
Introduction to Perovskite Solar Cells . . . . . . . . . . . . . . . . . .
8
2.5
vi
2.6
Solar cell efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3 VASP
10
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3.2
Inpu Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.2.1
INCAR file . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.2.2
POTCAR file . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.2.3
POSCAR file . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.2.4
KPOINTS file . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
Output Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.3.1
CONTCAR File . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.3.2
OSZICAR File . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.3
4 Semiclassical Model of Electron Dynamics
16
4.1
Introduction of the Semiclassical Model . . . . . . . . . . . . . . . . .
16
4.2
Filled Bands are Inert . . . . . . . . . . . . . . . . . . . . . . . . . .
17
4.3
Semiclassical Motion in Applied DC Electric Field . . . . . . . . . . .
18
5 Density Functional Theory (DFT)
21
5.1
What is DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
5.2
Fundamentals of DFT . . . . . . . . . . . . . . . . . . . . . . . . . .
22
5.3
Total Energy from DFT . . . . . . . . . . . . . . . . . . . . . . . . .
23
5.4
Exchange Correlation Approximations . . . . . . . . . . . . . . . . .
24
5.4.1
Local Density Approximation (LDA) . . . . . . . . . . . . . .
24
5.4.2
Local Spin Density Approximation (LSDA) . . . . . . . . . . .
25
5.4.3
Generalized Gradient Approximation (GGA) . . . . . . . . . .
26
6 Materials and Structure
6.1
27
Perovskite (CaTiO3 ) and its Structure . . . . . . . . . . . . . . . . .
vii
27
6.2
Lead Titanate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
6.3
Methylammonium Tin and Lead Halides . . . . . . . . . . . . . . . .
29
6.3.1
CH3 NH3 SnX3 . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
6.3.2
CH3 NH3 PbI3 . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
Perovskite-Based Solar Cells . . . . . . . . . . . . . . . . . . . . . . .
31
6.4
7 Results
36
7.1
Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . .
36
7.2
CH3 NH3 PbI3 Cubic Structure . . . . . . . . . . . . . . . . . . . . . .
37
7.3
Study of C-N bond in cubic CH3 NH3 PbI3
. . . . . . . . . . . . . . .
39
7.4
CH3 NH3 PbI3 Tetragonal Structure . . . . . . . . . . . . . . . . . . .
44
7.5
Study of PbI6 Octahedron with CH3 NH3 and Cs Cation . . . . . . . .
47
7.6
Band Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
7.7
Effective Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
8 Conclusions and Future Work
60
References
62
viii
List of Tables
7.1
Optimized lattice parameters and volume of cubic and tetragonal unit cells. 45
7.2
Effective mass of hole at the valence band maximum. . . . . . . . . . . .
58
7.3
Effective mass of electron at the conduction band minimum. . . . . . . .
59
ix
List of Figures
2-1 Basic Grid-Interactive PV System w/o Battery Backup.
. . . . . . . . .
5
2-2 Broad Classification of Solar Cells. . . . . . . . . . . . . . . . . . . . . .
8
2-3 Efficiencies of Various Solar cell Technologies. . . . . . . . . . . . . . . .
9
5-1 Algorithm for calculation of total energy from DFT . . . . . . . . . . . .
23
6-1 Basic ABX3 perovskite structure . . . . . . . . . . . . . . . . . . . . . .
28
6-2 Crystal Structure of PbTiO3 showing both cubic and tetragonal phases. .
29
6-3 CH3 NH3 SnCl3 in the triclinic phase . . . . . . . . . . . . . . . . . . . . .
30
6-4 CH3 NH3 SnI3 in cubic phase . . . . . . . . . . . . . . . . . . . . . . . . .
31
6-5 Rate of increase in perovskite solar cell efficiencies . . . . . . . . . . . . .
32
6-6 CH3 NH3 PbI3 /TiO2 heterojunction solar cell . . . . . . . . . . . . . . . .
34
6-7 Perovskite Solar cells performance parameters . . . . . . . . . . . . . . .
35
7-1 Structure of cubic CH3 NH3 PbI3 unit cell in the [1 1 1] direction . . . . .
37
7-2 Plot between volume and energy of unit cell with Birch-Murnaghan equation of state.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
7-3 CH3 NH3 PbI3 cubic unit cell with C≡N bond in [0 0 1] direction. . . . . .
39
7-4 CH3 NH3 PbI3 cubic unit cell with C≡N bond in [1 1 1] direction.
. . . .
40
7-5 CH3 NH3 PbI3 cubic unit cell with C≡N bond in [1 1 0] direction.
. . . .
40
7-6 CH3 NH3 PbI3 cubic unit cell in [1 1 -1] direction. . . . . . . . . . . . . . .
42
7-7 Figure showing rotation of C-N from [0 0 1] to [0 0 -1] direction in the
plane of C≡N bond. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
43
7-8 Energy variation with rotation and directional angles of C-N.
. . . . . .
43
7-9 Tetragonal structure of CH3 NH3 PbI3 in top view. . . . . . . . . . . . . .
45
7-10 Tetragonal structure of CH3 NH3 PbI3 in side view. . . . . . . . . . . . . .
46
7-11 CH3 NH3 PbI3 tetragonal structure in top view showing non-distorted octahedra and non-distorted Pb-I-Pb bond. . . . . . . . . . . . . . . . . . .
47
7-12 Tetragonal structure with cesium cation in top view. . . . . . . . . . . .
48
7-13 Variation of c/a ratio with distortion angle of PbI6 octahedron. . . . . .
49
7-14 Variation of average lattice constant (d ) with distortion angle of octahedron.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
7-15 Scheme of the Brillouin Zone for a simple cubic lattice. . . . . . . . . . .
53
7-16 Band structure of cubic CH3 NH3 PbI3 . . . . . . . . . . . . . . . . . . . .
54
7-17 Scheme of the Brillouin Zone for a simple tetragonal lattice. . . . . . . .
55
7-18 Band structure of tetragonal CH3 NH3 PbI3 . . . . . . . . . . . . . . . . . .
56
7-19 Band structure of tetragonal CsPbI3 . . . . . . . . . . . . . . . . . . . . .
57
xi
List of Abbreviations
VASP . . . . . . . . . . . . . . . . . . . . .
DFT . . . . . . . . . . . . . . . . . . . . . .
LDA . . . . . . . . . . . . . . . . . . . . . .
LSDA . . . . . . . . . . . . . . . . . . . . .
GGA . . . . . . . . . . . . . . . . . . . . .
MAPbI3 . . . . . . . . . . . . . . . . . .
Vienna ab initio Simulation Package
Density Functional Theory
Local Density Approximation
Local Spin Density Approximation
Generalized Gradient Approximation
Methylammonium lead iodide
xii
Chapter 1
Introduction
1.1
Motivation
Presently, fossil fuels are the primary sources of energy. Human activities are
responsible for the release of large quantities of carbon dioxide and various greenhouse
gases in to the atmosphere. Burning of fossil fuels to produce energy is the primary
reason for the release of these harmful gases. These greenhouse gases are responsible
for global warming which leads to climate change. Over the past century, earth’s
average temperature has risen by 1.4 ◦ F. This small change in average temperature of
the planet has led to many drastic changes in climate and weather. So replacement
of fossil fuels from the current energy supply is important.
Furthermore, fossil fuels coal, oil and natural gas energy sources are non-renewable.
About 85 % of world total energy supply is from fossil fuels [1]. But with existing
data of resources and reserves these fossils fuels are going to depleted in near future.
Recent research shows that total coal production in U.S is peaked in the years 2008
and 2010 [2] and decline gradually. Similarly all the existing fossil fuels are going to
reach peak productions in near future and decline eventually [3-5]. Since these fuels
are non-renewable and take millions of years to reform, the need of renewable sources
of energy is very high for the present world. Among renewable energy sources wind
1
and solar are the major contributors. In the design of solar cells we need to consider
parameters like maximum efficiency, life span and materials required. At present,
crystalline silicon and thin film solar cells are widely used. In crystalline silicon we
have both monocrystalline and polycrystalline silicon solar cells based on the crystal
structure orientation. These crystalline solar cells are more efficient when compared
to other solar cells but they are expensive. Thin film solar cells are made by depositing one or more thin films of photovoltaic material on different substrates. Thin film
solar cells are cheaper in practical but less efficient when compared to silicon cells.
However we need more technologies to have more efficient and cheaper solar cells,
this led to the study of perovskite photovoltaic material. Recent studies show that
this material is highly useful in solar cells [6-9]. This material along with the future
technology advancement has potential to make a new revolution in the field of solar
cells.
1.2
Thesis Outline
The thesis presented here is a theoretical study of structural and electrical properties of methyl ammonium lead iodide perovskite. The whole calculations and computations are done with the GGA approximations as a variant of density functional
theory (DFT).
Chapter 2 begins with definition of solar energy and explains basic semiconductor
physics related to solar cells. Then it includes different types and classification of
solar cells. A brief introduction related to perovskite solar cells is discussed in this
chapter.
