Rational Design of Hybrid Organic Solar Cells
by
Levi Lentz
B.S., Mechanical Engineering, San Diego State University (2012)
Submitted to the Department of Mechanical Engineering
I
MLr
MASSACHUPETFE:
0OF TECC
INSITTUTE
7
'
in partial fulfillment of the requirements for the degree
Master of Science in Mechanical Engineering
AUG 15 2014
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGYJune 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
Signature redacted
Department of Mechanical Engineering
May 9, 2014
Certified by....
Signature redacted ...............
Alexie Kolpak
Assistant Professor
Thesis Supervisor
Signature
redacted
Accepted by....
......... ....
low,
David E. Hardt
Chairman, Department Committee on Graduate Theses
2
Rational Design of Hybrid Organic Solar Cells
by
Levi Lentz
Submitted to the Department of Mechanical Engineering
on May 9, 2014, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
In this thesis, we will present a novel design for a nano-structured organic-inorganic
hybrid photovoltaic material that will address current challenges in bulk heterojunction (BHJ) organic-based solar cell materials. Utilizing first principles Density Functional Theory (DFT), we show that layered inorganic phosphates and tradition organic dyes can be combined to form a new class of bulk heterojunction photovoltaic
with high electron and hole mobilities with low exciton recombination, potentially
enabling very high efficiency with existing organic-based solar-cell molecules. We will
discuss the physical origin of these properties and investigate several approaches for
engineering the electronic structure of these materials. By using these methods, it
will be possible to engineer the transport and optical properties of these materials,
with potential applications beyond photovoltaics in areas from organic electronics to
photoactuators.
Thesis Supervisor: Alexie Kolpak
Title: Assistant Professor
3
4
Acknowledgments
Completion of a work such as a Master Thesis requires support from a wide variety
of individuals, both personal and professional.
In completion of this work, I have
learned a great deal about computational physics and chemistry, something foreign
to me at the start and more akin to a distant relative now. I would like to thank the
people who have helped me immensely along that journey.
Primarily, I would like to thank my adviser, Alexie Kolpak, for having the faith
to bring me onto her lab. My background was not in the field of research that I now
perform, yet she believed that I would be able to complete the work presented herein.
Without her faith and, many times, patience, I would not have had the opportunity
to excel at MIT the way I have.
My bearded friend, Brian Kolb. Hired on as a Post-Doctoral Researcher in our lab,
he was immesly helpful in guiding me along in the important aspects of computational
physics. From providing resources, to writing up tutorials, to just being the person
to bounce ideas off of, he created a lab environment inductive to groundbreaking
research. I wish him luck in his future career work.
My family, especially my brother Dillon, my father Kirk, and my step mother
Judy. Living far away from home can be trying on the soul and a core support group
was necessary to stymie the creeping thoughts of self-doubt. With out them, this
work would have taken significantly longer.
My MIT friends, too numerous to recount. They have been there for me to lament
research ills, solve psets, study for the qualifying exams, and enjoy a lazy afternoon
at the Boston Common. As we go forth from here to the corners of the world, I will
not soon forget the times and friendships we shared here at MIT.
The Show, and all my friends from undergrad. A personality is not singularly
formed and a great deal of mine came from the friendships that I formed at all
the sporting events attended with the wonderful people I love at San Diego State
University.
5
6
Contents
17
1.1
Solar Energy. . . . . . . . . . . . .
18
1.1.1
Solar-Thermal . . . . . . . .
19
1.1.2
Photovoltaics . . . . . . . .
20
.
.
.
Introduction
23
. . . . . . . . . . . . . .
26
Theory and Computational Principles
27
27
2.2
Density Functional Theory . .
29
2.3
Functionals
31
2.4
Pseudopotential Approximation
33
2.5
Electronic Properties from p[r]
34
2.6
DFT Bandgap Issue
. . . . .
36
2.7
DFT Code . . . . . . . . . . .
38
2.8
DFT Scaling . . . . . . . . . .
38
2.9
Conclusion . . . . . . . . . . .
39
.
.
Computational Tools . . . . .
.
. . . . . . . . . .
.
.
41
3.1
Overall Design
. . . . . . . . . .
41
3.2
M aterials
. . . . . . . . . . . . .
43
3.2.1
Organic Semiconductor . .
43
3.2.2
Inorganic Phosphate Group
46
.
.
Rational Design of Nano-structured Hybrid Photovoltaics
.
3
2.1
.
2
Objective
.
1.3
.
. . . . . . .
.
1.2 *Bulk-Heterojunctions
.
1
7
Exciton Separation . .
49
3.4
Mobility . . . . . . . .
52
3.5
Open Circuit Voltage .
52
3.6
Conclusion . . . . . . .
53
4 Atomic and Electronic Structure
55
.
.
.
.
3.3
55
4.2
Pseudopotential Testing . . . . .
57
4.3
Computational Results . . . . . .
61
.
.
.
Computational Details . . . . . .
..
Bulk Zr(HPO 3 )2
4.3.2
Bulk a - Ti(HP04) 2 - H20
62
4.3.3
Titanium Doping of Zr(HPO.:
63
4.3.4
Layered Organic Zr(HPO 3 ) 2
67
4.3.5
Layered Zr(HPO 3 )2 with Org
Photovoltaic
. . . . . . . . .
Scissor Operator
4.5
Organic MK2 Functionalization
.
4.4
.
71
73
. . . . . . . . . . . . . . . . . . . . . . .
Conclusion . .
68
72
.
Functionalized MK2
.
4.6
61
... .
4.3.1
4.5.1
76
77
5 Engineered Band Alignment
77
Functional Group Band Shifts
5.2
Titanium Doping Band Alignment
79
5.3
Corrected Band Alignment . . . .
81
5.4
Conclusions . . . . . . . . .. . . .
84
.
.
.
5.1
87
Transport Properties
87
6.1.1
Governing Equations . . .
88
6.1.2
DFT Calculations . . . . .
90
.
.
Mobility Derivation . . . . . . . .
.
6.1
Zr(HPO 3 ) 2 Mobility
. . . . . . .
91
6.3
Conclusion . . . . . . . . . . . . .
97
.
6.2
.
6
. . . . . . . . . . . . . . . . . . .
4.1
8
7
Conclusions
99
7.1
Efficiency Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
7.2
Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
A Molecule Figures
105
9
10
List of Figures
1-1
Solar spectrum at 1.5 air mass . . . . . . . . . . . . . . . . . . . . . .
18
1-2
Efficiency vs band gap . . . . . . . . . . . . . . . . . . . . . . . . . .
22
1-3
Schematic of the working of a BHJ material . . . . . . . . . . . . . .
25
1-4
Schematic of types of BHJ materials. Left, low recombination. Right,
high recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2-1
Relative computational cost of various methods. . . . . . . . . . . . .
28
2-2
Speedup associated with number of processors . . . . . . . . . . . . .
39
3-1
Schematic representing the proposed design. Alternating layers of organic and inorganic layers allow for efficient charge extraction
. . . .
42
. . . . . .
43
3-2
Schematic showing ideal band alignment in bulk structure
3-3
The MK2 molecule (2-Cyano-3-[5-(9-ethyl-9H-carbazol-3-yl)-3,3,3,4-tetran-hexyl-[2,2,5,2,5,2]-quater thiophen-5-yl] acrylic)
. . . . . . . . . . .
45
3-4 Different types of organic semiconductors studied . . . . . . . . . . .
45
ce-Zr(HPO 3 )2
3-5
Structure of
3-6
Structure of Zr(HPO 3 )2 layered with benzene rings
. . . . . . . . . .
48
3-7
Schematic of dipole separation . . . . . . . . . . . . . . . . . . . . . .
51
3-8
Exciton separation in an ordered nano-structured material with an
induced dipole
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
51
3-9
VOC in BHJ materials.
4-1
K-point convergence of Bulk Zr(HPO 3 )2
. . . . . . . . . . . . . . . .
57
4-2
Atom-Resolved Density of States for a - Zr(HPO 3 )2 . . . . . . . . . .
62
. . . . . . . . . . . . . . . . . . . . . . . . .
11
54
4-3
Comparison between hydrated and unhydrated Ti(HPO 4 )2 . . . . . .
64
4-4
Titanium doping of Zr(HPO 3 )2 . . . . . . . . . . . . . . . . . . . . .
66
4-5
Atom resolved density of states, spatially separated into each layer . .
68
4-6
Computational structure of Zr(HPO 3 ) 2 bound to organic MK2 . . . .
69
4-7
Spatially resolved DOS of Zr(HPO 3 )2 -MK2. . . . . . . . . . . . . . .
70
4-8
Schematic showing energetic shift due to polarized MK2
. . . . . . .
70
4-9
Computational modifications of MK2 molecule . . . . . . . . . . . . .
73
. . . . .
74
4-11 Schematic showing functionalization of the organic MK2 molecule . .
75
. . . .
80
4-10 Windowed-averaged electrostatic potential of modified MK2
5-1
Spatial resolution of the Valence and Conduction Band Edges
5-2
Structural form of Titanium Doped Zirconium Phosphate
. . . . . .
81
5-3
Titanium Doping of Zirconium Phosphate Layer . . . . . . . . . . . .
82
5-4
Scissor operator application to the bulk band structure . . . . . . . .
82
6-1
Directions of reciprocal space band structure
. . . . . . . . . . . . .
93
6-2
Band Diagram of Zr(HPO 3 )2 along reciprocal axes
. . . . . . . . . .
93
6-3
Plot of curve required to fit the elastic constant . . . . . . . . . . . .
94
6-4
Plot of curves required to calculate the deformation potential. ....
95
6-5
Effective Mass vs. Layers of Zr(HPO 3 ) 2
. . . . . . . . . . . . . . . .
96
7-1
Schematic representing the idea of the area fraction . . . . . . . . . .
100
7-2
Schematic for electron (hole) tunneling . . . . . . . . . . . . . . . . .
101
A-1 MK2 with Benzene interlinker . . . . . . . . . . . . . . . . . . . . . .
106
A-2 Chemical structures of MK2- Pyridine and Pyrimidine
107
. . . . . . . .
A-3 Chemical structures of MK2- Pyridine and Pyrimidine functionalized
with an OH group
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
108
A-4 Chemical structures of MK2- Pyridine and Pyrimidine functionalized
with two OH groups
. . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A-5 Chemical structures of Boron and Flourine functionalized MK2 . . . .
110
A-6 MK2 with both Fluorine and Boron . . . . . . . . . . . . . . . . . . . 111
12
A-7 BX shown with Benzene and Pyrimidine interlinkers . . . . . . . . . . 111
A-8 PX shown with Benzene and Pyrimidine interlinkers .... . . . . . . .
112
A-9 TPD shown with Benzene and Pyrimidine interlinkers . . . . . . . . .
112
13
14
3.2
Thiophene bandgap and chain length [5]
4.1
4.2
Pseudopoentials used in this study
.
58
4.3
Bond distance comparison for STP gasses
58
4.4
Bond distances for carbon, sulfur, hydrogen, and nitrogen pseudopo-
.
Thiophene property modifications [30]
. .
46
.
3.1
.
List of Tables
.
46
Zr(HPO 3 ) 2 experimental unit cell parameters
56
. . . .
59
4.5
Bulk zirconium lattice comparison
. . . .
59
4.6
ZrO 2 unit cell computational error
. . . .
59
4.7
Lattice constant error of Zr(HPO 3 ) 2
4.8
Experimental Structure of a - Ti(HPO 4 ) 2 - H 2 0
4.9
Similarities in Zr(HPO 3 ) 2 and Ti(HPO 4 )2 bond lengths in their bulk
.
.
.
tentials . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . .
.
.
. . . . . . . . . . . . . . . . .
60
63
65
4.10 Bangap Modification of Titanium doped Zirconium Phosphate . . .
4.11 Comparison between bulk structure and constitutive molecules.
.
.
69
4.12 Computational Optimization of MK2. Energies relative to Fermi Level
73
4.13 Voltage drop of functionalization
. . . . . . . . . . . . . . . . . . .
75
5.1
Comparison between bulk structure and constitutive molecules . . .
78
5.2
Corrected bandgap of system by applying two scissor operators . . .
83
5.3
Corrected bandgap of system by applying two scissor operators . . .
84
6.1
Electron and hole transport properties of Zr(HPO 3 ) 2 . . . . . . . .
95
.
.
.
.
.
.
65
.
.
structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
6.2
Comparison of mobilities between common semiconductors . . . . . .
97
7.1
Max efficiencies of select systems examined . . . . . . . . . . . . . . .
102
16
Chapter 1
Introduction
Lower carbon emissions and a drive for clean and long-term sustainable energy generation is driving the development of renewable resources. Solar energy has the potential
to replace the majority of conventionally generated electricity. In the United States
alone, the National Renewable Resource Laboratory (NREL) estimates that the total potential in the United States is 155GW [35]. The foremost way this energy is
captured is via silicon-based photovoltaic devices. Limited by the cost of manufacture, only recently have silicon photovoltaic materials become cost competitive with
conventional grid-level coal or natural gas-based electricity generation.
Organic-based photovoltaics are a promising alternative to silicon for the conversion of solar radiation into electrical power. These materials offer several advantages
over traditional inorganic-based materials. In particular, organic photovoltaics can
potentially be manufactured at scale using solution-based processing, which is significantly lower in cost [4] and more environmentally friendly than current semiconductor
manufacturing techniques. In addition, organic-based materials could enable flexible
photovoltaics, opening up possibilities for novel applications.