Chapter 3 includes the description about VASP software. Starting with the introduction of VASP, it also discuss about the important input and output files related
to the software tool.
2
Chapter 4 includes the discussion about basic semiclassical model of electron dynamics. It explains in detail about the filled and inert bands. The semiclassical
motion in an applied DC electric field is mentioned in this chapter.
Chapter 5 presented in this chapter discuss in detail about the density functional
theory (DFT). It begins with the introduction to DFT and its need. The approximations methods involved in DFT are discussed in brief.
Chapter 6 includes the discussion on basic perovskite compound and its structure.
Since the thesis presented here is a study of methyl ammonium halide perovskites. A
brief discussion on the structure of common methyl ammonium halide perovskite is
done in this chapter.
Chapter 7 presented in this thesis includes all the results. It begins with the
structure and structural properties of cubic structure of CH3 NH3 PbI3 . Results on
Carbon and nitrogen bond study are discussed. Discussion on lattice parameters
related to both cubic and tetragonal is presented. Design of tetragonal structure with
both methyl ammonium and cesium cation is included. Study on PbI6 octahedron
related to lattice parameters is presented. In the end, results on band structures and
effective masses are included in this chapter.
Chapter 8 has brief conclusions about the thesis.
3
Chapter 2
Solar Energy and Solar Cells
2.1
Solar Energy
Solar energy can be defined as the electromagnetic energy transmitted from the
sun. In other words it is the radiant light and heat from the sun. We can harness the
solar energy by different technologies [10]. In general all the solar technologies are
classified in to either passive or active solar. This classification is made depending up
on their way of capture, conversion and distribution of solar energy [10].
In case of active solar techniques they use photovoltaic panels and solar thermal collectors to harness the solar energy. The design of passive solar techniques
is completely different from active techniques. It depends up on the orientation of
the buildings, selection of different materials with required thermal mass and light
dispersing properties [11, 12].
2.2
Photovoltaic System and Need of PV
There is lot of difference between photovoltaic systems and other solar systems.
The PV systems uses PV cells made of semiconductor materials to convert sunlight
or solar energy to electricity [13, 14]. There are many other solar power systems
which use other phenomenon to convert the sunlight to electricity. For example
4
concentrated solar power systems (CSP) have the reflective devices to make energy
conversion possible [12].
In today’s world demand for energy is very high. In addition to this we are also
running out of fossil fuels, which are the main sources of energy for all of us in many
ways. At present we use coal, oil, natural gas as our primary resources in electricity
generation. The percent of electricity and energy contributed from renewable energy
sources is small when compared to fossil fuels. But these fossil fuels are non-renewable
sources which make our need for energy situation worse as the time passes. So we
need to search for other alternatives to generate the electricity.
The best alternatives are renewable energy sources [15] because they are renewable
and almost pollution free sources of energy. Hence solar and wind are the huge
contributors in renewables. With the scale of population we have today, we need to
use the best technologies to make sufficient energy generation. Among these systems
PV systems can be at the top of the table, which uses semiconductor materials.
Hence without argument the need of PV systems is very high today and will
increase in the near future.
Figure 2-1: Basic Grid-Interactive PV System w/o Battery Backup [16].
5
2.3
Basic Semiconductor Physics
Semiconductor materials have the conductive and electrical properties which fall
in between conductors and insulators [17, 18]. They are not considered as good
conductors or good insulators. Hence they are named as semiconductors. Initially
semiconductors have very few free electrons because of their crystalline pattern of
arrangement of atoms called as crystal lattice. But we can improve or increase the
conductivity of semiconductor by adding certain impurities to them.
We name this process of adding impurities to semiconductor materials as ‘Doping’
[17, 18]. We can control the strength of conductivity by this doping. These impurities
can result in the generation of electrons and holes based on which they are called as
donors or acceptors respectively.
Among all available semiconductor materials silicon is widely used. Silicon has
four electrons in the valence shell. Each silicon atom shares its valence electrons
with adjacent atoms because of which there are only few free electrons to conduct
electricity [19].
2.4
Types of Solar Cells
In general there are various types of solar cells based on technologies used and
absorption material used. Let us look at the broad classification of solar cells. In
broader perspective solar cells can be classified in to three types, they are:
2.4.1
Crystalline Solar Cells
Crystalline silicon solar cells can be further classified to monocrystalline and polycrystalline solar cells [20]. These two have different crystalline forms of silicon with in
them, which differs them. In present day generation these are most widely used types
6
of solar cells. Usually solar cells made of crystalline silicon are called as traditional
or conventional solar cells. In some other terms they can refer to first generation
or wafer based solar cells. Coming to the efficiencies these solar cells tops the list.
Monocrystalline have the highest lab efficiency (24.7 %) when compared to polycrystalline (20.3) and other generation solar cells [21, 22].
2.4.2
Concentrating Photovoltaic Solar Cells (CPV)
This type of solar cells uses the technology of optics such as lenses and different
type of curved mirrors to concentrate the sunlight into particular small area of PV cells
to generate electricity [23]. Its main advantage over other solar cells is it can reduce
the cost of solar cells because of the smaller area. But to make the concentration
of solar light we have to use expense mirrors and lenses which makes them hard
to maintain and costly. This makes them to have lesser usage compare to nonconcentrated solar cells. They can have very good efficiencies too, scientists are
trying to increase the usage of CPV by making them much cheaper to produce and
maintain.
2.4.3
Thin Film Solar cells (TFSC)
Thin film solar cells can be classified in to many types. Most common types are
CdTe, CIGS, CIS and Amorphous silicon. TFSC called as second generation solar
cells. They use one or more thin layers of PV material in making of cells [24, 25]. As
the name itself indicates these cells are thin, which makes them more flexible and light
weighted [26]. The production cost of thin film is also very cheap but the efficiencies
of these cells are less when compared to first generation solar cells [22].
7
Figure 2-2: Broad Classification of Solar Cells.
Figure adapted from
http://www.eai.in/ref/ae/sol/cs/typ/types of solar cells.html
2.5
Introduction to Perovskite Solar Cells
As our research is based on MAPbI3 lets know something about perovskite solar
cells. Perovskite solar cells use perovskite material as their absorber. Most commonly
used absorber is hybrid organic-inorganic lead or tin halide-based material [6, 7].
The physics beyond the perovskite materials is not fully understood now because
of the tricky lattice structures of these materials. But with the recent research and
technologies the lab efficiencies of these solar cells are increased quite significantly
making them the most debated solar material at present.
2.6
Solar cell efficiencies
Study of any solar cell efficiencies is very important. Because the importance and
usage of any solar cells predominantly depends on its efficiency. Efficiency and usage
are almost linear to each other. In addition to high efficiency if the production cost
8
and material cost of a solar cell is very less, it makes them best of all available solar
cells.
Figure 2-3: Efficiencies of Various Solar cell Technologies. Figure adapted
from NREL
We can see the efficiencies of different solar cells. Thin film solar cells are in competition with the conventional solar cells. The best efficiency of thin film cell prototype
is around 20.4 % making it comparable to the till date best known conventional solar
cell with the efficiency of 25.6% [21, 22].
9
Chapter 3
VASP
3.1
Introduction
VASP stands for Vienna Ab-initio Simulation Package [27]. It is a complex package
for performing simulations for quantum-mechanical molecular dynamics using pseudo
potentials or the projector-augmented wave method and a plane wave basis set [28].
VASP is based on the local density approximation. The variation quantity is free
energy here, with the exact evaluation of the instantaneous electronic ground state
at each time step.
Ultra-soft Vanderbilt pseudo potentials (US-PP) and Projector-augmented wave
(PAW) describes the interaction between ions and electrons [29, 30]. They also allow
for considerable reduction in number of plane-waves per atom for transition metals.
With the VASP we can calculate the forces and stress tensors which can be used to
relax the atoms into their ground states.
Solutions to the many body Schrodinger equation can be calculated by VASP
either by using density functional theory (DFT) [31, 32] or by Hartree-Fock (HF) [33]
approximation. In general VASP uses huge number of input and output files. Let
us discuss some of the most common input and output files used in almost all VASP
calculations or simulations.
10
3.2
Inpu Files
Following are the four input files which are required in almost all VASP calculations.
3.2.1
INCAR file
We can say INCAR as the central input file to VASP. It is responsible for ‘what
to do and how to do’. It contains large number of parameters most of which are by
default. In general a User who is unaware of all these parameters is not encouraged
to change them because of its complexity.
3.2.2
POTCAR file
The POTCAR file usually contains all the required pseudopotentials for each
atomic species which are used in the calculation. We need to concat the POTCAR
files if the number of species is more than one. From the version VASP 3.2 the
POTCAR file also contains the information regarding the atoms mass, their valence,
the energy of reference configuration for which pseudo potential was created and many
other features.
It also contains energy cutoff (ENMAX and ENMIN). Hence it is no need to
mention these values in INCAR file.
3.2.3
POSCAR file
The POSCAR file contains lattice geometry and the ionic positions. In some
special conditions they also have the starting velocities and predator-corrector coordinates for a MD-run. General syntax for POSCAR file is:
11
Cubic BN
3.57
0.0 0.5 0.5
0.5 0.0 0.5
0.5 0.5 0.0
1
1
In POSCAR file syntax usually first line is a comment line. Second line represents
lattice constant, which is also called as universal scaling factor used to scale all the
lattice vectors. Next three lines represent three lattice vectors which in turn define
the unit cell. The sixth line shows the number of atoms per species.
3.2.4
KPOINTS file
The KPOINTS file contains all the K-point coordinates and weights or the required
mesh size for creating the K-point grid. Two formats exist in normal.
Entering all k-points explicitly
Example file
4
Cartesian
0.0 0.0 0.0 1
0.0 0.0 0.5 1
0.0 0.5 0.5 2
0.5 0.5 0.5 3
Tetrahedra
1 0.183333333333333
6 1 2 3 4
12
The first line is a comment line. Second line provides the number of k-points.
Where as the third line need to specify whether the coordinates are in Cartesian or
reciprocal. Next three lines shows the coordinates followed by weight for each kpoints. If the tetrahedron method is not taken in to account the KPOINTS file ends
after these six lines. If in case it is used the next line should start with ‘T’ or ‘t’
which is called as control line. This should be followed by line which represents the
number of tetrahedra and the volume weight for each tetrahedron and next line with
the weight as well as four corner points of each tetrahedron.
Strings of k-points for band structure calculations:
By connecting specific points of the Brillouin zone [34], to create string of k-points
the third line of the KPOINTS file must start with an “L” for line-mode:
K-points along high symmetry lines
10 ! 10 intersections
Line-mode
cart
0.0 0.0 0.0 ! gamma
0.0 0.0 1.0 ! X
0.0 0.0 1.0 ! X
0.5 0.0 1.0 ! W
0.5 0.0 1.0 ! W
0.0 0.0 1.0 ! gamma
The coordinates of the k-points can be either in Cartesian or reciprocal (fourth line
starts with either c or r).
13
K-points along high symmetry lines
10 ! 10 intersections
Line-mode
rec
0.0 0.0
0.0 ! gamma
0.5 0.5
0.0 ! X
0.5 0.5
0.0 ! X
0.5 0.75 0.25 ! W
0.5 0.75 0.25 ! W
0.0 0.0
3.3
0.0 ! gamma
Output Files
VASP uses lot of output files. But let us discuss couple of important output files
used in VASP.
3.3.1
CONTCAR File
This file has a similar format as POSCAR. This file is usually used for continuation jobs. It contains the information regarding actual coordinates, velocities and
predictor-corrector coordinates which are needed for next MD-runs. It also contains
the positions of the last ionic step of the relaxation for relaxation runs. In the case
of un converged relaxation run we need to copy CONTCAR to POSCAR before
continuing. CONTCAR is similar to POSCAR for static calculations.
14
3.3.2
OSZICAR File
2 OSZICAR File Information regarding the convergence speed and about the
current step is written in to this file.
A typical OSZICAR file syntax is:
N
E
dE
d eps
ncg
rms
rms(c)
CG: 1 -.13238703E+04 -.132E+04 -.934E+02 56 .28E+02
CG: 2 -.13391360E+04 -.152E+02 -.982E+01 82 .54E+01
CG: 3 -.13397892E+04 -.653E+00 -.553E+00 72 .13E+01 .14E+00
N denotes the number of steps, E represents the free energy of that state, dE
stands for change in free energy, d eps shows the change in band structure energy.
15
Chapter 4
Semiclassical Model of Electron
Dynamics
4.1
Introduction of the Semiclassical Model
Semiclassical model of electron dynamics explains the motion of electron in a
lattice in presence of external field. Usually electrons in crystalline solids are assumed
in the form of Bloch wave functions. The semiclassical model deals with the dynamics
of Bloch electrons [35, 36]. Paul Drude assumed that electrons collide with the fixed
ions. This picture cannot account for very long mean free paths that had been found
in metals, as well as their temperature dependence. On the other hand, the Bloch
theory would have predicted infinite conductivity since the mean velocity of a Bloch
state, is nonvanishing. This can be traced to the fact that Bloch states are stationary
solutions to the Schrodinger equation incorporating the full crystal potential [37, 38].
So, the interaction between the electron and the fixed periodic array of ions has been
fully accounted for and the ions can no longer be sources of scattering [39].
However, in reality no solid is a perfect crystal. Furthermore, there are always
impurities, missing ions or other imperfections that are responsible for the scattering
of the electrons. In fact, it is these that limit the conductivity of metals at very low
16
temperatures. At high temperatures, we have thermally excited lattice vibrations
producing deviations from the perfect crystal structure, which can scatter electrons
and limit conductivity [40]. While the above reveals that Drude’s picture of electronion scattering is inappropriate, by substituting the scattering events by the realistic
ones his approach for formulating the electron dynamics is still valid. The formulation
presented below describes the motion of the Bloch electrons in between collisions [41].
The semiclassical model predicts, in the absence of collisions, how the position r
and wave vector k of an electron evolve in the presence of externally applied electric
andmagnetic fields, assuming knowledge of the electron’s band structure.
4.2
Filled Bands are Inert
A filled band is one in which all the energies lie below the Fermi energy. Such
bands cannot contribute to an electric or thermal current. To see this, notice that an
infinitesimal phase space volume element dk about the point k will contribute dk/4Π3
electrons per unit volume, with velocity v(k) = (1/h)(∂ε(k)/∂k) to the current [39].
Summing this over all k in the Brillouin zone, the total contribution to the electric
and energy current densities from a filled band are:
Z
j = (−e)
Z
jε =
dk 1 ∂ε
4Π3 h ∂k
dk
1 ∂ε
1
ε(k)
=
3
4Π
h ∂k
2
Z
dk 1 ∂(ε(k))2
4Π3 h ∂k
(4.1)
(4.2)
Since the integral over any primitive cell of the gradient of a periodic function must
vanish, and ε(k) is periodic both of these integrals value are zero. Therefore, only
partially filled bands need to be considered in calculating the electronic properties
of a solid [42]. This explains why Drude’s assignment to each atom of a number of
17
conduction electrons equal to its valence had been successful.
Clearly, a solid in which all bands are completely filled or empty will be an electrical insulator. Since the number of levels in each band is twice the number of primitive
cells in the crystal (due to the two degenerate spin states of electrons), all bands can
be filled or empty only in solids with an even number of electrons per primitive cell
[39].
4.3
Semiclassical Motion in Applied DC Electric
Field
The solution to the semiclassical equation of motion for k in a uniform dc electric
field is:
k(t) = k(0) −
eEt
h
(4.3)
In the similar manner the wave vector of every electron changes by same amount.
That is:
v(k(t)) = v(k(0) −
eEt
)
h
(4.4)
If the band is completely filled, this constant shift in the wave vector of all the
electrons can have no effect on the electric current. This is in contrast with the free
electron case, where v is proportional to k, and would thus grow linearly with time
[36].
Here we shall provide a detailed account for how the transport properties of electrons in some cases can be described by that of positive charges called ”holes”. The
contribution of all the electrons in a given band to the current density is by:
18
Z
j = −e
dk
v(k)
4Π3
(4.5)
The integral is over all the occupied levels in the band. But the integral of above
equation over the entire band should be zero. So, the above equation is equal to:
Z
j = +e
dk
v(k)
4Π3
(4.6)
This corresponds to the unoccupied levels of the band. It follows that the current
produced by occupying a specified set of levels with electrons is precisely the same
as the current that would be produced if the specified levels were unoccupied and all
the other levels in the band were occupied, but with particles of charge +e. Such
fictitious particles of positive charge are called holes [35, 36].
In the presence of electric and magnetic fields, the motion of electrons and holes
is given by:
1
hk = (−e)(E + v × H)
c
(4.7)
The above equation describes how the occupied orbitals evolve with time and the
unoccupied orbitals have to evolve in the same manner because a newly occupied
orbital is necessarily accompanied by a newly emptied orbital.