In order to realize these possibilities at scale, however, a number of challenges must
be overcome to improve the efficiency of organic-based photovoltaics. In the remainder
of this chapter, we first discuss these challenges in the context of traditional, organic,
and hybrid photovoltaics (PV). We then review the state-of-the-art in organic-based
bulk heterojunction (BHJ) solar cells, and conclude with an outline of the main
17
Ground-Level Reference Spectra
-
ASTM G173-03 Reference Spectra
-
Blackbody at 5777K
15-
0.5--
0
0
1000
2000
3000
4000
5000
6000
Wavelength [nm]
Figure 1-1: Solar spectrum at 1.5 air mass
objectives of this work, in which we use first-principles computations to design a
novel class of nano-structured organic-based photovoltaic materials to overcome these
challenges.
1.1
Solar Energy
The solar spectrum illuminating the earth's surface provides an abundant amount of
energy to supply nature's energetic needs with enough to spare to provide for the
majority of humanity's endeavors. The energy that we receive from the sun comes
in quantized packets called photons, following the Plank's distribution as if the sun
emits as a black body. Because of the large distance between the earth and the sun,
it can be estimated to emit at approximately 6000 C. NREL provides comprehensive
measurements of the incident ground irradiation, shown in Figure 1-1. Overlaid is
the exact solution to a black body with a solid angle of 6.87 x 10-
steradians, the
average solid angle of the sun-earth distance.
In Figure 1-1, the solar spectrum was plotted at what is commonly referred to as
18
"air mass 1.5," or AM1.5. The definition of airmass is given as
AM =
,
(1.1)
Lo)
where L is the length the light traveled through the atmosphere, and Lo is the length
that light would travel, normal to the earth's surface at sea level. In this way, the
length that light will have to travel through the atmosphere can be quantified. This
is important because the longer that light is traveling through the atmosphere, the
more scattering that it will experience, and hence the more the light will deviate from
the perfect black body. AM1.5 is commonly used because it is the distance that light
will have to travel in most populated areas. This is an industry standard and is used
for the design of many solar devices.
By integrating both curves represented in Figure 1-1, the black body spectrum
yields a incident radiation of 1367 W/m2 while the ground-level spectrum yields a
incident radiation of 872 W/m 2 . This loss in solar potential is due primarily to
atmospheric interference that interacts with the solar spectrum.
In this way, the
sun deviates from being a perfect black body while viewed from the surface of the
earth. By concentrating this spectrum, heat can be generated in a solar-thermal
collection process to use for low-grade hot water generation or for high-grade steam
generation. Alternatively, the light can be extracted directly from the photoelectric
effect to directly provide energy in the form of electricity. Both of these processes
will be described in the following sections.
1.1.1
Solar-Thermal
One way to capture the sun's power is by capturing the light in the form of heat, which
can be used for a variety of applications from rooftop water heaters to grid-level energy
plants generating megawatts of electricity from high-level flux concentrators. This
thesis will not be studying the applications of solar-thermal applications; however,
knowledge of this area of solar applications warrants a brief overview.
Solar thermal acutally provides the largest amount of energy generation from
19
solar in the world in the form of solar heat. This energy is traditionally harnesses
in the form of solar water heaters, although it is rarely seen in the United States,
accounting for only 0.4% of installed water heaters [21]. Israel, the world's larges
producer of solar hot water per capita, produces many times amount, significantly
offsetting the overall energy usage [21]. Because of this, solar thermal water heating
could significantly offset the electrical/gas uses by homes around the US to warm
their houses.
The other form of solar thermal is obtaining high-grade heat by concentrating the
solar flux and using it to generate steam. In general, solar concentrators are capable
of generating a concentration factor of tens to hundresds of suns (equivalent to a 10100x increase in the solar flux [~-. 104- 10 5W/m2 ]). This allows for a higher amount
of energy density per area, creating the possibility of high-grade heat necessary for
electrical generation.
Normally these systems have to be grid-scale and centrally
located due to the fact that the energy then takes a 30-50% loss in the thermalelectric conversion process. In addition, large amounts of water and land are required
to make even small thermal-electric power plants.
1.1.2
Photovoltaics
Solar cells rely on the principle of the photovoltaic effect, described originally by
Einstein in the Annus Mirabilis papers [10]. This effects relies on the particle-wave
duality-of light, allowing energetic photons to impart their energy to electrons, generating charge carriers and providing a photocurrent that can be harvested as electricity.
These devices offer a very unique way to generate electricity as they can be modular
in nature, providing the opportunity to be small in form factor.
Utilizing both the photoelectric effect as well as band theory, inorganic semiconductors can efficiently extract electricity from the solar spectrum. Conventional
inorganic solar cells contain a p-n junction in the material, creating a built-in electric
field that drives the charged carriers towards their respective electrodes. By utilizing
the photoelectric effect, light creates a bound hole-electron pair, a quasiparticle is
traditionally called an exciton, and the p-n junction is able to overcome coulumbic
20
interaction between the hole and electron pair, separating the charge and driving the
photocurrent.
This photoelectric effect can be observed in both metals as well as semiconductors.
In the case of metals, the excited charges are simply used in the form of heat, whereas
in semiconductors, this charge is in the form of useable electricity. This is because
semiconductors have a built-in bandgap to the material, or a range of energies that
electrons, or holes, cannot occupy. Bandgaps are composed of both a conduction
band where excited electrons reside, and a valence band, where the holes reside.
What this physically means is that a material that has a perfect absorptivity can
absorb all photons with an energy greater than the bandgap of the material. When
incident light has an energy greater than the bandgap, it photoexcites an electron
to an energy state greater than the conduction band edge of the material. It then
radiatively loses energy, in the form of heat, until it enters the conduction band of
the material. Because of this radiative emission, there is actually a loss from the
photon to electron that implies that there is an optimal bandgap associated with the
material. Any photon that has a lower energy than the bandgap of the materials is
not absorbed, a further loss in the system.
For the purpose of photovoltaics, there exists two types of semiconductors: direct
gap and indirect gap. This refers to the nature of the gap; a direct gap semi conductor
has the conduction band minimum and valence band maximum at the same position in
reciprocal space where as in a indirect gap semiconductor, there is a spatial separation
of these two points. Direct gap semiconductors are materials traditionally used in
lasers as well as photodiodes and have many uses as photovoltaics. This is desirable
because the transition between the CBM and VBM is only energetic in nature. In
an indirect gap semiconductor, momentum must be conserved, and in order for an
electron to transition between the VBM and CBM, momentum must be imparted
from the crystal into the electron, physically meaning that the material must be
thicker to impart momentum to the system.
While silicon is an indirect bandgap absorber, preventing it from being used as
a thin film absorber, its abundance in nature as well as the industry know-how to
21
0.4
O0.301
0O.2 -1
0O.1-
0
0
0.5
I
lI
1
I
1.5
I
2
2.5
3
EEgap
gp(eV)
Figure 1-2: Efficiency vs band gap
process it from traditional electronic applications have propelled it to the forefront of
the solar energy industry. Additionally, Silicon has a bandgap of ~1.1eV, occupying
an almost ideal spot in the Shockley-Queisser limit, the efficiency limit of a singlejunction p-n junction [7]. The limit of efficiency vs bandgap is show in Figure 1-2,
calculated following this limit.
With all of these advantages in mind, silicon-based solar cells have become the
main material used in solar energy capture. However, significant processing of the
material is necessary as amorphous silicon suffers from severly degregaded efficiency
over crystalline materials. Because of this, other alternatives that decrease the processing requirements as well as the cost per watt of electrical generation are of great
interest.
Even with these disadvantages, the solar-cell design offers many unique opportunities for solar-energy capture in the form of distributed electrical generation. In
this way, modular units based off of photovoltaic cells can be used on rooftoops of
homes or businesses to provide energy in a non-centralized way. This offers freedom
22
for both the individual as well as the corporation to create small-scale installations
in residential areas or to create grid-scale solar capture plants based off of effecient
solar cell devices. Consequently, a large amount of research is being done in both
inorganic and organic materials.
1.2
Bulk-Heterojunctions
One alternative to silicon and other inorganic photovoltaics is in the realm of organic
semiconductors. Rather than being comprised of a single material doped for either por n-type carriers, two different materials are joined together, using energetics of the
band alignments at their interfaces to drive exciton separation. This type of composite
photovoltaic is known as a bulk-heterojunction (BHJ). BHJs are normally organic in
nature, but hybrid composites comprised of both organic and inorganic materials also
exist. BHJs, in general, have a lower overall efficiency compared to single materials,
but their low cost of manufacture as well as low environmental impact drives research
into this area of organic-based PV materials.
The material considerations that limit the efficiencies of BHJs are primarily low
carrier mobility and high exciton binding energy. The mobility is an indication of how
fast charge carriers (electrons and holes) can move within the material. The lower the
mobility, the slower charge carriers can move, thereby increasing the probability of
recombination. The binding energy of the exciton is the the energy required to separate the bound hole-electron pair. This is a problem in all semiconductors, however
in inorganic materials, the binding energy is of the order of the thermal energy at
room temperature, while in organic materials, exciton binding energies much higher,
on the order of 1.0eV. As a result, organic materials are observed to have much higher
recombination of the electron and holes in the material. The exact nature of exciton
binding energies will be described in Section 3.3.
Recombination is simply when a bound hole-electron pair recombine or when a
free charge carrier meets another charge carrier of the opposite charge. In both of
these scenarios, that charge is then lost, leading to a decrease in the overall efficiency
23
of the system via loss of the photo-excited charge carrier. To overcome the issue
of the large binding energy of the exciton, this design of the heterojunction was
born. This design comes from the need to have an energetic difference large enough
to separate the two bound charges. To this end, two organic materials are joined
together that have a energetic difference between their conduction band edges that is
large enough to encourage separation of the exciton. The main issue with this has to
do with decreasing the effective bandgap of the material, thereby lowering the open
circuit voltage of the material (Voc ). This is due to the fact that the efficiency in
photovoltaics is defined as:
FF x Vc x Isc
E xAc
ExA~
(1.2)
where FF is the fill factor, an indication of how much power is utilized over ideal
condictions; VOC is the open circuit voltage; ISc is the short circuit current, the max
current that can go through the device; E x A, is simply the amount of energy per area
time the total cross-sectional area (the total energy incident on the material). From
this, it is apparent that controlling the VOC is extremely important to the efficiency of
the material. To this end, the two disperate materials are carefully chosen in the BHJ
to ensure that the exiton is properly separated without lowering the Voc significantly.
This process is schematically shown in Figure 1-3.
The other limiting factor of most bulk heterojunctions is the mobility issue. In
these materials, the mobility is of the order 10- 4cm 2 /Vs [24], whereas traditional
semiconductors have mobilities of 500 - 1000cm 2/Vs [49]. The extremely low mobility
in organic materials contributes to the recombination of charge due to the long amount
of time required for the charge carriers to travel to their respective electrodes. If the
electron and holes could be guaranteed to never interact with each other, this would
not be extremely detrimental. However, minimization of such interactions implies
that the BHJ would have to be highly ordered. BHJs are generally solution-processed,
leading to high degree of disorder in the material to maximize the surface area between
the two materials, represented in Figure 1-4.
In Figure 1-4a, a highly ordered BHJ material can be seen. This design takes
24
Exciton Separation
ductio' -..
ncin
Banj
1
.and
-*--
-----..
rt
OF
=Ce
alenceBn
..-------- Donor
9
Acceptor
Figure 1-3: Schematic of the working of a BHJ material
Anode
Anode
Cathode
Cathode
(a) Highly ordered BHJ schematic
(b) Conventional BHJ schematic
Figure 1-4: Schematic of types of BHJ materials. Left, low recombination. Right,
high recombination
25
significant processing that leads to a lower amount of recombination.
Conversely,
Figure 1-4b shows how a real BHJ material would be made [46]. This design, while
significantly easier to make, is more inducive to higher recombination. A material
that could have the structure of Figure 1-4a while have the ease of processing as
Figure 1-4b would combine the best of both material structures.
Because of the intrinsic value of organic BHJ materials, a significant amount of
work is being conducted to address the limiting factors preventing wide-scale acceptance of these materials. If these material issues can be addressed, cheap and
environmentally friendly organic solar cells could become a reality.
1.3
Objective
In this work, we will address the issues with traditional bulk heterojunction designs.
These materials, because of their low cost of manufacture offer a unique opportunity
for large scale solar capture the world over. By using first principle calculations, new
materials will be explored, modified, and designed in such a way that the issues of
recombination and low Voc can be alleviated or fixed.
The remainder of this thesis is organized as follows: computational methods used
in this work will be described from their mathematical foundations, the proposed
material design that will address the issues with traditional BHJ will be described,
the results of this study will be presented, concluded with discussion of the results
as well as future design and modification of the proposed design. In this way, this
work will provide a framework for highly efficient BHJ materials capable of producing
cheap, and efficient, solar power.