In classical treatment, the RHS of above equation is the force acting on a charged
particle due to E and B, and would have been set equal to the mass of that particle
multiplied with dv/dt. It is more often the case that dv/dt is directed opposite to
dk/dt when the k orbital is unoccupied. This may be perceived from the following:
At equilibrium or near equilibrium the unoccupied levels usually lie near the top of
the band. If the band energy ε(k) has its maximum value at k0 , say, then if k is
sufficiently close to k0 then we can expand ε(k) about k0 [39]. The linear term in
(k − k0 ) vanishes because k0 is a maximum point. If we assume that k0 is a point
19
with sufficiently high symmetry, then
ε(k) ≈ ε(k0 ) − A(k − k0 )2
(4.8)
Where A is positive since ε is maximum at k0 . Hence it is conventional to define
a positive quantity m* with the dimensions of mass by:
h2
=A
2m∗
(4.9)
For the energy levels with wave vectors near k0 ,
v(k) =
h(k − k0 )
1 ∂ε
≈−
h ∂k
m∗
(4.10)
d
h
v(k) = − ∗ k̇
dt
m
(4.11)
Hence we have,
a=
This shows that the acceleration a of states near the top of a band is opposite to
k̇. By substituting the acceleration wave vector relation in to equations of motion, we
find that as long as an electrons orbit is close to band maximum for the expansion to
be accurate, the electron responds to the external field as if it had a negative effective
mass -m∗ . By changing the signs we can also describe the motion of positively charged
particles with a positive effective mass m* [39].
20
Chapter 5
Density Functional Theory (DFT)
5.1
What is DFT
It is a computational quantum mechanical method used to investigate or determine
the electronic structure in particular atoms and molecules [31, 32]. Where electronic
structure can be defined as the state of motion of the electrons in electrostatic field
created by stationary nuclei. Here we use functionals (functions of another function)
to determine the properties of atoms or crystals. The usual function we consider here
is the spatially dependent electron density. Hence we call it as density functional
theory.
In most of the cases DFT results agrees with the experimental data or results.
Computational costs for DFT calculations are low when compared to other traditional
methods [43]. This is the reason why DFT is widely used in recent years.
In general nucleus is heavy and assumed to be stable (Born-Oppenheimer approximation) when compared to atom electron. On the other hand electrons are light in
weight and volatile to overlapping. Hence electronic structure is considered as quantum mechanics problem. It is very complex in nature to understand. Functional can
be defined as the formulae which can intake a function and map it to a number. This
kind of function is called as functional.
21
5.2
Fundamentals of DFT
Any kind of quantum problem associated with the matter or waves can be understood with the start of Schrodinger equations. So let us discuss some of the basics
regarding this equation.
In general we have time-dependent and time-independent Schrodinger equations,
but for the time being let us focus only on the time-independent equations [44, 45].
These equations are good at predicting the wave functions can result in standing
waves which are called as stationary states (also called as ”orbitals”). General time
independent Schrodinger equation can be written as [44, 45]:
EΨ = ĤΨ
(5.1)
Where Ψ is the wave function, E is the energy and H is the Hamiltonian operator
[45]. The above equation can be defined in words as: when the Hamiltonian operator
acts on particular wave function Ψ and the result is proportional to same wave function Ψ, thenΨ is called as stationary state and corresponding E is the energy of the
state Ψ.
The Schrodinger wave equation for a single particle moving only in electric field
is given by [46]:
−h̄2 2
)∇ + V (r) Ψ(r)
EΨ(r) = (
2m
(5.2)
In the case of many body electronic structure calculations, where nuclei is treated
as fixed quantity which create static external potential V. Then we have the timeindependent Schrodinger equation as:
h
i
ĤΨ = T̂ + V̂ + Û Ψ = EΨ
22
(5.3)
5.3
Total Energy from DFT
Figure 5-1: Algorithm for calculation of total energy from DFT. Figure
adapted from Payne et al. [47].
23
Above figure illustrates the procedure to obtain the total energy of system using
DFT method. At the first step we need to construct the potential from the given
atomic positions. Later we need to make initial guess for electron density. Then
we should calculate the effective potentials using this density functional theory. In
the following step we can calculate the Kohn-Sham equations [48, 49] using effective
potentials, which can be used to calculate new electron densities. This is how we can
obtain the total energy through sequence of steps.
5.4
Exchange Correlation Approximations
We have few problems with the DFT calculations, where we don’t know the exact
functionals for exchange and correlation. To solve this problem we have few approximations which can be used in certain property calculations. Most widely used
approximations are local-density approximation (LDA) and generalized gradient approximation (GGA). Let us discus few points regarding these two approximations.
5.4.1
Local Density Approximation (LDA)
As most of the properties of solids or systems depend on total energy calculations,
it is very essential to get accurate or very approximate values of total energy. But
we know the calculation of exchange-correlation energy is complex process. So we
have certain approximations which help us in estimating the exchange-correlation
energies. As we know from the Hohenberg-Kohn theorem every parameter including
exchange-energy is a function of electron density [50]. So the easy way of describing
the exchange-correlation energy is to adopt local density approximation (LDA). These
days this approximation is universally used for pseudopotential calculations.
In LDA approximation [50] we assume that the exchange-correlation energy of
electron at distance point r in the electron gas, εXC(r), is equal to the exchange24
correlation energy per electron in homogenous electron gas that has same density as
the electron gas at point r [47]. So we have the equation relating these terms which
is described as below.
LDA
EXC
Z
[n(r)] =
εXC (r)n(r)d3 r
(5.4)
and
δEXC [n(r)]
∂ [n(r)εXC (r)]
=
δn(r)
∂n(r)
(5.5)
εXC (r) = εhom
XC [n(r)]
(5.6)
Where,
Here LDA approximation assumes that exchange-correlation energy is totally local. It also ignores the effect of nearby inhomogeineties in electron density on energy.
LDA calculations best suit for energy of non-spin polarized system, hence global
energy minimum of energy system can be located by this scheme. However there
exists more than one minimum in case of magnetic materials. In this case in order to
perform energy calculation the simulation costs would be very high.
5.4.2
Local Spin Density Approximation (LSDA)
The local spin density approximation is almost similar to LDA approximation but
which also includes the spin of electron [50, 51].
We can express this approximation in mathematical terms as:
LSDA
EXC
Z
[n↑ , n↓ ] =
d3 rn(r)εXC (n↑ (r), n↓ (r))
(5.7)
Where εXC (n↑ , n↓ ) is the exchange-correlation energy of single particle with uni25
form spin densities. Above equation is only valid when spin densities vary slowly
over the given space. But above discussed approximations are not so good in case of
energy difference calculations. Hence we go for GGA approximations [52].
5.4.3
Generalized Gradient Approximation (GGA)
Coming to GGA approximation it is also a local approximation but it also considers the gradient factor of electron density at the same location. We can say for sure
that it accounts for inhomogeneities when compare to LDA approximations.
GGA
EXC
Z
[n↑ , n↓ ] =
εXC (n↑ , n↓ , ∇n↑ , ∇n↓ )n(r)d3 r
(5.8)
There are many other functional for GGA approximations. Here the ∇ terms refer
to the gradient terms of electron density which is unseen in previous approximations.
We can also express this functional in other form as:
GGA
EXC
[n]
Z
= −Cx
4
n 3 F (s)dr
(5.9)
Where
s=
∇n
2kF n
(5.10)
s is the measure of inhomogeneity. Where kF is the wave vector of homogenous
electron gas with density n. So these are few mostly used approximations in DFT.
26
Chapter 6
Materials and Structure
6.1
Perovskite (CaTiO3) and its Structure
General Perovskite is a calcium titanium oxide. The chemical formula of this
mineral is CaTiO3 . This mineral was first discovered in Russia in the regions of Ural
Mountains by Gustav Rose. Perovskite is named after famous Russian mineralogist
Lev Perovski [8].
Different classes of compounds which have the same crystal structure as of CaTiO3
(ABX3 ) are referred to perovskite structure materials. This particular perovskite
crystal structure was first explained by Goldschmidt during his work on tolerance
factors. Later it was published by scientist named Helen Dick Megaw.
Ideally perovskite can be represented by the simple ABX3 model, where B is the
metal cation and X an oxide or halide anion. In perovskite structure they together
form a BX6 octahedral arrangement where B is at the center of octahedra surrounded
by six nearest X atoms. Species A is also a cation which fills the space formed by
eight adjacent octahedra in three-dimensional space. In case of CaTiO3 calcium
corresponds to A species. In case of organic-inorganic metal halides this species A is
replaced by the organic cation.
27
Figure 6-1: Basic ABX3 perovskite structure showing BX6 corner sharing
octahedra [53].
Let us look at some of the common compounds which have chemical formula as
ABX3 .
6.2
Lead Titanate
Lead titanate is an inorganic compound. Its chemical formula is PbTiO3 . At very
high temperatures the structure of this compound is cubic perovskite structure. As
the temperature drops down it undergoes different phase transformations. At 760 K,
it undergoes a second order phase transition to tetragonal perovskite structure. This
tetragonal structure has the ability of ferroelectricity. Like all other lead compounds
28
it is also toxic. It mostly affects the skin with irritation and rashes.
Figure 6-2: Crystal Structure of PbTiO3 showing both cubic and tetragonal
phases.
6.3
Methylammonium Tin and Lead Halides
Solar cells are usually defined or classified based on their absorber layer. So solar
cells which use perovskite material as their absorber layer are called as perovskite
solar cells. In general mostly used perovskite materials are organic-inorganic lead
and tin halide materials. These materials are cheap in production and manufacture.