26
Chapter 2
Theory and Computational
Principles
As we push our technology to higher and higher levels of sophistication, it becomes increasingly important to create materials that serve very specific applications, allowing
novel technologies to exist in ways never thought possible. In this drive, experimental
advancement have allowed for more accurate measurement of how materials behave
in real applications. However, this drive has been matched by advancements in computing power, allowing theorist to contribute to the design and prediction of highly
advanced materials.
The realm of computational material design allows contribution to several material
science areas including material property prediction, explanation of observed physical
behavior, and effient prediction of new material properties. By accurately solving
the many-body Schr6dinger equation, many material properties can be calculated
accurately and quickly.
2.1
Computational Tools
Computational modeling is a tool to understand how systems physically behave much
like any number of physical tools are used to measure systems properties. Whereas
in physical systems you have to balance cost of measurement with accuracy, in com27
Low cost/constituent, large
length scales, long times
Cn
+--
~Coarse-
Continuum
mdeling
m
lengt hw ies n luna Empiricalmehd
potentials
Tigh
0
Many ikexcoted
Denitybinding
functional
theory
Decreasing
accuracy
Figure 2-1: Relative computational cost of various methods.
putational methods you have to balance the cost of computation with the accuracy
of the modeling. In this way, there exists a hierarchy of the methods available. Some
of these methods are shown in Figure 2-1.
In this work, it is necessary to compute the atomic structure and electronic properties of materials.
Numerous methods exist to model these properties, including
molecular dynamics, density functional theory (DFT), Hartree Fock methods, and
Beyond-DFT. Each of these has their own level of electronic accuracy possible to
them.
Material properties such as melting temperature, diffusion constants, and
other molecular properties related to atomic motion can be modeled with MD methods.
However, electronic properties such as band alignment and carrier transport
require explicit quantum mechanical treatment of the electrons. As such, DFT and
more advanced methods are required to model the electronic interactions.
As computational resources have grown exponentially in recent years, methods
for carrying out the solution to the exasperative Shr6dinger equation have been developed. One of the primary tools that enables solution of this equation is DFT. Al-
though exact in principle, in practice DFT relies on several important approximations,
which will be described in Section 2.2. More accurate methods have been developed
28
such as Beyond-DFT methods that do not rely on these assumptions. However, these
methods are extremely computationally expensive, and would be near impossible to
use on the systems considered in this work. Considering the time fram available to
this work, traditional DFT has been employed to compute and measure the electronic
properties of new BHJ designs.
DFT can accurately model many different system properties. By solving for the
electron density of the system, many system properties can be calculated. These
properties include ground-state atomic structure, band-strcuture, local and atomresolved density of states, and other properties necessary to analyze a PV material.
2.2
Density Functional Theory
In general, the Schr6dinger eduation is a relationship describing the total energy of a
quantum system. Originally pushed in his seminal paper "Quantisierung als Eigenwertproblem" [47], this equation has opened the realm of describing the subatomic
meaning of the universe. In its most general form, the time-independent Schr6dinger
equation takes the form of Equation 2.1. This equations allows relates the total energy operator to the total energy of the system. The total energy operator, called the
Hamiltonian, can be formed to take the physics of the system into account.
ET = HTW
(2.1)
Equation 2.1, however, is extremely trying to solve in a real system of greater
complexity than the hydrogen atom. This is due to the fact that this equation involves
3N variables, the x-, y-, and z- components of the N electrons. This makes solving
large system very intractable as N grows; DFT allows this system to be solved.
DFT is based on the theory presented by Pierre Hohenberg and Walter Kohn in
1964 [19]. This theory provides that all electronic properties can be solved in terms of
the charge density, p, alone. This turns a 3N problem into an Nth order computation.
The largest power is that this provides a methodology to write the total energy in the
following functional form: E0 = Eo[po]. Further, this threorem allowed for all other
29
material properties to be written as a function of the charge density alone, provided
the functional relationship is known.
Hohenberg and Kohn developed their equation in terms of an external potential
V(r). By rewriting Equation 2.1 in dirac notation we can write it in the following form
H IIF) = E I'). In the ground state, the ground-state energy can be calculated with
the ground state wavefunction Eo = (TI H IT). By the variational principle, Kohn
and Hohenberg were able to prove that for any external potential, all the ground state
electronic properties were contained within the electron density.
However, this raised a new set of problems as finding the ground state energy
from the charge density is still not a trivial problem. Hodenberg and Kohn were able
to show that for every trial charge density, ptrial (r) ;> 0 gives f Ptrial(r)dr= N, the
number of electrons, and Eo <; Ev[ptria] where Ptrial is the ground state charge density
for some other potential with wavefunction <trial. Kohn and Sham where then able
to expand this into the method of Density Functional Theory (DFT) for finding po
and for finding EO = Eo[po].
Kohn and Sham were able to fully expand this idea by writing expanding Equation
2.1 in DFT space with the foreknowledge of E = E[p], yielding [25].
E
=
I
2
2
T,[p] + Vext(r)p(r)dr + e -
p(r)p(r')
, drdr' + Exc[p],
|r - r|
(2.2)
where T,[p] is the kinetic energy of a non-interacting, inhomogeneous electron gas,
Vext(r)p(r)dr is the energy imparted by the external potential, the third term is the
intergal form of the Hartee energy, and Exc[p] is the exchange-coorelation energy.
The first three can be exactly solved if the charge density, p is known, where the
Exchange-coorelation term is not exactly known, rather approximations have to be
used to calculate it.
Each of these terms were exactly solved by Kohn and Sham in terms of functionals
with the charge density. For the kinetic energy, T,[p]
that p(r) = E>
#(r)#i
(r). Where the
#i
= -
J
>j f q V20,(r)dr, such
are the one-electron Kohn-Sham obritals.
With these equations, it is possible to solve the energy of the whole system. Since
30
these are of little physical significance, solving can be instead be done by varying
q(r) as shown in Equation 2.3.
{ 2V2 + Vext(r) + VH (r) + Vxc(r)
2mI
Oi(r) = E6ji(r),
where the only unknown is the then the exchange-coorelation.
(2.3)
This comprises
four components: the kinetic correlation energy, the exchange energy, the correlation
energy, and the self-interaction correction.
2.3
Functionals
Since the expression for the exchange-correlation is not known, there are a variety of
approximations that can be used for the exchange-correlation functional. These include the Local Density Approximation (LDA), the local spin-density approximation
(LSDA), and the generalized gradient approximation (GGA).
The most simple approximation to account for the exchange-coorelation is the
LDA approximation.
This approximation was developed by Kohn and Sham that
follows the functional form of Equation 2.4, below. This functional was originally
developed by Kohn and Sham in 1965 [25] and yields very good results despite its
simplicity. Because of this simplicity, and low cost of computation, it is one of the
most commonly used functionals to approximate the exchange-correlation.
E CA
j p(r)Exc[p(r)]dr
(2.4)
In this equation, EcX is the exchange correlation energy of a homogeneous electron
gas with density p. As a simple way of explaining this, this exchange correlation is generated from the electron density at a given point, as if that point was surrounded by
the same density electron gas. From this, it is a purely local approximation. Despite
the simplicity of the approximation, this is one of the most widely used coorelations in
part because it maintains the correct sum rule for the exchange-correlation hole. [22]
In this work, however, the PBE functional has been used. This functional is in
31
the family of GGA functionals [31,32,39,42], and does not rely on empirical fitting.
This was developed to enhance the issues associated with the LDA approximation
when systems have significant variances in the density. In this way, PBE corrects
LDA by including information about both the density at a position as well as the
gradient of the density at a position. PBE was developed by John Perdew, Kieron
Burke, and Matthias Ernzerhof in the late 1990s [40]. The functional form of PBE is
shown below in Equation 2.5.
Ex
[P., Pd..n]=
f (Pup(r),Pdown(r), I Vpup(r) 1, 1 pVdmn(r)|)dr (2.5)
This functional was chosen for the exchange-coorelation because it is more apt
at modeling the local charge variation in systems that have large changes in their
charge density.
In the systems that will be modeled, both organic and inorganic
systems are combined, leading to significant charge fluctuations and localizations. It is
computationally more expensive to model, however the gain in accuracy is necessary.
Also important to accurate modeling in this work is the forces due to the van
der Waals interactions. These interactions include forces not due to covalent bonds
between materials or from hydrogen bonds, but are due to charge fluctuations and
dipoles. In this work, the so-called London Force has been used that accounts for
forces between induced dipoles. This allows forces from charge variation to be accounted for.
By using the method developed by Stefan Grimme [16], the energy including the
London Forces is now
EDFT-D = EKS-DFT
(2.6)
+ Eisp
where this is now the existing Kohn-Sham energy combined with the dispersion energy
given by
Nat-
Nat
Edip = -86
i=1 j~i+1
6' ffdmp(Rij),
23
where Nat is the number of atoms in the system, C
between atom i and
j, 86
(2.7)
is the dispersion coefficient
is a global scaling factor that is dependent on the density
32
functional that is used, and Rj3 is the interatomic distance.
fdmp
is a dampening
function used to avoid near-singularities given by
1
famp(Rij) = 1 + edU4/1).
(2.8)
In this equation R, is the sum of atomic vdW radii, derived from the radius of the
O.O1a- 3 electron density contour from ROHF/TZV computations of the atoms in the
ground state carried out by Grimme [15]. By including these interactions, a more
accurate representation of the systems can be included in all modeling.
2.4
Pseudopotential Approximation
The other main approximation used in DFT is the pseudopotential approximation,
which relies on the idea that the core electrons do not significantly contribute to
the system away from the nucleus. The reason for this approximation is that these
electrons require many plane-wave to be represented accurately because of the sharp
nature of the wavefunction within the core. To this end, a so-called pseudopotential
is used so that the core electrons can be smoothed out, while the valence electrons are
treated explicitly. This significantly decreases the computational cost with minimal
loss of physical meaning.
These pseudopoentials were developed and tested to accurately model a variety
of systems. By testing these pseudopotentials in a wide variety of chemical environments, they can be shown to accurately describe the physics of the systems that they
model. To develop a pseudopotential, forst and all-electron calculation is performed
to get the all electron potential, VAE, as well as the all-electron wavefunction, 0$+ iAEI
and the eigenvalues, O4 E. Further, the construction of this pseudopotential is subject
to the following constraints:
33
f
1.
EA =EAE
2.
Of?
4
3.
*>S P/(r)4ps(r)dr
4.
J
r > rc, where r, is the chosen cutoff radius
=
(r) IAE(r)dr
~PAE
'90"
r=rc
"Ops
-arm
=
r
Ir=rc
The above constraints imply that the pseudopotential must be made for each orbital. By using this information, a pseudopotential can be constructured for each
element in the periodic table. To this end, many pseudopotentials have been developed and tested. This work will not develop unique pseudopoentials, rather developed
pseudopotentials were tested to verify their accuracy, as described in Section 4.2.
2.5
Electronic Properties from p[r]
The extent of this work is to examine the electronic properties and how they relate
to the operation of a solar-cell material. To this end, once the ground-state system
has been determined, several electronic properties are measured including the charge
density, the density of states (DOS), and the electrostatic potential.
All of these
properties can be directly calculated from the DFT electron density. These properties
directly indicate how the system will behave and act on a macroscopic scale; they
are very important to calculate as accurately as possible soas to ensure accurate
prediction of material properties.
While these properties are generally built-in functions of the DFT code used,
understanding how they relate to the charge density is important to undertand the
base-line predictions that the code outputs. In this section, the mathematical relations that dictate the various properties are outlined.
The number of states at a given energy level is indicative of several material
properties including band-gap and mobility.
The local density of state (LDOS) is
a measure of the total number of states at a given energy level. This is defined in
Equation 2.9.
34
N
n(r, E)
1i(r)26(E - ci)
=
In the above euqation, Oi(r) are the Kohn-Sham orbitals.
(2.9)
To obtain the total
density of states (DOS), integration over r is required as shown in Equation 2.10.
n(c)
n(r, c)dr
=
(2.10)
Further, the DOS can be decomposed into the atomic orbitals, particularly useful
to determine the chemical nature of bonds or the composition of band-edges. This is
shown in Equation 2.11.
fla(E)
I (OPil
=
kta) 126(E
_
60)
(2.11)
Where a are the 1 and/or m quantum numbers. By integrating the LDOS over
all c, the total charge density can be determined as in Equation 2.12, shown below.
This can also be spatially resolved to show how the charge density varies throughout
the system.
p(r)
n(r, c)dc
=
(2.12)
The electrostatic potential will also be of use in calculating several material properties of the material including the induced electric field as well as isolating any effects
that surface states may have. This is shown in Equation 2.13.
V 2 (r) = 47rp(r)
(2.13)
All of the above equations will be exploited to determine the accurate electronic
properties of all materials explored in this work.
35
2.6
DFT Bandgap Issue
With all of the power of DFT, one issue that plagues it is the so-called "DFT band
gap issue." This issue arises from the fact that DFT underestimates the bandgaps of
a material by up to 50%. This has wide and reaching implications, especially with
regard to lowband gap systems, where underestimating the bandgap can actually
cause the system to become metallic.