These materials have the ABX3 structure similar to perovskite material. Among all
of these existing materials mostly used material is methylammonium lead trihalide
(CH3 NH3 PbX3 ).
29
6.3.1
CH3 NH3 SnX3
We know that by decreasing temperature perovskite material undergoes phase
transitions. As the temperature falls down CH3 NH3 SnCl3 usually go through three
phase transitions. At very high temperature this material will be in cubic phase.
Around 463 K temperature we can see the phase transition from cubic to rhombohedral, around 331 K to monoclinic and around 307 K to triclinic phase [53].
From Fig. 6-3 and Fig. 6-4 we can clearly see the deformation of unit cell as the
temperature changes. This is how the transition of phases is explained. In the case
of cubic phase if we replace the organic cation with the inorganic Cs cation, we can
see slight distortion of the octahedron structure [53].
It is also observed that at high temperatures (cubic phase), the orientation of
methylammonium cation is random. As the temperature is decreased the orientation
of cation is more random. The deformation is very less compared to cubic phase.
Figure 6-3: CH3 NH3 SnCl3 in the triclinic phase. Figure adapted from [53].
30
Figure 6-4: CH3 NH3 SnI3 in cubic phase. Figure adapted from [53].
6.3.2
CH3 NH3 PbI3
In the case of methylammonium lead iodide perovskite the transition temperature
is 327.4 K for cubic to tetragonal. The tetragonal phase is transformed to orthorhombic below 162.2 K [54]. In the cubic phase of MAPbI3 organic methylammonium
cation is surrounded by eight PbI6 octahedra. But the size of octahedra formed in
this material is larger when compared to general perovskite material. As a result of
this methylammonium cation can move freely in cubic phase at high temperatures.
6.4
Perovskite-Based Solar Cells
Importance of photovoltaic (PV) cells which convert sunlight directly into electricity is increasing at greater rate in renewable world. At present about 85% of
31
PV installations use crystalline silicon and remaining belong to polycrystalline thin
film solar cells. Among thin films widely used materials are cadmium telluride and
cadmium sulfides. The main disadvantages of these materials are they are rare in
nature and few of them are toxic. But use of perovskite material as light harvesters
has rapidly reached conversion efficiencies of greater than 15% [55] in short time.
In organic-inorganic perovskite semiconductor materials CH3 NH3 PbI3 tend to have
higher charge carrier mobilities. Diffusion lengths of charges are also higher when
compared to other perovskite materials. The rate of increase in perovskite solar cell
efficeinces is shown in Fig. 6-5.
Figure 6-5: Rate of increase in perovskite solar cell efficiencies. Comparison
can be seen with amorphous Si (a-Si), dye sensitized (DSSC) and
organic (OPV)[55].
32
These perovskite solar materials are usually deposited by low temperature solution
methods. The reported diffusion lengths of charge carries (both holes and electrons)
are about 100 nm in CH3 NH3 PbI3 [55]. These values are very reasonable with the low
temperature diffusion methods. These higher diffusion lengths are one of the primary
reasons for higher quantum efficiencies. The other important characteristic of these
materials which leads to better efficiencies is its high open circuit voltage (VOC ) values.
In the case of CH3 NH3 PbI3 , VOC is near to 1 V and for hybrid CH3 NH3 Pb(I, Cl),
VOC is slightly higher than 1.1 V [56].
The higher absorption coefficient of CH3 NH3 PbI3 than normal dyes favors them to
use as sensitizers in solid state dye sensitized solar cells. Power conversion efficiency
(PCE) of 9.7 % was reported with the use of methylammonium as a light observer
deposited on titanium oxide (TiO2 ) film [57]. This particular device have also reported
the short-circuit photocurrent density (JSC ) value as 17.6 mA cm−2 , VOC value as
888 mV. Fill factor (FF) for this solar cell is 0.62 with reasonable long stability
[57]. It is also observed that the fill factor value decreases with the increase in TiO2
thickness. This reduction is because of the increase in dark current and electron
transport resistivity.
Later it was discovered that CH3 NH3 PbI3 can act as both light harvester and hole
transport material (HTM) [58]. A HTM-free solid state mesoscopic CH3 NH3 PbI3 /TiO2
heterojunction solar cell is fabricated and its performance parameters are calculated.
The calculated values are JSC = 16.1 mA cm−2 , VOC = 0.631 V, FF = 0.57 and PCE
= 5.5% at full sun with 400 nm thick TiO2 . The value of PCE is increased to 8%
with the use of 300 nm thick TiO2 film.
33
Figure 6-6: CH3 NH3 PbI3 /TiO2 heterojunction solar cell. a) Device configuration, b) Energy level diagram, c) J-V characteristics, d) IPCE.
Figure adapted from [57].
There are three main considerations that will effect these perovskite solar cells.
First is the energy conversion efficiency. But with the values ranging from 10% to
15% in the very short span of time makes this material very considerable. Second is
cost, which is more complex because of both energy and materials cost. But, present
fabrication techniques explain the low energy costs involved in the manufacture of
these solar cells. The cost and availability of materials is also in good shape when
compared to other existing thin film solar cells. The third consideration is stability,
but material being new only few studies are conducted till date. One of the study
34
shows, for sealed cell at the temperature of 45 C the efficiency is only decreased by
20% in 500 hours.
Figure 6-7: Perovskite Solar cells performance parameters. Figure adapted
from [57].
Even with the above mentioned considerations for the realization of these solar
cells, it is by no means unrealistic to expect development and production of these
perovskite solar cells. To make this happen we need further study related to various
properties of these materials. With CH3 NH3 PbI3 as primary interest we produced
certain results related to structural and electrical properties of this material, explained
in chapter 7.
35
Chapter 7
Results
7.1
Computational Method
All computations and calculations are performed by Vienna Ab initio simulation
package (VASP) [59-61], with the generalized gradient approximation (GGA) [52, 62]
and density functional theory [49, 63]. All the potentials used in the calculations are
PAW potentials [30, 64]. In general PAW potentials are more accurate than ultrasoft pseudo potentials. K-points were generated according to Monkhorst-Pack scheme
[34]. The core radii used in PAW potentials are smaller when compared to ultra-soft
potentials and these potentials can reconstruct the exact valence wave function with
all nodes in core region.
The wave functions are described by large plane-wave basis set and plane-wave
coefficients are adjusted until the ground state is obtained. Cell relaxations are done
in order to calculate equilibrium lattice parameters. This lattice constant was varied
and fit into parabolic equation as a function of energy to obtain total minimum energy.
For band structure calculations we have done self-consistent and non-self-consistent
runs at desired K points [34].
36
7.2
CH3NH3PbI3 Cubic Structure
In general CH3 NH3 PbI3 crystals exhibit cubic and tetragonal structures at room
temperatures. With the experimental values as initial guess we calculated a cubic
structure of CH3 NH3 PbI3 . We started with series of runs to get lattice parameters
and energy values for the relaxed structure. The optimized lattice parameters for this
calculated unit cell are a0 = b0 = c0 = 6.466Å. The C≡N bond of methylammonium
cation shown in Figure 7-1 is oriented along [1 1 1] direction. The total number of
3
atoms in this particular unit cell are 12. The volume of this cubic cell is 270.406 Å .
Figure 7-1: Structure of cubic CH3 NH3 PbI3 unit cell in the [1 1 1] direction.
Dark gray: lead atoms; purple: iodine atoms; brown: carbon
atom; light grey: nitrogen atom; white: hydrogen atoms
37
With the all available energy and volume values, we have used Birch-Murnaghan
[65, 66] equation of state to plot the relationship between energy and volume of unit
cell. This helps us indicating the lowest internal energy of unit cell.
9V0 B0
E(V ) = E0 +
16
(
)
2
h V 32
i3 0 h V 23
i2 h
i
3
V
0
0
0
( ) − 1 B0 + ( ) − 1 6 − 4( )
V
V
V
(7.1)
3
Total energy of cubic unit cell is -50.87 eV at the volume of 270.40 Å . Calculated
bulk modulus B0 = 10.36 GPa.
Figure 7-2: Plot between volume and energy of unit cell with BirchMurnaghan equation of state.
38
7.3
Study of C-N bond in cubic CH3NH3PbI3
Study on carbon and nitrogen bond is done in this thesis, to get the complete
picture of methyl ammonium lead iodide perovskite structural properties. To make
this happen C≡N bond is rotated in two possible ways. First C≡N is rotated around
its own axis and the angle of rotation is called as rotation angle. Second, C≡N is
rotated in the plane of bond and the angle of rotation is called as directional angle.
Figure 7-3: CH3 NH3 PbI3 cubic unit cell with C≡N bond in [0 0 1] direction.
Blue plane represents the plane of C≡N bond. Orange plane
represents plane perpendicular to bond.
39
Figure 7-4: CH3 NH3 PbI3 cubic unit cell with C≡N bond in [1 1 1] direction.
Figure 7-5: CH3 NH3 PbI3 cubic unit cell with C≡N bond in [1 1 0] direction.
40
The primary reason for this carbon and nitrogen bond analysis is to get more
relaxed cell structure with the orientation of hydrogen and iodide atoms. We can
see from the chemical formula (CH3 NH3 PbI3 ) of methyl ammonium lead iodide perovskite, methyl ammonium cation comprises of total six hydrogen atoms. Three
hydrogen atoms are bonded to carbon atom and other three to nitrogen atom. The
orientation of these hydrogen atoms with corresponding iodine atoms differ with C≡N
directional angle. The three hydrogen atoms bonded with carbon atom can be at facing or crossing with other three hydrogen atoms. Similarly they can be at facing or
crossing with iodine atoms. To have more stable and relaxed cell structure, hydrogen
atoms should be in a crossing position which leads to less repulsive forces between
hydrogen atoms. Further-more they energetically prefer a facing position with iodide
atoms which leads to more electrostatic force of attraction between cation (H + ) and
anion (I − ).
As shown from Figure 7-6, C≡N bond is rotated directionally in the plane of
bond to [1 1 -1] direction. In this particular direction hydrogen atoms are rotated
such that they are in crossing with each other and they are in phase with I − anion.
The variation in the orientation of hydrogen atoms and iodide atoms can be clearly
seen with the change in directional and rotational angles of C≡N bond. The bonds
represented here in between hydrogen and iodide atoms are most strong possible
bonds with least electrostatic energy involved, when compared to other directions of
C≡N bond.
41
Figure 7-6: CH3 NH3 PbI3 cubic unit cell in [1 1 -1] direction. Hydrogen atoms
in crossing position with other hydrogen atoms. Hydrogen atoms
in facing position with iodide atoms.
As shown in Figure 7-7, the C≡N bond is rotated directionally from [0 0 1] direction to [0 0 -1] direction. This rotation is done in the same plane where carbon and
nitrogen bond exists. This is a total of 180 degree rotation with step of 30 degree. On
the other end from Figures 7-3, 7-4, 7-5, C≡N bond is rotated around its own axis in
[0 0 1], [1 1 1] and [1 1 0] directions. This is a total of 60 degree rotation done with
the step of 15 degree. The variation of energy values in both the process is clearly
plotted in Figure 7-8.
42
Figure 7-7: Figure showing rotation of C-N from [0 0 1] to [0 0 -1] direction
in the plane of C≡N bond.
Figure 7-8: Energy variation with rotation and directional angles of C-N.
43
From Figure 7-8, when C≡N bond is in [0 0 1] direction the values of energy
doesnt change much as bond is rotated from 0 to 60 degrees. In similar manner when
C≡N bond is in [1 1 0] direction, the change in energy values with rotation is very
less and also energy values are bit higher at each rotation angle. But when C≡N
bond is in [1 1 1] direction initially the energy values are higher, but gradually goes
on decrease with increase in rotation angle. At the rotation angle of 45 degree, we
can get the least possible energy in this position. In other words, as we increase the
rotation angle the hydrogen atoms are in facing position with iodine atoms. From
same figure we can see that at 120 degree of C≡N directional angle the value of energy
is lowest. This particular angle corresponds to [1 1 -1] direction in the [0 0 1] to [0
0 -1] directional change. The rotation of C≡N dimer in [1 1 1] direction reduces the
energy values which are comparable to least energy and most stable [1 1 -1] direction.
7.4
CH3NH3PbI3 Tetragonal Structure
Tetragonal structure of CH3 NH3 PbI3 perovskite crystal is designed based on the
lattice parameters of cubic cell. The lattice parameters for tetragonal structure are
assumed as shown in Table 7.1. Calculated volumes of relaxed cell structures are
also mentioned in the table. The computational approach followed in formation of
tetragonal cell structure is similar to cubic structure. Tetragonal structure of methyl
ammonium lead iodide perovskite is shown in Figure 7-9. It can be called as 4 fold
cell structure, since its lattice vectors are derived from 4 cubic unit cells at top and
bottom. This leads to lattice vector in c axis of tetragonal cell is twice as of cubic
unit cell. We can also see the PbI6 octahedra from top view in Figure 7-9.
44
Table 7.1: Optimized lattice parameters and volume of cubic and tetragonal
unit cells.
Lattice
Structure
Parameter
(Å)
Cubic
Tetragonal
Lattice
Parameter (Å)
(Å)
Lattice
Volume
Parameter (Å)
(Å)
a0 = 6.466
√
a1 = 2a0
b0 = 6.466
√
b1 = 2b0
c0 = 6.466
= 9.145
= 9.145
= 12.933
c1 = 2 c0
3
(Å )
270.406
1081.627
Figure 7-9: Tetragonal structure of CH3 NH3 PbI3 in top view.
45
Figure 7-10: Tetragonal structure of CH3 NH3 PbI3 in side view. The two
equal lattice parameters can be seen in a-axis and b-axis. Lattice parameter in c-axis is twice that of cubic lattice. Total
of number of atoms in this unit cell are 48. Each octahedron
has eight faces formed with one lead atom and six iodine atoms.
Figure shows non-distorted view of PbI6 octahedrons as result of
which when seen from top view, top most iodine atoms superimpose with corresponding lead atoms. We can see non-distorted
or non-stretched Pb-I-Pb bond.
46
Figure 7-11: CH3 NH3 PbI3 tetragonal structure in top view showing nondistorted octahedra and non-distorted Pb-I-Pb bond.
7.5
Study of PbI6 Octahedron with CH3NH3 and
Cs Cation
In reality on the scale of super-cell or bulk we need to have most symmetric
structure rather than random orientation of atoms or cations. Because either at low
temperature the cations are randomly oriented, or at high temperature they can freely
change their orientation. Hence for this particular analysis the methyl ammonium
cation was replaced with cesium cation to simulate the bulk-level higher symmetry.
The reason for choosing cesium atom is because of its size and spherical symmetry. It’s
47
size is very similar to methyl ammonium cation. In addition to this it has very good
spherical symmetry which compensates the random orientation of methyl ammonium
cation.
Figure 7-12: Tetragonal structure with cesium cation in top view. Figure
shows distorted view of PbI6 octahedra and also distorted view
of Pb-I-Pb bond.
As per the experimental data as the temperature is increased the distortion of PbI6
is increased [54] in tetragonal phase. This is also an indicator of displacive character
of octahedron at the cubic to tetragonal transition [54]. But in our calculations there
is no temperature parameter, so to justify this distortion of octahedron a detailed
analysis is done on lattice parameters variation with angle of rotation of octahedron
(distortion angle).
48
Figure 7-13: Variation of c/a ratio with distortion angle of PbI6 octahedron.
As shown in above plot, a and c are the lattice parameters of tetragonal crystal
structure. Initially when the distortion angle is 0 degree, the value of c/a ratio
is equal to 1.414. As the distortion angle is increased the c/a ratio also increases
gradually. Distortion angle is varied from 0 to 15 degree and corresponding c/a ratio
is plotted. The red line from above figure corresponds to c/a ratio variation with
methyl ammonium cation. Blue line represents the variation in lattice parameter
ratio with cesium cation. Black line represents the experimental variation. It is clear
that, as the rotation of PbI6 octahedron is increased the c/a ratio increases. This
shows the transition of crystal from cubic structure to tetragonal. The change in c/a
ratio with CH3 NH+ 3 is smaller because of less spherical symmetry when compared
with Cs cation. This also shows that the parameter c increases monotonically and
49
parameter a decreases monotonically as the crystal transform in to tetragonal from
cubic structure.
Figure 7-14: Variation of average lattice constant (d ) with distortion angle
of octahedron.
To have complete understanding about the structure of crystal and volume of
the cell, further analysis was done on the average lattice parameter (cubic root of
the formula unit cell volume). The variation and dependence of this average lattice
constant with the distortion angle of octahedron is plotted. As the angle of rotation
of octahedron increases, the value of d decreases gradually. This represents that
the volume of the crystal cell is decreased with the increase in distortion angle in
tetragonal crystal. The blue line indicates the decrease of parameter d with cesium
atom. So this monotonically decreasing graph indicates the reduction in the volume
50
of cell in tetragonal phase with the increase in octahedron rotation. As mentioned
from above figure the total energy of crystal (Etotal ) is also decreased gradually with
the increase in distortion angle. The green line in above picture shows the reduction
in energy with Cs cation.
These plots conclude the transition of cubic to the tetragonal phase with c/a
variation as a function of octahedron distortion.
7.6
Band Structures
In general calculation of band structure is done in two steps. In the first step
we perform a self-consistent run to get the charge density. Once we have the charge
density, we run a non-self-consistent run at desired K points to get the band structure.
In this second step we use the charge density from first step.
We used DFT technique and VASP software for calculating the band structure.
The required input files for VASP are INCAR, POTCAR, KPOINTS and POSCAR
files. Once we have all these files we command VASP to run a self-consistent run.