It can be shown [38], that for a N-electron system in an external potential, there
arises a funamental bandgap given by
(2.14)
Egapud = I - A,
where I is the ionization potential and A is the electron affinity given by
I
=
Ev(N
1)
-
-
E(N)
(2.15)
A = Ev(N ) - Ev(N - 1),
which can also be written as a derivate for N electrons in the following form
Egap
deri =
aN N6
9
ON N-
6
1
(2.16)
In the case where the exchange coorelation is an explicit form of the charge density
such as LDA or GGA, the above can be rewritten as
deriv =KS
gap
gap
KS
=
6
LUMO
KS
~
(2.17)
EHOMO-
where these values come from the KS equations. Cohen and Wang were able to show
that DFT predicts too high of EHOMO and too low ELUMO. This is due to delocalization
error associated with the exchange correlation that gives rise to a overal convex energy
curve [38].
Even with this error in computing the bandgap, the generally accepted view is
that trends in the bandgaps remain the same. In this way, even if the exact value
36
of a bandgap is not strictly right, any modification of the system will have the same
relative error, if the same pseudopotentials and exchange correlation are used. To this
end, several methods for fixing the bandgap have been proposed including the "scissor
operator" in which the overall conduction band is shifted up by the delta-energy that
is required to have the proper bandgap [11, p. 196]. This, in general, requires an
exact knowledge of the experimental bandgap of the system, something that is not
always known. This implies that the overall electronic character of the system will
remain consistant, just shifted by an error that will remain consistant across similar
systems. In this way, trends, rather than absolute bandgaps are studied in this work.
There exists options to more precisely calculate the bandgap in systems, however
these rely on an advanced type of DFT termed Time-Dependent DFT (TDDFT).
One method is by using the GW approximation (GWA) that is an approximation
that allows for an alternative way to calculate the self-energy of a many-body system
of electrons. Lars Hedin proposed this method in his seminal paper, relies on approximation that the expansion of the self-energy into the single-particle Green's function
(G) and the screened Coulomb interaction (W) [17]. In this way, Hartree Fock can
be expanded to accurately describe the screening nature of the bandstructure. This
approximation is very computationally expensive, as such, it was used sparingly in
this work. However, this approximation has been shown to accurately calculate the
bandgap of many materials [50]. In this way, the scissor operator can be defined
formally as
AEsciSor = EGap- ESGT.
(2.18)
This formalism allows for any interacting system to have the real bandgap be estimated as
Eap = EDT + AESCissor.
(2.19)
This will become important in calculating the overall bandgap of the systems used
throughout this work.
37
2.7
DFT Code
In the preceding sections, the overall DFT theory was established from a mathematical
approach.
These principles are employed into several prepackaged codes that are
optimized for highly parallel environments, allowing for computations of significant
system size. Without the work of these groups, this work would not be possible
within the time frame of a Master's Degree. The code used through out this work
is an open source DFT code called Quantum Espresso, and formerly referred to as
PWscf [14]. This code was chosen because of the "over-the-counter" nature of the
open-source movement.
Additionally, because of this open source nature, a wide
variety of support is available as well as the ability to directly access the source for
any modifications that may be necessary.
2.8
DFT Scaling
Despite the power that DFT offers, it has one main limitation, the scalability of the
system. As computational resources have grown, larger systems have been able to be
modeled, however DFT codes do not scale linearly with the number of atoms in the
system or the number of processors used. In this way, there are still systems that are
not able to be calculated in a reasonable amount of time. One way of quantifying
this is by looking how the code scales with the number of processors used for a fixed
number of atoms. On the computational resources that were used in this work, the
speed up with the number of processors used was calculated. This is shown in Figure
2-2.
In this scalability study, a vanadium oxide supercell was used with 112 atoms
represented. As it can be seen from this graph, even increasing the total number
processors by 64 times only increases a speedup of approximately five times in total
time. Similarly, the code scales poorly with the total number of atoms present. In this
work, systems have atomic sizes of 32-300 atoms, all with varying chemical complexity.
Scalability was a huge concern and some more advanced systems could not be exactly
38
6
II I I I I I I I I I I I I I I I I I I I I I I I I I I I I II
5-
2-0
20
40
60
80
100
120
Number of Processors
Figure 2-2: Speedup associated with number of processors
calculated because of the scaling issues associated with DFT. The computational
limitations of DFT have put limits on the size of systems that can be modeled in
this work; modifications to systems were performed to allow for advancement in
the modeling and understanding of complex BHJ devices. In all cases the chemical
character of the systems was preserved.
2.9
Conclusion
Density functional theory is a powerful tool that affords several ways in which to
calculate the properties of complex materials. By exploting the work of Hohenberg,
Kohn, Sham, and others, advanced materials can be designed and qualified in an
efficient computational environment. This work will exploit this theory extensively
to determine the material properties of complex hybrid solar accepting materials.
39
40
Chapter 3
Rational Design of
Nano-structured Hybrid
Photovoltaics
The main objective of this work is to employ first principle computational tools to
rationally design a novel class of nano-structured hybrid materials that overcome the
issues inherent to traditional bulk heterojunction photovoltaic materials. This chapter
will outline the the general approach, focusing on engineering materials to enhance
exciton separation and decrease recombination in a hybrid organic-inorganic nanostructured photovoltaic material. By designing to address these issue, an efficient
BHJ hybrid material can be realized.
3.1
Overall Design
Addressing the issues with traditional BHJ materials requires a layered structure;
by alternating layers of organic and inorganic regions, a piecewise material can be
designed in which photons are absorbed in the organic material, generating excitons
which are separated into electrons and holes by an intrinsic dipole field. The electrons
and holes are driven by an intrinsic electric field to the inorganic layers at lower and
higher potentials, respectively, as illustrated schematically in Figure 3-1.
41
3A
Anode
S18.5 A
Figure 3-1: Schematic representing the proposed design. Alternating layers of organic
and inorganic layers allow for efficient charge extraction
The design shown in the figure has the ability to effectively separate bound excitons.
In the organic regions, a local molecular dipole is engineered to provide
sufficient energetic potential to separate excitons into free holes and electrons and
drive the free charge carriers away from each other, decreasing the overall probability
of recombination.
Once the free carriers reach the high-mobility inorganic regions,
they are rapidly advected out of the material to the electrodes. As Figure 3-1 shows,
the dipole changes direction in alternating organic regions suck that the separated
electrons and holes are at different energy levels; thus, each inorganic region contains
carriers of a single type. Furthermore, once carriers are in an inorganic region, there
is no thermodynamic driving force for recombination with carriers of opposite charge
in adjacent inorganic layers. Together, these properties dramatically decrease recombination. It is also worth noting that the alternating dipoles prevent any macroscopic
polarization, or electric field, from being generated.
In an ideal system, such as that shown schematically in Figure 3-2, there will be no
energetic losses at the organic-inorganic interfaces. Bound electron-hole pairs would
be generated in the organic region, then separated by the intrinsic local dipole into
42
Figure 3-2: Schematic showing ideal band alignment in bulk structure
free carriers, which would be effectively removed into the inorganic regions before they
could recombine. Thus, this design would increase efficiency in two ways: First, due
to the presence of the local dipole in the organic material and the nanoscale distance
between inorganic regions, charge carriers will only remain in the organic region for a
short period before being driven into high-mobility inorganic layers and transported
to the electrodes, thereby minimizing recobination. Second, since no energetic losses
are required to separate excitons, the Voc can be maximized to obtain optimal power
output per absorbed photon.
3.2
Materials
The nature of the design proposed in Section 3.1 requires two different materials to
be joined together: an organic semiconductor and a layered inorganic material. In
this section, we discuss the types of materials we have explored for each component
in the system. Of utmost interest are polar organic materials and inorganic materials
that readily form layered structures with organic materials and also posses a higher
mobility than typical organic semiconductor.
3.2.1
Organic Semiconductor
The organic semiconductors used throughout this study are derived from structures
that are experimentally used and studied. These materials are traditionally molecules
43
from a specific subset of BHJ materials called dye sensitized solar cells (DSSC).
These materials normally are liquid-suspended, with the other material used in the
heterojunction being liquid in nature, where a redox reaction sustains the charge in
the system. Because these materials are extensively studied, they were chosen as a
baseline type of organic semiconductor to examine. In this way, we are able to readily
compare our computational results to experimentally available data. A group of these
DSSC semiconductors that is widely studied is based on polythiophene.
Thiophene is a sulfer-based molecules that readily polymerizes; the degree of polymerization create the ability to tune the energetic properties of the BHJ material.
In addition, thiophene can be functionalized with other orgnaic molecules to tune
the bandgap of the semiconductor.
As a result, several experimentally available
compounds are created with thiophene acting as the charge generator.
An exam-
ple that was explored heavily in this work is 2-Cyano-3-[5-(9-ethyl-9H-carbazol-3-yl)3,3,3,4-tetra-n-hexyl-[2,2,5,2,5,2]-quater thiophen-5-yl] acrylic, commonly referred to
as MK2. This was chosen because the organic structure is both linear, compact, and
polar in nature.
MK2 is one of several organic molecules based on polythiophene. This includes
MK1-MK6, each manufactured in such a way as to have better absorptance cooeffiencients, or to have an optimized bandgap. Organic MK2 is shown in Figure 3-3. From
experiment, MK2 has a bandgap of 1.85eV, a peak absorptance at 480nm, and a
incident photon conversion effiency of 5.01% while bound to titanium dioxide [51].
From the Shottkey-Quassier limit, MK2 itself is predicted to have a max efficiency
of 22.56%, relying on the assumption tha there is no recombination; the low efficiency in the experimental results is likely due to the inherent non-directionality of
the TiO 2 - MK2 heterojunction design, as well as the relatively small amount of MK2
covering the surface. By combining this material with a high-mobility inorganic material in a directional fashion as the proposed design dictates, overall recombination
should be limited.
In addition to organic thiophene, several other semiconductors were mixed with
thiophene to change the optoelectical properties of a single thiophene molecule. They
44
\
CN
HOOC
N
Figure 3-3: The MK2 molecule (2-Cyano-3-[5-(9-ethyl-9H-carbazol-3-yl)-3,3,3,4-tetran-hexyl-[2,2,5,2,5,2]-quater thiophen-5-yl] acrylic)
S
NNN
N
I
Thiophene
N
N
0
N
NN
5-methyl-4H-thieno[3,4-cpyrrole-4,6(5)-dione
benzo(c(1,2,51oxadlazole
[1.2.5]oxadiazolo-(3,4-cpyridine
Figure 3-4: Different types of organic semiconductors studied
were also chosen to tune the energy levels within the inorganic layers. These include BX (benzo[c][1,2,5]oxadiazole), PX ([1,2,5]oxadiazolo-[3,4-c]pyridine), and TPD
(5-methyl-4H-thieno[3,4-c]pyrrole-4,6(5H)-dione), which are shown in Figure 3-4.
Specifically, each of these different types of materials are chosen because of their
bandgaps, and their electron affinity when bound to thiophene. The electron affinity
is an indication of the electron-acceptor strength of the material, or how easily it it
can transfer electrons out of the material. Previous work has used TDDFT to evaluate the total band energies of these molecules bound between a thiophene group in
the following structure: thiophene-A-thiophene, where A is either BX, PX, TPD or
thiophene [30]. The results of this study are shown in Table 3.1.
45
Table 3.1: Thiophene property modifications [30]
Molecule
Tri-Thiophene
Thiophene-TPD-Thiophene
Thiophene-BX-Thiophene
Thiophene-PX-Thiophene
Bandgap
3.27
3.15
2.56
2.44
Electron Affinity
-1.69
-2.19
-2.7
-3.01
Table 3.2: Thiophene bandgap and chain length [5]
00
Computational Gap [eV]
5.51
3.65
2.93
1.71
Experimental Gap [eV]
5.37
4.12
3.52
-
Thiophene Chain Length
1
2
3
The large bandgap of the tri-thiophene is to be expected. Both experimentally and
computationlly, the bandgap of polythiophene depends on the number of thiophene
units, decreasing significantly from over 5eV for a thiophene molecule to less than 2eV
for above six linked thiophene units. In another computational work that accurately
calculated the band gap of thiophene chains [5], the trend in bandgap decrease can
be observed, shown in Table 3.2.
In this way, the bandgap of these materials can be controlled by changing the total
number of thiophene repeats present in the organic photovoltaic. This will become
important in Chapter 4 when this nature will be exploited to make this work more
computationally streamlined.
3.2.2
Inorganic Phosphate Group
Transitional metal phosphates are a group of materials that readily form bulk layered
strucutres. These structures form in a number of different chemical compositions with
different phosphate moieties, providing a range of electronic and chemical properties.
Similar to the more familiar graphene and transition metal dichalocogenides that
are currently of interest for numerous applications, these phosphates are composed
of stacks of weakly interacting two-dimensional sheets held together mainly via van
46
der Waals interactions; a mixture of strong covalent/ionic bonding occurs within
the plane of each 2D sheet. Even more interestingly, it has been experimentally
demonstrated that these materials can be chemically modified to insert ordered arrays
of covalently boud organic molecules between the 2D sheets. These properties make
the 2D transition metal phosphate materials extremely useful to rectify several of the
issues of traditional BHJ materials.
These materials have the following two different categories of chemical formulation:
M(HPO 3 ) 2 as well as XH 2 P 3 0 10 where M can be materials in +4 such as zirconium,
hafnium, and titanium while X is materials in +3 such as Al, Ga, Fe, Mn, V, and Cr
[44]. All of these structures have been experimentally explored for various electronic
properties.