Once this step is completed we should have CHGCAR file which contains charge
density. With this CHGCAR file we do non-self-consistent run at each desired K
point which results in energy and K point dispersion, which is band structure. In
the process of doing this second step we need to make some changes in INCAR and
KPOINTS file. The only change we do in INCAR file is we alter the ICHARG tag to
ICHARG =11. This particular change helps in reading the same CHGCAR during
the subsequent run.
In calculating the band structure of cubic MAPbI3 the following KPOINT file is
used in step one. Then we modify KPOINT file for step two.
51
The KPOINT file:
Automatic Mesh
0
Monkhorst Pack
7 7 7
0 0 0
The Modified KPOINT file:
R-G-X-M-G
20
Line-Mode
Cartesian
0.5 0.5 0.5
0.0 0.0 0.0
0.0 0.0 0.0
0.0 0.5 0.0
0.0 0.5 0.0
0.5 0.5 0.0
0.5 0.5 0.0
0.0 0.0 0.0
This modification should be made to specify the particular direction to calculate
the energy. Usually we specify high-symmetry direction to VASP in energy calculation. The line two in above KPOINT file represents the number of K points along
each high-symmetry lines. Here we considered 20 K points along each line. By specifying the line mode in above file, it commands VASP to interpolate between the
52
required K POINTS. By having these two changes we run a non-self-consistent run,
which calculates energy for each K point in between R-G, X-G and M-G.
Figure 7-15: Scheme of the Brillouin Zone for a simple cubic lattice. Where
G (0, 0, 0), R (1/2, 1/2, 1/2), X (0, 1/2, 0), M (1/2, 1/2, 0) are
respective coordinates. Figure adapted from Bilbao Crystallographic Server.
53
Figure 7-16: Band structure of cubic CH3 NH3 PbI3 .
Then we have the required output in EIGENVAL file, which has the energy values
of different bands at all the required K points. Then the band structure of cubic
structure is visualized with the help of plotting tool as shown in Fig. 7-16.
54
Figure 7-17: Scheme of Brillouin Zone for a simple tetragonal lattice. Where
G (0, 0, 0), Z (0, 0, 1/2), M (1/2, 1/2, 0), A (1/2, 1/2, 1/2), R
(0, 1/2, 1/2), X (0, 1/2, 0) are respective coordinates. Figure
adapted from Bilbao Crystallographic Server.
55
Figure 7-18: Band structure of tetragonal CH3 NH3 PbI3 .
Similarly band structure of MAPbI3 tetragonal cell is also calculated as shown
in Fig. 7-18. As discussed above we have specified 20 K points along each high
symmetry lines. We have used Brillouin zone which is the reciprocal cell of simple
tetragonal cell. The band structure calculation is also done with cesium cation as
shown in Figure 7-19. These calculations are done based on PAW-GGA. To justify
the replacement of cesium cation with methyl ammonium, we have calculated the
band structures in both cases with same lattice constants and same c/a ratio (2).
We used PAW-GGA to compare the difference in the band gaps and the difference is
indeed small. Calculated band gap is 1.68 eV in MAPbI3 [67].
56
Figure 7-19: Band structure of tetragonal CsPbI3 .
7.7
Effective Masses
In general electrons have rest mass (m0 ). But in solids, in the presence of external
electric or magnetic fields always electrons accelerate with different mass known as
effective mass (m* ) of electron. This effective mass can be positive or negative based
on the direction of movement of electron in the external field. If electrons have
negative effective mass it implies the mass of valence band holes. From the calculated
band structure we can say that there exists a conduction band energy minimum
and valence band energy maximum. And we also know that effective mass value is
57
constant for parabolic bands. Hence we used parabolic approximation to find the
effective mass (m* ) of charge carries in this material.
In general effective mass is expressed as;
∗
2
m = h̄
h ∂ 2 ε(k) i−1
(7.2)
∂k 2
Where (k) represents band edge eigenvalues and k is the wavevector. As we know
the photogenerated holes and electrons in MAPbI3 usually relax to the top of valence
band and bottom of the conduction band respectively. Effective masses of both holes
(m∗h ) and electrons (m∗e ) in CH3 NH3 PbI3 are calculated with above equation and
band structure. Based on the tetragonal band structure and existing maximum and
minimums, we have calculated effective mass of hole and electron. These calculations
are related to tetragonal band structure at gamma point. Comparing the calculated
effective masses with the other semiconductors like silicon, values are very comparable.
Effective mass of electron in silicon is estimated as 0.19m0 . Effective mass of hole vary
from 0.16m0 to 0.53m0 based on light and heavy holes respectively [68]. This makes
methyl ammonium lead iodide perovskite material as suitable material for solar cells.
Table 7.2: Effective mass of hole at the valence band maximum.
Different lines of symmetry
Effective mass
m∗
( mh0 )
Z-G
-0.320
M-G
-0.358
X-G
-0.322
58
Table 7.3: Effective mass of electron at the conduction band minimum.
Different lines of symmetry
Effective mass
∗
e
)
(m
m0
Z-G
0.140
M-G
0.139
X-G
0.213
Average Effective masses:
(
m∗h
)avg = −0.333
m0
(7.3)
m∗e
)avg = 0.164
m0
(7.4)
(
There are certain difficulties in finding effective masses in our calculations. These
difficulties are mainly because of different bands and directions. Potential flavors used
and arbitrary choice of curve fitting range. The effect of orbital angular momentum
and individual spin angular momentum (L-S coupling) is also a reason. But still our
calculations and approximations are very comparable.
59
Chapter 8
Conclusions and Future Work
An ab initio study of methyl ammonium lead iodide perovskite (CH3 NH3 PbI3 )
material has been presented. The importance of density functional theory, approximation techniques are discussed in detail. Study on structural and electrical properties of MAPbI3 is presented. Detailed analysis of both cubic and tetragonal phases
is explained. Lattice constant tests are performed to obtain relaxed cubic cell structure with optimized lattice constants. The importance of carbon and nitrogen bond
orientation is explained in cubic phase. The rotation and directional angle study of
C≡N is performed to acquire more stable cubic structure.
Tetragonal structure of MAPbI3 is designed based on the lattice parameters of
cubic structure. Replacement of methyl ammonium cation with cesium atom is performed in tetragonal structure for the symmetry. The variation of c/a ratio with
respect to PbI6 octahedron distortion is studied to understand the cubic to tetragonal transition of the material. Relationship between average lattice parameter and
distortion angle of octahedron is discussed.
Band structures are calculated for both cubic and tetragonal structures. Effective
masses of both holes and electrons are reported.
The results presented here have demonstrated the detailed analysis of CH3 NH3 PbI3
perovskite structural and electronic properties. This work maybe extended by con-
60
ducting optical property simulations of solar cells with this material used as the
absorber layer. Electrical stimulation of the entire solar cell may also be carried out
with input effective masses computed in this thesis. An alternative path would be to
relate the octahedron distortion and structural properties, to the orthorhombic phase
transition with DFT calculations. The results and work presented in this thesis can
be used as groundwork for these future works.
61
References
[1] Boehman, A.L., et al., FOSSIL AND RENEWABLE FUELS CHALLENGES
FOR THE 21st CENTURY. Prepr. Pap.-Am. Chem. Soc., Div. Fuel Chem, 2004.
49(2): p. 724.
[2] Reaver, N.G. and S.V. Khare, Imminence of peak in US coal production and overestimation of reserves. International Journal of Coal Geology, 2014. 131: p. 90-105.
[3] Hubbert, M.K., Energy from Fossil Fuels. Science, 1949. 109(2823): p. 103-9.
[4] Mackey, B. and D. Lindenmayer, Fossil fuels’ future. Science, 2014. 345(6198): p.
739-40.
[5] Lincoln, S.F., Fossil fuels in the 21st century. Ambio, 2005. 34(8): p. 621-7.
[6] Heo, J.H., et al., Efficient inorganic-organic hybrid heterojunction solar cells containing perovskite compound and polymeric hole conductors. Nature Photonics,
2013. 7(6): p. 486-491.
[7] Malinkiewicz, O., et al., Perovskite solar cells employing organic charge-transport
layers. Nature Photonics, 2014. 8(2): p. 128-132.
[8] Navrotsky, A. and D.J. Weidner, Perovskite: a structure of great interest to geophysics and materials science. Washington DC American Geophysical Union Geophysical Monograph Series, 1989. 45.
[9] Rhodes, C.J., Perovskites and their potential use in solar energy applications. Sci
Prog, 2014. 97(Pt 3): p. 279-87.
62
[10] MacKay, D.J., Solar energy in the context of energy use, energy transportation
and energy storage. Philos Trans A Math Phys Eng Sci, 2013. 371(1996): p.
20110431.
[11] Chauvin, A.S. and M. Gratzel, Solar energy harvesting. Chimia (Aarau), 2013.
67(3): p. 113.
[12] Mller-Steinhagen, H. and F. Trieb, Concentrating solar power. A review of the
technology. Ingenia Inform QR Acad Eng, 2004. 18: p. 43-50.
[13] Buresch, M., Photovoltaic energy systems: Design and installation. 1983.
[14] Green, M.A., Solar cells: operating principles, technology, and system applications. 1982.
[15] Johansson, T.B. and L. Burnham, Renewable energy: sources for fuels and electricity. 1993: Island press.
[16] Sonnenenergie, D.G.F.,Planning and installing photovoltaic systems: a guide for
installers, architects and engineers. 2007: Earthscan.
[17] Hamaguchi, C. and C. Hamaguchi, Basic semiconductor physics. Vol. 212. 2001:
Springer.
[18] Neamen, D.A. and B. Pevzner, Semiconductor physics and devices: basic principles. Vol. 3. 2003: McGraw-Hill New York.
[19] Chuang, S.L. and S.L. Chuang,Physics of optoelectronic devices. 1995.
[20] Burgelman, M., P. Nollet, and S. Degrave, Modelling polycrystalline semiconductor solar cells. Thin Solid Films, 2000. 361: p. 527-532.
63
[21] Zhao, J., et al., 19.8% efficient honeycomb textured multicrystalline and 24.4%
monocrystalline silicon solar cells. Applied Physics Letters, 1998. 73(14): p. 19911993.
[22] Green, M.A., et al., Solar cell efficiency tables (Version 45). Progress in photovoltaics: research and applications, 2015. 23(1): p. 1-9.
[23] Andreev, V.a.c.M., V.A. Grilikhes, and V.D. Rumiantsev,Photovoltaic conversion of concentrated sunlight. 1997: Wiley.
[24] Bloss, W., et al., Thin film solar cells. Progress in Photovoltaics: Research and
Applications, 1995. 3(1): p. 3-24.
[25] Chopra, K.L. and S.R. Das, Why Thin Film Solar Cells? 1983: Springer.
[26] Aberle, A.G., Thin-film solar cells. Thin Solid Films, 2009. 517(17): p. 47064710.
[27] Kokalj, A., Computer graphics and graphical user interfaces as tools in simulations of matter at the atomic scale. Computational Materials Science, 2003. 28(2):
p. 155-168.
[28] Hafner, J., Ab initio simulations of materials using VASP: Densityfunctional
theory and beyond. Journal of computational chemistry, 2008. 29(13): p. 20442078.
[29] Laasonen, K., et al., Implementation of ultrasoft pseudopotentials in ab initio
molecular dynamics. Physical Review B, 1991. 43(8): p. 6796.
[30] Blchl, P.E., Projector augmented-wave method. Physical Review B, 1994. 50(24):
p. 17953.
[31] Dreizler, R.M. and E. Engel, Density functional theory. 2011: Springer.
64
[32] Parr, R.G., Density functional theory. Annual Review of Physical Chemistry,
1983. 34(1): p. 631-656.
[33] Slater, J.C., A simplification of the Hartree-Fock method. Physical Review, 1951.
81(3): p. 385.
[34] Monkhorst, H.J. and J.D. Pack, Special points for Brillouin-zone integrations.
Physical Review B, 1976. 13(12): p. 5188.
[35] Nenciu, A. and G. Nenciu, Dynamics of Bloch electrons in external electric fields.
II. The existence of Stark-Wannier ladder resonances. Journal of Physics A: Mathematical and General, 1982. 15(10): p. 3313.
[36] Nenciu, A. and G. Nenciu, On the dynamics of Bloch electrons in constant electric
fields. Physics Letters A, 1980. 78(1): p. 101-102.
[37] Chaikin, P.M. and T.C. Lubensky, Principles of condensed matter physics. Vol.
1. 2000: Cambridge Univ Press.
[38] Ziman, J.M., Principles of the Theory of Solids. 1972: Cambridge university
press.
[39] Ashcroft, N.W. and N.D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New York, 1976). There is no corresponding record for this reference, 2005.
[40] Cahill, D.G. and R. Pohl, Lattice vibrations and heat transport in crystals and
glasses. Annual Review of Physical Chemistry, 1988. 39(1): p. 93-121.
[41] Hofstadter, D.R., Energy levels and wave functions of Bloch electrons in rational
and irrational magnetic fields. Physical review B, 1976. 14(6): p. 2239.
[42] Shklovskii, B.I. and A.L. Efros,Electronic properties of doped semiconductors.
Moscow Izdatel Nauka, 1979. 1.
65
[43] De Proft, F. and P. Geerlings, Conceptual and computational DFT in the study
of aromaticity. Chemical reviews, 2001. 101(5): p. 1451-1464.
[44] Feit, M., J. Fleck, and A. Steiger, Solution of the Schrodinger equation by a
spectral method. Journal of Computational Physics, 1982. 47(3): p. 412-433.
[45] Shirley, J.H., Solution of the Schrodinger equation with a Hamiltonian periodic
in time. Physical Review, 1965. 138(4B): p. B979.
[46] Wannier, G.H., Wave functions and effective Hamiltonian for Bloch electrons in
an electric field. Physical Review, 1960. 117(2): p. 432.
[47] Payne, M.C., et al., Iterative minimization techniques for ab initio total-energy
calculations: molecular dynamics and conjugate gradients. Reviews of Modern
Physics, 1992. 64(4): p. 1045.
[48] Seidl, A., et al., Generalized Kohn-Sham schemes and the band-gap problem.
Physical Review B, 1996. 53(7): p. 3764.
[49] Hohenberg, P. and W. Kohn, Inhomogeneous electron gas. Physical review, 1964.
136(3B): p. B864.
[50] Dahl, J.P. and J. Avery, Local density approximations in quantum chemistry and
solid state physics. 1986, Plenum Press, New York, NY.
[51] Perdew, J.P., Density-functional approximation for the correlation energy of the
inhomogeneous electron gas. Physical Review B, 1986. 33(12): p. 8822.
[52] Perdew, J.P., K. Burke, and M. Ernzerhof, Generalized gradient approximation
made simple. Physical review letters, 1996. 77(18): p. 3865.
[53] Borriello, I., G. Cantele, and D. Ninno, Ab initio investigation of hybrid organicinorganic perovskites based on tin halides. Physical Review B, 2008. 77(23): p.
235214.
66
[54] Kawamura, Y., H. Mashiyama, and K. Hasebe,Structural study on cubicTetragonal transition of CH3 NH3 PbI3 . Journal of the Physical Society of Japan, 2002.
71(7): p. 1694-1697.
[55] Hodes, G., Perovskite-based solar cells. Science, 2013. 342(6156): p. 317-318.
[56] Lee, M.M., et al., Efficient hybrid solar cells based on meso-superstructured
organometal halide perovskites. Science, 2012. 338(6107): p. 643-647.
[57] Kazim, S., et al., Perovskite as light harvester: a game changer in photovoltaics.
Angewandte Chemie International Edition, 2014. 53(11): p. 2812-2824.
[58] Etgar, L., et al., Mesoscopic CH3 NH3 PbI3 /TiO2 heterojunction solar cells. Journal of the American Chemical Society, 2012. 134(42): p. 17396-17399.
[59] Kresse, G. and J. Hafner, Ab initio molecular dynamics for liquid metals. Phys
Rev B Condens Matter, 1993. 47(1): p. 558-561.
[60] Kresse, G. and J. Furthmller, Efficiency of ab-initio total energy calculations for
metals and semiconductors using a plane-wave basis set. Computational Materials
Science, 1996. 6(1): p. 15-50.
[61] Kresse, G. and J. Furthmuller,Efficient iterative schemes for ab initio totalenergy calculations using a plane-wave basis set. Phys Rev B Condens Matter,
1996. 54(16): p. 11169-11186.
[62] Perdew, J.P., et al., Applications of the generalized gradient approximation for
exchange and correlation. Phys Rev B, 1992. 46: p. 6671.
[63] Kohn, W. and L.J. Sham, Self-consistent equations including exchange and correlation effects. Physical Review, 1965. 140(4A): p. A1133.
67
[64] Kresse, G. and D. Joubert, From ultrasoft pseudopotentials to the projector
augmented-wave method. Physical Review B, 1999. 59(3): p. 1758.
[65] Birch, F., Finite elastic strain of cubic crystals. Physical Review, 1947. 71(11):
p. 809.
[66] Birch, F., Finite strain isotherm and velocities for singlecrystal and polycrystalline NaCl at high pressures and 300 K. Journal of Geophysical Research: Solid
Earth (19782012), 1978. 83(B3): p. 1257-1268.
[67] Umari, P., E. Mosconi, and F. De Angelis, Relativistic GW calculations on
CH3 NH3 PbI3 and CH3 NH3 SnI3 perovskites for solar cell applications. Scientific
reports, 2014. 4.
[68] Giorgi, G., et al., Small photocarrier effective masses featuring ambipolar transport in methylammonium lead iodide perovskite: A density functional analysis.
The Journal of Physical Chemistry Letters, 2013. 4(24): p. 4213-4216.
68