A wide variety of materials is beneficial to qualify if there will be a
material that will display the properties desired: high mobility and proper band
alignment with organic material.
The primary inorganic material studied in this work is Zr(HPO 3 )2 . This material
has been studied primarily for the ionic exchange properties that it posses [8]. It has
been used in a variety of fields from nanocomposites to nuclear applications, yet its
electrical and optical properties have not been well-studied, likely because of its large
bandgap.
Originally chosen because of the experimental evidence that it forms layered structures with organic material, we wil show that Zr(HPO 3 ) 2 also has excellent optoelectronic properties necessary for the proposed novel BHJ material design. This structure forms both a and y phases, and has been experimentally shown to' form both in
Zr(HPO 3 ) 2 and Zr(HPO 4 ) 2 structures [1,36]. For both of these studies, the unit cell
parameter did not vary significantly as the additional oxygen was terminated normal
to the layers. The simpler Zr(HPO 3 ) 2 structure was used throughout this work, shown
in Figure 3-5.
This material also forms layers with organic materials [1, 9], such as the structure
shown in Figure 3-6. This was explored experimentally in order to form structures
with controlled porosity for ionic transport applications, but few experiments have
been performed to explore its electronic or transport properties.
47
oH
p
0
@0
Zr
OZr
Figure 3-5: Structure of a-Zr(HPO 3 )2
op
x@0
L
e r
Figure 3-6: Structure of Zr(HP03)2 layered with benzene rings
48
The nature of forming layered structures with organic materials means that the
main issue will be determining if this material will behave in such a manner as to
increase the optoelectronic properties of the organic MK2 molecule.
This will be
discussed in Chapter 4.
3.3
Exciton Separation
The idea of an exciton was originally proposed by Frenkel in his governing paper that
described the interaction of absorption of light into heat in crystals [13]. In this,
he described how light can generate a bound hole and electron pair caused by the
interaction of light with the material. The generated charge would leave behind in
its space a positive charge normally called a hole. These two particles would form
what is classically called an exciton. The interaction between these two is given by
the coulomb potential energy
e2
Ec =
Er
,
(3.1)
where e is the relative permittivity (dielectric constant) in the material, e is the
fundamental charge of an electron, and r is the distance between the charges. In
this way, both the size of the exciton as well as the material dielectric constant
affects the exciton binding energy.
These exictons are classically termed Frenkel
Excitons [33]. Other types of excitons exist such as Wannier-Mott excitons, however,
Frenkel excitons will be considered primarily in this work because organic materials
generally have low dielectric constants. For example, benzene (C6 H6 ) has a dielectric
constant of ~3 [37] whereas silicon has a relative permittivity of 11.4 (ab initio) or
12.7 (exp) [3]. Further types of excitons can be observed, especially when there are
systems with free charge, that can screen the Coulomb potential, yielding
Escreened =
--
e 2r
exp
(
(3.2)
where A is the screening length. Organic materials do not have a significant amount of
free charge to effectively screen the charge. In this way, in typical organic materials,
49
the binding energy of the exciton can range between 0.1-1.5eV.
As discussed in Chapter 1, exciton separation in traditional BHJ materials is accomplished via conduction band-edge alignment; two materials are joined together
because their conduction bands have an energetic difference AECB greater than the
energetic binding energy of the exiton. However, exitonic binding energies are normally large, requiring a large AECB that decreases the Voc by the same amount.
Inaccurate control of the band alignment leads to low open circuit voltages or low
probability of exciton separation because of processing requirements, material properties, and the dual nature of a traditional BHJ design. Additionally, this method
relies on electron/hole diffusion with no built-in potential driving the separation; the
lack of directionality increases the probability of free carriers recombining, rather than
leave the organic region at the electrodes.
In contrast to traditional BHJ operation, this work proposes a design in which a
local dipole is responsible for the separation of the exciton. With such a design, it
is possible to addresses the two afformentioned issues with BHJ materials: First the
dipole allows for a maximized Voc, as the excitonic sepatation is not due to an energy
loss, but rather to an electric field. Due to the dipole-induced field in the organic
region, this design allows for the exciton separation without the need of the Voc to
decrease to achieve this separation. Second, the dipole can provide a driving force to
separate the charge carriers, decreasing the chances of recombination in the material
before extraction. Referring to Figure 1-3, the lowing of the Voc can be observed in
the material. Rather, in Figure 3-7, if a molecule has an inheirant material dipole,
a directionality can be given to the exiton. In Figure 3-8, the exitonic separation
in the proposed system is shown; a dipole in the organic region, as indicated in the
change in the vacuum level (Xorganic), and the energetic alignments of the organic and
inorganic region are only necessary to extract the exciton from the organic region.
In this way, a near perfect band alignment between can be realized as very small
energetic differences are necessary for the transfer of charge between the organic and
inorganic region.
This idea of exitonic separation relies on an intrinsic dipole that can generate an
50
N
Dipole of Molecule
Figure 3-7: Schematic of dipole separation
*
4
MU
Inorganic
XOrganic
-Inorganic
EC
I
~m.
NEMENE :ONEENNE
U
Ev
0000 1
i
Figure 3-8: Exciton separation in an ordered nano-structured material with an induced dipole
51
electrostatic potential greater than that of the exciton binding energy. Further, in
order for this to effectively separate the exciton with little recombination, an ordered
structure is required; a dipole alone cannot create the separation of the exiton, a
layered structure is required.
3.4
Mobility
The second major bottleneck in tradition BHJs is the mobility issue. The mobility of
a material is a measure of the speed at which an electron, or hole, can move through
a material in the presence of an electric field. This can be written as
Vd =
where Vd [cm 2 /s] is the drift velocity,
/.
(3.3)
pLE,
[cm 2 /(V
.
s)] is the mobility, and E [V/cm]
In this way, a material with a high mobility will be able to
is the electric field.
move charge faster with the same applied electric field. This is of interest because
when the material, such as organic materials, has a high exciton binding energy,
the longer it is in the material, the higher the chances are for recombination. As
an example, the mobility in [6-6]-phenyl C71 butyric acid methyl ester (PC6oBM)
has a mobility of 1.5 x 10- 2 cm2 V-IS-1 [24] where traditional silicon has a mobility
of 102
-
10 cm 2 /(V
.
s) [49]. This very small mobility in the organic material, high
recombination of charge can be observed, and low photon conversion efficiency (PCE)
can be observed. Calculation of the mobility will be described in Section 6.1.
3.5
Open Circuit Voltage
Besides recombination, the open circuit voltage Voc in BHJ materials can also be
observed to be quite low compared to traditional inorganic solar cells.
52
The open
circuit voltage is determined in the following way
Voc = AkbT 1n
e
JC+1),
Js
(3.4)
where Jsc is the current generated in the short circuit, Js is the saturation current
(reverse bias), and A is the diode quality factor for a p-n junction [23]. While this
was originally developed for p-n junction solar cells, it has been used to describe the
nature of BHJ organic materials.
In an idealized state, the VOC is the total bandgap of the material. While this
cannot be realized in a true system because of atomic defects in a real crystal. The
total bandgap of a material is used as a rubric for judging the overall material. BHJ
devices rely on decreasing the overall bandgap of the system to separate the exciton,
shown schematically in Figure 3-9, decreasing the overall Voc. However, the bandgap
that interacts with the solar spectrum is still the large bandgap of the accepting
material in the BHJ. In this way, both the effeciency of the device is decreased by the
nature absorption nature of the optical bandgap (described by the Shottkey-Quisser
limit) as well as taking a decrease in the operational effeciency described by
FF x Vc x Isc
ExA=
E x A.
(3.5)
This implies that the effiency of the overall material can also be increased simply
by increasing the Voc. BHJ materials do not exploit this effectively because of the
requirement of separating the exiton via band energy differences.
3.6
Conclusion
In this chapter the existing issues with traditional BHJ designs was discussed and
a new design was proposed. By combining a layered structure of organic and inorganic materials it may be possible to create a piece-wise system that overcomes the
issues with traditional BHJ designs. This design may also make using existing DSSC
materials possible because of the inherently directional nature to the 2D structure.
53
4
Egap-1
-Photon
r
j
Egap-2
-I
Figure 3-9: VOC in BHJ materials.
In the following chapters, the system is quantified from a calculation standpoint to
determine the viability of such a design as well as any improvements that can be
made to this system.
54
Chapter 4
Atomic and Electronic Structure
In previous chapters, the framework for the ideal BHJ material design has been
proposed. The organic DSSE material was explained as well as a proposed framework
for the using Zr(HPO 3 ) 2 as an inorganic inter-linker to convect charge away. In this
section, the results of studying this bulk structure will be discussed. This will provide
a framework for discussing how to effectively modify the structure in ways that further
optimize the optoelectronic properties of the bulk structure.
4.1
Computational Details
In DFT, like most computational paradigms, certain computational values must be
converged.
In the case of DFT, the plane-wave basis set must be fully converged
in order to accurately compute the electronic properties of the system. In order to
properly converge this basis set, two key values must be converged, firstly the energy
cutoff (Ece,) of the plane waves, and secondly the k-point mesh.
This energetic
convergence will be described in this section.
The convergence test of the energies and k-points were done on the smallest system in this study, represented by the bulk structure of Zr(HPO 3 ) 2 . The experimental
unit-cell parameters of this system are shown in Table 4.1. This system was chosen
precisely because smaller unit cells have the most strict convergence criteria principally because these plane-waves are evaluated in k-space, because of the Fourier
55
Table 4.1: Zr(HPO 3 ) 2 experimental unit cell parameters
Value
Parameter
a
b
c
a
5.42000 A
5.42000 A
5.57950 A
900
900
1200
nature of the calculations. The reciprocal space system is defined as
a=
bxc
27r a.(bxc)
b* = 2
cxa
a.(bxc)
(4.1)
axb
C* = 27r a.(bxc)
where a, b, c are the prinicple lattice vectors, and a*, b*, c* are the reciprocal lattice
vectors. It is possible to see that the smaller the overall volume of the real-space unit
cell, the larger the reciprocal lattice will be. Organic molecules generally have very
large unit cells and superstructures that will be studied also have very large lattice
constants. Bulk Zr(HPO 3 ) 2 has the smallest overall volume; any system converged in
this structure will be converged in larger unit cells.
First the cutoff energy was converged to within 20 meV. Increments of 5Ry
(~68.028eV) were used in calculating the convergence values. The convergence criteria is a reverse looking criteria in that AEc,,, = E, - E,_ 1 < 20meV. In a similar
fashion, the k-point mesh was also converged. This mesh is defined as Na, Nb, N,
where N is the number of k-points along the a*, b*, and c* lattice vectors. This was
converged after the total energy of the structure was converged and was also found to
be converged in a reverse looking criteria within 20meV. This is plotted in 4-1. From
this criteria, it was found to be converged at a 3,3,3 kpoint mesh.
To this end, all other structures were ran at minimum energy cutoff of 50Ry
=
680.28eV, and a maximum k-point mesh of 3, 3, 3. As unit cell lengths grew in real
56
-4484.4
I
I
I
I
I
I
-4484.6-
& -4484.8
-4485S
0
1
I
I
I
I
I
2
3
4
5
6
7
I
I
8
9
10
K-point Mesh (Na ,Nb ,Nc)
Figure 4-1: K-point convergence of Bulk Zr(HPO 3 ) 2
space, the k-point mesh along that direction could be relaxed, however the dense
k-point mesh was used throughout this work.
4.2
Pseudopotential Testing
Quantum Espresso comes with a variety of prepackaged pseudopotentials as well as a
plefora of other pseudopotentials developed by other research groups. In general, the
pseudopotentials that are employed by the Quantum Espresso library are as simple as
need be, without the addition of more complex approximations that hybrid functionals
use. The pseudopotentials used throughout this work are shown in Table 4.2.
All pseudopotentials were used from the Quantum Espresso pseudopoential library, generated from PBE exchange-correlation [40,41,43]. A general pseudopotential without these added complexities was desired to eliminate the addition of artifacts
from the hybrid nature of other pseudopotentials. Where possible, the pseudopoetentials were tested to guarantee that they returned sensical answers.
For the standard temperature and pressure (STP) gasses (02, N 2 , H 2 ), the pseudopotentials were tested to guarantee that they calculated the correct bond length of
57
Table 4.2: Pseudopoentials used in this study
Species
C
0
N
H
S
P
Zr
Ti
F
B
Pseudopotential Config
2s2 2p2 3d-2
2s2 2p4
2s2 2p3
1si
3s 3p
3s 3p
4s 4p 4d 5s 5p
3s 3p 4s 3d
2s 2p
2s 2p 3d
Table 4.3: Bond distance comparison for STP gasses
Bond
Gas
02
O
O
N2
H2
N
H
N
H
Experimtal Distance [A]
Computational Distance [A]
1.2075
1.0977
0.7414
1.2374
1.1109
0.7521
Error [%]
2.48
1.20
1.44
the material. These pseudopotentials have been extensively used and tested in other
works, so in this way, this was the primary test used to test these pseudopotentials.
Oxygen, Nitrogen, and Hydrogen gas experimentally form bond length structures
of 1.2075A, 1.0977A, and 0.7414A, respectively [20]. The computational results of
isolated molecules are shown below in Table 4.3 and shows superb agreement with
experiment. These pseudopotentials will also be compared to how they interact with
carbon to further verify their validity.
Carbon and sulfer form a wide variety of materials in nature, so to quantify
these pseudopotentials, the carbon-carbon distance in benzene and the carbon-sulfur
distance in thiophene was used as a benchmark for the pseudopotential's viability. In
benzene, the carbon-carbon distance is 1.3970A and in thiophene, the carbon-sulfur
distance is 1.714A; additionally, the carbon-hydrogen distance is 1.079A [18]. Further,
in pyridine, the experimental N-C distance was found to be 1.340A. Once again, the
computational results can be found in Table 4.4; near perfect bonding distances can
58
Table 4.4: Bond distances for carbon, sulfur, hydrogen, and nitrogen pseudopotentials
Bond
C
S
C
N
C
C
H
C
Experimtal Distance [A]
1.397
1.714
1.079
1.34
Computational Distance [A]
1.393
1.72
1.089
1.338
Error [%]
0.29
0.35
0.93
0.15
Table 4.5: Bulk zirconium lattice comparison
Experimental
Computational
% error
a [A]
c[A]
a/c
3.225
3.230
0.169%
5.134
5.17
0.695%
1.592
1.60
0.521%
be observed.
Bulk zirconium forms a hexagonal close-pack structure with lattice constants of
a = 3.225A, c = 5.134A, and c/a = 1.592 [12]. The comparison of this with compu-
tational detail is shown in Table 4.5. It can be seen to have great agreeance with the
experimental values found in literature.
The pseudopotentials were also tested against a common material in zirconium
dioxide (ZrO 2 ). This structure forms a monoclinic unit cell with paramaters shown
in Table 4.6 [52] along with the computational error of this unit cell. The total error
was found to be very small with the used pseudopotentials.
As an additional test of the pseudopotential accuracy, the enthalpy of formation
was calculated for ZrO 2 . The enthalpy of formation is simply the energy required to
Table 4.6: ZrO 2 unit cell computational error
Lattice Parameter
a
b
c
a
Experimental
5.1501A
5.2077A
5.3171A
90.000
0
7
99.2240
90.000
59
Computational
5.1200A
5.216A
5.281A
90.000
99.010
90.000
Error [%]
0.58
0.16
0.68
0.00
0.22
0.00
Table 4.7: Lattice constant error of Zr(HPO 3 ) 2
Lattice Parameter
a
b
c
Experimental Value
5.418A
5.418A
5.579A
900
Computational Value
5.517A
5.517A
5.660A
900
0
900
1200
900
1200
-y
Error
[%]
1.83
1.83
1.45
0.00
0.00
0.00
form 1 mole of a compound from the constitutive elements defined as
AH;
=
Z(vH; )roducts
-
(4.2)
Z(vH)reactants,
where v is the number of moles for each element in the reaction and HY is the enthalpy
of elements. In this way, the formation of ZrO 2 is given by
+ 0 2 -+
ZrO
(4.3)
2
.
Zr
From a computational standpoint, the enthalpies are taken as the total energy of the
consititutive elements. In this way, Equation 4.2 is simplified to be
A Ef
=
Z(vEf),roducts
-
Z(vEf)
(4.4)
reactants-
Experimentally, the enthalpy of formation has been found to be -1097.46kJ/mol [6].
In this way, the total enthalpy of formation can be calculated from Ezr02 Ezr = -11.57eV = -1116.6lkJ/mol.
E02-
This gives an error of only 1.744%, an excellent
agreement with experiment.
These pseudopotentials were also tested against the bulk phase of Zr(HPO 3 ) 2 from
experimental values [36]. The results of this calculation are shown in Table 4.7.
Other pseudopotentials were likewise tested to verify that they matched experimental properties. They were also chosen from the well-tested quantum espresso
pseudopotential library and have been used in many other computational works.
60
The results of this section show that the chosen pseudopotentials have great agreement with the existing experimental systems. Moving forward, these pseudopotentials
will predict accurate optoelectronic properties of the tested systems and will be used
over a wide variety of systems to verify the design proposed in the previous chapter.
4.3
Computational Results
In this section, we present the computational results of this work. The overall structures used will be described as well as their ground-state electronic properties. These
properties will be used to iterate, on the design of this BHJ. The backbone of this
design will rely on Zr(HPO 3 )2 as well as the organic MK2 molecule.
4.3.1
Bulk Zr(HPO 3 )2
In the realm of layered phosphates, one of the most studied is a-Zr(HPO 3 )2 . It is
primarily used as an ionic transport material in experimentally realized structures.
Of utmost interest is the fact that this structure forms layered structures with organic materials, allowing for solution processing of the bulk structure, limiting the
processing cost associated with creating the photovoltaic structure.
Additionally,
layered phosphate compounds form with zirconium, titanium, and halfnium with almost identical unit cell parameters, to be discussed later. This structure would allow
for a highly ordered layer structure with a piecewise mobility increase, addressing
both of the primary issues with traditional BHJ designs. The optimized DFT lattice
parameters of Zr(HPO 3 )2 are shown in Table 4.7.
By calculating the atom-projected density of states (pDOS), both the energetic
properties of the band edges as well as their character can be examined. The pDOS
of Zr(HPO 3 )2 is shown in Figure 4-2. Although this figure shows that Zr(HPO 3 )2 is a
large bandgap semiconductor with a DFT-gap of 4.41eV, it is clear from the pDOS
that the bonding is not fully ionic, but has significant covalent nature to the bonds. In
particular, the valence band is dominated by oxygen p-states but there is significant
hybridization (overlap) of Zr d-, H s-, and P p-states at lower energies away from
61
-
IIIZ
-
4
H-s
-
--
P-p
o-p
2,-
Q
-4
-2
0
2
4
E - Ef [eV]
Figure 4-2: Atom-Resolved Density of States for a - Zr(HPO 3 ) 2
the band edges. The empty conduction band states also display this same trend of
hybridization with the conduction band being made up of hybridized Zr d-, 0 p-, and
P p-states. The nature of these hybridizations contribute to overall system properties
such as the mobility of the system, to be discussed in subsequent chapters.
4.3.2
Bulk a - Ti(HPO 4 ) 2 . H 2 0
In order to understand the nature of, and potentially modifying, the bandgap of
Zr(HPO 3 ) 2 , other transition metal phosphates were explored. a - Ti(HPO 4 ) 2 - H 2 0
forms a similar reduced unit cell as Zr(HPO 3 ) 2 [45] with titanium offering different
electrical properties compared to zirconium. The experimentally realized structure is
hydrated, with water molecules occupying the space between the independent layers.
These water molecules generally do not serve to change the electrical properties of a
material, but further processing is generally required to fully remove them from the
structure. The 'hydrated' unit cell parameters of a - Ti(HPO 4 ) 2 - H 2 0 are shown
below in Table 4.8. A hydrated structure simply means that there are unbound water
62
Table 4.8: Experimental Structure of a - Ti(HPO 4 ) 2 - H 2 0
Lattice Parameter
a
b
C
a
0
7
Experimental Value
8.611A
4.99330A
16.15070A
900
110.2060
900
molecules within the super structure. This is normally due to the processing or other
experimental creation of the material.
The DOS was calculated for hydrated titanium phosphate and is shown in Figure
4-3a. The character of the band edges are now comprised of titanium states, where
before the states were made up of zirconium states. It has a DFT-bandgap of 2.88eV,
significantly less than the bandgap of the Zr(HPO 3 )2 structure potentially providing
a way to control the bandgap of Zr(HPO 3 ) 2 as well as controlling the band alignment
in the bulk structure, to be described later.
This titanium phosphate structure can be represented as an unhydrated layer,
yielding the chemical formula a - Ti(HPO 4 ) 2 , where water molecules are not suspended inside the layer. This now has a much closer lattice constant to the Zr(HPO 3 ) 2
lattice that was experimentally shown. In this unhydrated layer, the lattice constants
become a=5.054A, b=4.984A, c = 6.O1A, a = 0 = 90.0', -y = 60.00. These lattice
constant are almost identical to the values found in Table 4.7. The DOS was again
plotted, and this is shown in Figure 4-3b. In this way, the overall DFT-gap can be
seen to be 2.77eV, significantly less than the overall DFT-gap of Zr(HPO 3 ) 2 (4.41eV).
4.3.3
Titanium Doping of Zr(HPO 3 ) 2
The nature of the lower bandgap in both hydrated and unhydrated Ti(HPO 4 ) 2 , the
possiblity of doping the Zr(HPO 3 )2 1ayer with titanium atoms becomes an interesting
way to control the bandgap of Zr(HPO 3 )2 . In particular, they both share close lattice
constants and have similar bond-lengths in the their bulk form as shown in Table 4.9.
63
5
1
1
II-
4-
3
-
4-0
P
P-P
Ti-d
H-s
2-
-4
-2
0
2
4
E-Ef [eV]
(a) Atom-resolved DOS of a - Ti(HPO 4 )2 H 2 0
4-
Ti-d
--
O-p
H-s
-
2
0
0
-4
-2
2
0
4
E - Ef [eV]
(b) Atom-resolved DOS of Ti(HPO 4 )2
Figure 4-3: Comparison between hydrated and unhydrated Ti(HPO 4 ) 2
64
Table 4.9: Similarities in Zr(HPO 3 ) 2 and Ti(HPQ 4 ) 2 bond lengths in their bulk struc-
tures
Length [A]
1.961
2.068
1.586
1.513
Bond
Ti0
Zr 0
Ti
P
Zr
P
The primary reason to dope the Zr(HPO 3 ) 2 layer would be to change the energies
and character of the band endges. Additionally, the titanium may control other bulk
system properties such as mobility
To explore these system properties, doping of the zirconium structure with a range
of titanium atoms was performed. These dopant levels go as Ti:Zr, and were performed for the following ratios: 1:1, 1:3, and 1:5. This limit was reached because after
a 1:5 ratio, computational time became excessive, and a trend could be established.
The variation of the system DOS with the ratio of titanium doping levels are in Figure
4-4.
It is apparent that the doping level does not significantly change the character
of the band edges with respect to eachother.
The band gap remains the same to
the third significant figure, and can seen to be 0.7eV, significantly less than the
bandgap observed in zirconium phosphate of 6.9eV. This is mainly because of the
gap state, or the titanium states present between the valence band and the higher
energy conduction band, lower the effective bandgap the material will project. These
energetic states slowly decrease in width as Ti:Zr ratio decreases and the material
becomes more Zr(HPO 3 ) 2 like. This can be seen in the Table 4.10.
Table 4.10: Bangap Modification of Titanium doped Zirconium Phosphate
Dopant Level (Ti:Zr)
1:1
1:3
1:5
Bandgap [eV]
1.67
1.7
1.7
Titanium Gap-State Width [eV]
0.8
0.7
0.6
From Table 4.10, it is apparent that the width of the titanium gap states decreases
65
Z,-d
CA
2-
0E
-
0-4
-2
2
0
4
E-E, [eV]
(a) 1:1 doping of Ti:Zr
O-p
-4
0
2
4
E-Ef [eV]
(b) 1:3 doping of Ti:Zr
2-
4
-Ti
-d
ZT-d
-O-p
Pp
0
2-
0
V"
4
-2
2
0
4
E-Ef [eV]
(c) 1:5 doping of Ti:Zr
Figure 4-4: Titanium doping of Zr(HPO 3 ) 2
66
in a predictable way, yet the bandgap remains constant. In this way, the bandstructure of the zirconium phosphate can be controlled as long as the titanium gap states
remain present. As the level of titanium doping decreases, the overall structure behaves more like Zr(HPO 3 ) 2 , while still having a modified bandgap. This presents
a unique tool that can help with controlling any band alignment issues in the bulk
structure.
4.3.4
Layered Organic Zr(HPO 3 ) 2
As was mentioned previously, Zr(HPO 3 )2 readily forms benzene-ring interlinkers between layers, primarily used for ionic transport. The large bandgap of Zr(HPO 3 )2 makes
it impractical for typical photovoltaic materials and the optoelectronic properties of
Zr(HPO 3 ) 2 have not been studied in depth. It is a small jump to imagine a structure
where organic solar materials are used as the interlinking material instead of simple
benzene rings. In this way, a layered structure with optically-optimized organic interlinking molecules can be imagined and realized. In order to design this structure, it
is first worth exploring the experimental structure, comprised only of benzene rings.
This structure was shown in Figure 3-6.
Complete testing of this structure was performed, and an atom-projected DOS
was plotted as described before and is shown in Figure 4-5. Due to symmetry, the
unit cell had two layers of zirconium phosphate as well as two layers of dibenzene
to account for the directionality that the zirconium layer has. Because of this, this
calculation has the following chemical formula: (Zr(P0 3 ) 2 ) 2 Hl6 C 24 , representing a
rather large structure in DFT-space.
Figure 4-5 shows that there is no significant optical properties in the experimentally realized layered Zr(HPO 3)2 structure.
Inside of the benzene region, the
DFT-band gap is 2.58eV, making this region inefficient at capturing solar radiation.
Further, any exitonic generation will quickly recombine without leaving the organic
region due to the fact that there is no favorable energetic band alignment between
the two materials making up this heterojunction.
Additionally, dibenzene has no
directional dipole, preventing the separation of the exciton.
67
As such, this cannot
Zr-d
P-p
o-p
-C-p
C6#
-4
-2
0
2
4
E-Ef [eV]
Figure 4-5: Atom resolved density of states, spatially separated into each layer
be considered a solar material. However, very interesting material properties manifest when the polar molecules that were discussed in Chapter 3 are used in lieu of
dibenzene. This will be discussed in detail in the following section.
4.3.5
Layered Zr(HPO 3 )2 with Organic Photovoltaic
To design a more efficient BHJ material, the dibenzene layers were replaced with
organic materials used as dyes in dye-sensitized solar cells. The inorganic zirconium
phosphate structures are able to control the directonality of the organic interlacers
by acting as the backbone of the material. This system is shown in figure 4-6. By
replacing the dibenzene layers with truly organic photoacceptors, the bulk structure
will be shown to have optoelectrical properties necessary for efficient light capture.
As a first computation, the unfunctionalized MK2 molecule was inserted between
the inorganic phosphate layers. The DOS for the structure shown in Figure 4-6 is
shown in Figure 4-7. Comparing this result to the results in Figure 4-5 shows a large
shift in the valence band edges are shifted by approximately 0.82eV. This band shift
68
C
0
N
S
P
H
Zr
Figure 4-6: Computational structure of Zr(HPO 3 ) 2 bound to organic MK2
Table 4.11: Comparison between bulk structure and constitutive molecules.
MK2-Layer
CB [eV]
0.11
3.36
0.35
Shift [eV]
N/A
N/A
0.24
VB [eV]
-0.71
-0.95
-0.47
Shift [eV]
N/A
N/A
0.24
DFT-Gap [eV]
0.82
4.31
0.82
(Zr(HPO 3 ) 2 )Top
2.02
-1.34
-1.87
-0.92
3.89
(Zr(HPO 3 ) 2 )Bottom
1.28
-2.08
-2.71
-1.76
3.99
Structure
MK2-Molecule
(Zr(HPO 3 ) 2 )ulk
is due to the total dipole, or electric field, inherent to the polar nature of organic
MK2. Table 4.11 shows the energetic shifts in the bulk structure when bound with
unfunctionalized MK2, this trend is shown schematically in Figure 4-8.
The band shift observed is indicative of two trends: first it shows that the MK2
molecule's polarity can drive effects in the inorganic region (namely the band shifts),
second this dipole is an indication of how much energy the dipole is inducing into
the exciton. If this shift can be pushed higher, band alignment between the band
edges can be perfected. If a perfect band alignment can be achieved, where perfect
means there is no energetic gap or barrier at the interface of the organic region and
the inorganic layer, the Voc of the material can be optimized as the effective band
gap of the material is not hindered under a perfect band-alignment scenario.
69
o-p
-
N-p
S-p
-
0
C-p
Zr-d
P-p
0
0
0
-4
-3
-2
-1
1
0
2
3
4
E-Ef [eV]
Figure 4-7: Spatially resolved DOS of Zr(HPO 3 ) 2 -MK2.
ECB
Eg, Zr(HPO 3)2
EF E
Eg, Zr(HPO 3)2
Eg, Zr(HPO 3)2
E9, Org
E., Org
E9,
EVB
Un-polarized dibenzene system
Zr(HPO 3)2
Polar MK2 system
Figure 4-8: Schematic showing energetic shift due to polarized MK2
70
4.4
Scissor Operator
As described in Section 2.6, the scissor operator is a way in which to correct the
overall bandgap of a system. In this system, there are two scissor operators present,
the scissor operator for the inorganic layer, and the scissor operator of the organic
region. This was chosen in such a way as to describe the physics of both the organic
and inorganic system better.
The bandgap of Zr(HPO 3 ) 2 was not readily available in literature, however this
system is small enough to calculate the exact bandgap with accurate GW methods
as described in Section 2.6. From performing a fully converged GW calculation with
the TDDFT functionality of Vasp [26-29], an exact band gap of Zr(HPO 3 ) 2 was able
to be calculated. By performing this calculation, a bandgap of 6.9eV was found for
Zr(HPO 3 ) 2 . In this way, the scissor operator defined in Equation 2.18 is found to be
AEscissor-Inorg =
2.49eV.
(4.5)
For the organic MK2 molecule, the experimental bandgap was known to be 1.85eV
[51]. A "molecule in a box" calculation was performed on an isolated MK2 molecule
converged in such a way that the periodic boundary did not allow the adjacent MK2
molecules to interact with one another. The bandgap of this computation yielded a
DFT-bandgap of 0.663eV. The scissor operator is then found to be
JAEscissororg = 1.187eV.
(4.6)
All bandgaps repersented herein are underestimated by AEsciSSO. Since the effect of
the scissor operator is to merely move the conduction band up in energy by AEsci, 8 r,
conceptually and fundamentally all band caracteristics and shifts will remain constant
with the main effect being on the overall bandgap of the system. By using these scissor
operators, the exact band gap of the system can be calculated.
71
4.5
Organic MK2 Functionalization
One of the goals of this design is to maximize the force that will separate the exciton
in the organic region. To this end, organic MK2 has been functionalized with electron activating and deactivating groups. The directionality of the molecule allows a
coherent dipole to be created across molecule, providing the driving force that can
separate the bound hole and exciton pair. After establishing the ability to modify the
energy levels of the inorganic region, it becomes necessary to verify that this induces
a potential change sufficient to separate the exiton out of the organic region.
The organic MK2 molecule is extremely large and represents a computational limit
on the study of the bulk structures. The majority of these atoms are used to change
the bandgap of the material, and modify the electronic properties of the system.
However, this study will be further modifying the structure to see how functional
modification will change the electronic character of the molecule, implying that the
majority of these functional atoms may be unnecessary.
This is possible because
Thiophene is the main contributor to the exciton generation; the other atoms are
there to facilitate charge transfer. Further, polythiophene serves mainly to decrease
the bandgap of the material.
To increase the computational efficiency, the MK2
molecule was decreased in sized to the bare minimum amount of atoms to represent
the electronic character of the molecule. This is shown in figure 4-9.
What we are most interested in is how the band gap, and edges, change with the
number of total molecules and the total number of thiophene groups. The results
are shown below in Table 4.12. It is apparent that the electronic properties are not
significantly changed as the number of atoms are decreased. The main take-away is
that the molecule does behave as is to be expected, as the number of thiophene-units
decreased the bandgap also decreases. By limiting the number of atoms present in
MK2 system, the overall computational time can be decreased without significant loss
in overall electronic character of the system.
72
HN
N
HN
S
~~8
s
CN
S
ON
CN
-9
144 Atoms
66 Atoms
44 Atoms
Increasing Computational Time
Figure 4-9: Computational modifications of MK2 molecule
Table 4.12: Computational Optimization of MK2. Energies relative to Fermi Level
Structure
Experimental
MK2
MK2 without
alkyl chains
MK2 with one
# Thiophene Groups CBM [eV]
Band Gap [eV]
4
-0.35
0.38
0.73
4
-0.69
0.15
0.84
1
-1.04
0.43
1.47
thiophene group
4.5.1
VBM [eV]
Functionalized MK2
Functionalization of the organic region is the driving force behind the energetic shifts
observed in the Zr(HPO 3 ) 2 layers. Additionally, these functionalizing groups were
chosen to induce a dipole across the organic layer, creating a potential field strong
enough to separate the exiton. Ideally, functional groups that created this energetic
shift in the phosphate layers will also create the dipole necessary to separate the
exciton.
To calculate the induced electric field, and dipoles of the material, the same unit
cell as the bulk layered zirconium phosphate structure was used. However, only the
73
0
-
-0.2
-0.4-0.6-
-
-0.8
0
20
40
60
80
Length along z-direction [Bohr]
Figure 4-10: Windowed-averaged electrostatic potential of modified MK2
relaxed organic molecules occupied the unit cell. This allows the electric field across
the organic molecule to be observed directly.
In order to calculate the voltage drop across the material, a dipole correction was
induced into the calculation. This was applied far in the vacuum and is required to
guarantee no field from the periodic calculation. A windowed-average, with window
size of 2.5Bohr (the C-C distance), of the electrostatic potential was then taken [2].
This average is shown in Figure 4-10. The difference of the discontinuity that is seen
in the graph is the change in vacuum level, representing the total voltage drop as an
electron travels across the material.
This calculation was carried out over a wide variety of functional groups. In this
way, we can verify that the molecule has enough potential to separate the exciton
generated by the thiophene groups (0.18eV [30]). Through functionalization as shown
in Figure 4-11, both the voltage drop across the molecule as well as the band shifts in
the inorganic region can be controlled. The results of these calculations are shown in
Table 4.13. It is apparent that several of these have a dipole-potential large enough to
overcome this binding energy in the material. It is worth noting that MK2 without
74
0
S
\
/
OH
NH2
N
H
Functional Group I
Functional Group 2
Figure 4-11: Schematic showing functionalization of the organic MK2 molecule
Table 4.13: Voltage drop of functionalization
Molecule
MK2
MK2
MK2
MK2
MK2
BX-Thiophene
BX-Thiophene
PX-MK2
PX-MK2
TPD-MK2
TPD-MK2
Functional Group 1
Pyrimidine
Boron
Flourine
Flourine
Benzene
Pyrimidine
Benzene
Pyrimidine
Benzene
Pyrimidine
Functional Group 2
Boron
-
Total Voltage Drop [V]
0.102
0.194
0.418
0.160
0.115
0.582
0.119
0.00
0.123
0.223
0.173
functionalization is unable to separate the exciton by itself, so functionalization is
necessary.
By functionalizing the organic region, the voltage drop across this region can be
increased over the baseline voltage drop of MK2. It appears that the combination of
a pyridine ring actually lowers the overall voltage drop, while combining the material
with atoms with a variety of electronegativity shows an overall trend in the MK2
voltage drop. As an example, boron seemed to have the largest voltage drop, but also
of interest is how simply adding molecules such as BX, PX, or TPD can change the
voltage drop over a single thiophene. This functionalization is a way to control how
strongly the exciton is separated. Functionalization with simple atoms with varying
75
electron negativity can increase the overall potential drop, effectively increasing the
overall energetic efficiency of the material.
4.6
Conclusion
In this chapter several results were laid out. The pseudopotential testing results
were presented with these pseudopotentials being used throughout this work. The
experimental layered organic Zr(HPO 3 ) 2 structure was examined, but no optical properties were seen. However, by using organic MK2 and the inter-linker between the
Zr(HPO 3 ) 2 layers, it is easy to band shifts within the inorganic region become evident
and imply that this structure could be further modified to create a truly efficient organic heterojunction. By examining the total vacuum level change across the organic
molecules, organic MK2 needed to be functionalized in such a way to create a potential
drop necessary to separate the exciton. The electronic properties of Zr(HPO 3 ) 2 were
also modified by doping the structure with titanium. All of this knowledge will be
used to iterate on our design in the next chapter to alleviate these issues.
76
Chapter 5
Engineered Band Alignment
In Chapters 4 and 6, the material properties of a layered Zr(HPO 3 )2 solar material was
proposed and the electronic properties of the system were measured. Even though minor band shifts were observed, the energetic barrier in the MK2-Zr(HPO 3 )2 structure
are too large for efficient exciton separation., In this chapter, the band alignment will
be engineered through chemical modifications to the superstructure.
5.1
Functional Group Band Shifts
Calculation similar to those described in Chapter 4 were carried out for the molecules
with functional groups added as electron activating/deactivating groups. This was
implemented to explore the effects that the overall dipole induced by the molecule
dipole. Additionally, several molecules were sustituted as an alternative to thiophene.
These molecules were chosen for their differences in acceptor strengths (correlated
with the LUMO energy); specifically these were BX, PX, and TPD.
In general, by combining these structures with simple functional groups, it is
possible to significantly alter the relative energy levels of the zirconium phosphate
band edges in the two distinct inorganic layers. By enabling this control over band
alignment, it may be possible to design a nanostructured photovoltaic material with
optimal band alignment, thereby minimizing recombination and maximizing both
carrier extraction and Voc , leading to significant improvement in power efficiency
77
Table 5.1: Comparison between bulk structure and constitutive molecules
Base
Molecule
MK2
MK2
MK2
MK2
MK2
MK2
MK2
MK2
MK2
MK2
BX
BX
PX
PX
TPD
TPD
R1
R2
-
-
Pyrimidine
Pyridine
Pyridine
Pyridine
Pyrimidine
Pyrimidine
Flourine
Flourine
Boron
Benzene
Pyrimidine
Benzene
Pyrimidine
Benzene
Pyrimidine
-
OH
20H
OH
20H
-
Boron
-
Hole
Barrier
1.13
0.32
0.37
0.34
0.18
0.37
0.53
0.25
0.42
1.35
0.96
0.37
1.92
0.42
0.57
0.37
Electron
Barrier
0.86
0.64
0.37
0.44
1.95
0.50
0.18
0.62
0.86
0.59
1.97
2.37
2.11
2.09
1.28
1.63
relative to traditional BHJ photovoltaics.
In Table 5.1, the band shifts of the systems as well as their energetic barriers are
shown. The name of each molecule corresponds to a molecule represented in Appendix
A. Of most importance is the difference between the band edges between the organic
molecule and the phosphate layer. In Table 5.1 the hole barrier and electron barrier
represents the energetic barrier between the the organic layer and the inorganic layer
for the holes and electrons. This barrier prevents charges from leaving the organic
region, leading to recombination.
The selection of the functional groups were chosen to explore the effects that
electronegativity has on shifting the bands of the zirconium phosphate layer. In this
way, the dipole of the molecule can be controlled as well as separating the exiton.
The combination of these functional groups still present a large energetic barrier at
both the VB and the CB, yet it is apparent that by combining these functional groups
78
the energetic barrier can be controlled. Interactions with these functional groups and
the the Zr(HPO 3 ) 2 layers can also be contributing to these energetic barriers. These
barrier are still excessive and will prevent efficient carrier extraction, to be discussed
in the next chapter.
The spatial resolution of the conduction and valence conductions bands is indicative of how the exciton will be separated. The spatial resolution of these bands are
plotted below in Figure 5-1. The location of the conduction band and valence band
edges are indicative of how the charge is being separated in the material. This separation means that the exciton is immediately separated into locations that are not
spatially near, encouraging the dissociation of the exciton. The energetic barriers
described in Table 4.11 still prevent the charge from fully leaving the organic region,
however, with other modifications to the structure, these barriers can be reduced and
eliminated. Once this is eliminated, the dipole across the molecule will allow the the
charge to freely leave the organic region, as indicated by the spatial separation of the
band edges.
5.2
Titanium Doping Band Alignment
The large bandgap of the zirconium phosphate layer makes band-alignment with
the small band-gap organic semiconductor very difficult. As described in Section
4.3.2, titanium phosphate also forms in a very similar structure as zirconium phosphate. However, this structure has not been experimentally explored to form the
same layered structure as Zr(HPO 3 )2 with organic structures. To this end, the bulk
Zr(HPO 3 )2 structure was doped with titanium with the following chemcial formula
in the phosphate layer TiZr 3 (PO 3 )s. This level of doping (1:3) is extremely high and
was done because of the size of the computational structure. This dopant level was
chosen to examine the interaction of the titanium and zirconium without needing an
excessive amount of computational resources.
The bi-layer nature of the computation allowed the other layer to remain as undoped zirconium layer. This allows the effect of the disperate layers to be observed.
79
0r
(a) Spatially resolved Valence Band
(b) Conduction Band
Figure 5-1: Spatial resolution of the Valence and Conduction Band Edges
80
Figure 5-2: Structural form of Titanium Doped Zirconium Phosphate
Between the layers, MK2 functionalized with pyrimidine is used. This structure is
shown in Figure 5-2. This allows direct comparison to the values in Table 4.11.
The DOS of this calculation is shown below in Figure 5-3. From this figure, the
titanium doping significantly modifies the conduction band character of the doped
zirconium layer; addressing the band alignment through chemical modification of
the phosphate layer. This is particularly useful because of the myraid of chemical
species that form the same phosphate layer. In this structure, the doped layer has
a DFT-bandgap of 2.63eV. Additionally, while this layer addresses the conduction
band alignment issue, there still represents a energetic barrier of 0.61eV in the valence
band. It was not explored in this study, but by combining other functional groups, the
energetic barrier at the valence band can be addressed allowing the titanium doping
to correct the conduction band alignment.
5.3
Corrected Band Alignment
As was mentioned in Section 4.4, both the organic region and the inorganic region
have bandgaps that are not the true gaps that would be observed in experiment. In
this way, by applying a scissor operator to both region's conduction band energies,
the correct band gap can be estimated. The exact nature of this shift can be seen
schematically in Figure 5-4.
81
Ti-d
Zr-d
P-p
-
--
O-p
S-p
-
C-p
r-
0n
-4
-2
0
E - E [eV]
2
4
Figure 5-3: Titanium Doping of Zirconium Phosphate Layer
EDFT+Scissornor(
EDFT+Scissor
EDFT
EDFT+Scissorg
"EDFT.
EDFT
Figure 5-4: Scissor operator application to the bulk band structure
82
Table 5.2: Corrected bandgap of system by applying two scissor operators
Structure
Accurate Bandgap
AEscss[eV]
Zr(HPO 3 ) 2
6.9
4.77
1.85
2.56
2.44
3.15
2.49
1.97
1.187
1.83
1.33
2.07
Ti(HPO 4 ) 2
MK2
BX
PX
TPD
In the inorganic region, the scissor operator was used as a constant 2.49eV (Equation 4.6). In the organic region, the scissor operator varied based on the character of the light-absorbing molecule.
For MK2, the scissor operator was used as
1.187eV(Equation 4.5). For BX, PX, and TPD, the scissor operator was determined
from the computational work performed by K6se [30], these operators are given as
Eso,
= 1.83eV
ESPLSO, = 1.33eV
ESTP,
(5.1)
= 2.07eV
This was done for select systems that had large voltage drops across the system. The
DFT gap, the experimental (or GW band gap), and the scissor operator are shown
in Table 5.2. These shifts are used to accurately calculate the bandgap of the system
as well as the energetic barriers between the organic and inorganic layers. Even with
these shifts, there are still energetic barriers between the organic and inorganic layers.
These are represented in Table 5.3.
Since the overall band alignment and structure is now exactly known, the exact
barriers can be examined. The differences in the scissor operators actually make
the energetic barriers greater than with the underestimated DFT bandgaps. The
energetic barriers at both of the inorganic interfaces means that the total bandgap
is in the organic region. This bandgap will be used to estimate the overall efficiency
of the material, to be discussed later. This information combined with the voltage
drops represented in Table 4.13, can be used to further estimate the efficiency of the
83
Table 5.3: Corrected bandgap of system by applying two scissor operators
Organic Molecule
MK2
MK2
MK2
BX
BX
PX
PX
TPD
TPD
MK2
FG1
CB-Difference [eV]
1.95
Pyrimidine
1.89
Flourine
1.92
Benzene
2.63
Pyrimidine
2.61
Benzene
3.31
Pyrimidine
3.35
Benzene
1.78
Pyrimidine
1.95
Titanium Doped
Pyrimidine
0.37
Bandgap [eV]
2.78
2.54
2.99
2.84
2.82
2.60
1.95
3.77
2.78
2.74
material.
Ideally, there would be no energetic barrier at the interface between the organic
and inorganic interfaces. These gaps imply that the energetic shift in the inorganic
region is not due entirely to the voltage drop across the organic region. Other reasons
for these barriers would be due to an interface dipole or charge sharing between the
atoms at the surfaces.
Further exploration would be required to develop a more
accurate model of the driving effects of these barriers.
5.4
Conclusions
In this chapter, a way to energetically shift the conduction band and valence bands
of zirconium phosphate has been discovered. By combining functional groups with
the organic molecule, significant band shifts in the inorganic region can be observed.
These functional groups were only able to shift the energetic levels in Zr(HPO 3 )2 to a
certain extent, and other methods for achieving perfect band alignment were explored.
By doping the Zr(HPO 3 )2 layer with titanium, the band alignment of the conduction
band was able to be addressed, and the barrier for electrons was removed. However,
the hole barrier was still present. Further exploration of functional groups is necessary
84
to fully remove these barriers as they represent the largest obstacle to an efficient
material.
85
86
Chapter 6
Transport Properties
Key to an efficient solar cell material is the overall mobility of the structure. Given the
low mobility of organic photovoltaics, the designed BHJ presented in Chapters 4 and
5 addressed the issue of band alignment in the composite material. In this section
the mobility of the Zr(HPO 3 )2 layer will be calculated and quantified.
Further, a
piecewise model for the system mobility will be presented.
6.1
Mobility Derivation
The mobility of charge carriers is an excited-state material property and thus is
time-dependent, a property that cannot be captured dirctly with DFT because it
solves the time-independent Schr6dinger equation. Nevertheless, mobility is directly
related to the band structure of the material and can be well approximated in the
limit of a perfect crystal by determining the electronic properties as a function of
deformation.
Perfect crystals have the highest mobility possible as experimental
systems have crystallographic defects that serve to lower the overall mobility by acting
as scattering centers in the material.
Due to the relatively low concentratioi of
defects in real materials, as well as the large number of defect combinations within
the material, realistic inclusion of defects in DFT requires both large system sizes
and numerous calculations, defects are normally computationally intractable. The
calculation described in this section are an upper limit to the mobility of any system.
87
6.1.1
Governing Equations
In this section, the derivation of the mobility calculation formulated Zhigang Shuai,
et al, [48, p.67-88] will be described. By following this derivation, the mobility can
be calculated using very little computational time as the final form of the mobility
relies on energetic properties.
By starting with the Boltzmann's tranport equation, all properties of the mobility
can be determined. We start by assuming the distribution function f(f, k, t) describes
how a particle in phase space evolves in time. Taking the time derivative of this gives
df
dt
Of
t
Of dr Of dk
= -- + +
Or dt Ok dt
.(6.1)
this can be simplified using the knowledge that df/dt = v is the packet velocity and
the external force acting on the particle is h , yielding:
(6.2)
(
df
af Of
Of F(r)
.
+
_v(k)
+
=
dt
Ot
Or
Ok
This is now treated as the scattering time of the system, if it is assumed that the
external force is purely mechanical in nature. This equation is still complicated to
deal with because of the time-dependent nature of the system. This can be further
simplified by rewriting the above equation if the distribution function is a FermiDirac distribution at equilibrium, fo = 1/{exp{[E(k) - EF]/kBT} + 1}. For charge
transport, and group velocity, this equation becomes
Of
a;:
at sct
-eoEv(k)
Ofo
09E
,
(6.3)
where E is the electric field, and afo/E represents the derivative of the Fermi-Dirac
distribution with respect to the energetic bands, E.
In order to get this to a tractable equation, several approximations must be made:
the relaxation time approximation and the deformation potential. Using the relaxation time approximation this derivative is converted into a probability distribution
relating the probability to transition between states k and k'. By making this approxi88
mation, the final mobility equation can be derived as performed by Shuai [48, p.67-88].
This is shown below in Equation 6.4.
f r(i, k)v'(i, k)exp[cFc(k)/kBT]dk
E
e(h)
-
eo
iE(CB)(VB)
kT
a
E
f exp[-ej(k)/kBT]dk
iE(CB)(VB)
This equation holds for both the hole and electronic mobility of the material,
where the required properties that need to determined are the relaxation time, T(k),
the group velocity va(i, k), and the band energy, ei(k) (the minimum or maximum
in the band energies for the CB and VB, respectively). The work by Long [34] made
calculation of the group velocity trivial from the band energies. To this end, once the
band structure is known, both the band energy and the group velocity is known from
va(i, k) = V" (k)
(6.5)
h
and Shuai was able to show that the relaxation time could then be approximated by
1
v(k') eEl~
27rkBTE2
- c(k )]
- Vkk
T(k)1
(6.6)
This introduces two new constants, E1 and cq, the deformation potential and the
electric constant. However, this form of the mobility and the relaxation time requires
an extremely dense kpoint mesh. As an example, Long used a 64x64 kpoint mesh
in the plane of a graphene sheet. This quickly becomes computationally difficult for
complex systems that would required many more kpoints to get the physics of the
system correct.
To alleviate the reliance on a dense k-point mesh, one last approximation must
be made.
The effective mass approximation allows the mobility to be solved by
only knowing the band diagram of a system. Using a parabolic band argument, the
dispersion relation can be written as
(k - ko) 2
c(k) = co + (
89
(6.7)
where the given approximation comes from m* = h2 (d2 E/dk2 )-', giving an "effective"
mass of the electrons and holes. As the equation indicates, these masses come from
the curvature of the conduction and valence bands, respectively. This allows us to
write v(k) = hk/m*, simplifying the relaxation time to
1
r(k)
2E2kBTm*
kh cq
1
3(6.8)
Due to the anisotropic nature of the hybrid systems studied in this work, we
compute the 2D mobility fo these systems (specifically the mobility within the plane
of the phosphate structures). The final equation for the 2D mobility is then given by.
__2eohocq
1-2D
-
2EOkB3
q'(6.9)
2
2kBTm*
2E
Now that a concise equation has been developed to describe the mobility, only
three constants need to be calculated, the deformation potential (E2), the elastic
constant (cq) and the effective mass (m*).
6.1.2
DFT Calculations
Within the effective mass approximation, the three required constants come from the
band structure alone. In this section, we will describe the physical meaning of these
constants as well as how they are computed.
The deformation potential is given as
El =
AV.
,
Al/10
(6.10)
where this fitting comes from straining the unit cell. AV is the change in energy
of the conduction band (electrons) or valence band (holes) and Al/lo is simply the
strain along one of the lattice directions. By straining the unit cell at various different
increments (in general, at 1% increments from +3%), a curve fit to this strain can
give the deformation potential.
90
The elastic constant is given as
_cQI'Ll'\
- - =
VO
2
lo
(6.11)
,
'
AE
where V is the original unit cell volume, lo is the lattice constant along the a direction,
AE is the change in total energy for the strained unit cell, and Al/10 is the strain
along the a direction. It is worth noting that V takes a different character for either
1D, 2D, or 3D mobility. In this work the 2D mobility was calculated, so V is actually
the cross sectional area perpendicular to the required axis.
The effective mass is defined as
m* = h2( d 2 -.
dk
(6.12)
The only term of computational interest in this equation is (d 2 E/dk?)-1, which simply
is the inverse of the curvature of the conduction band (electrons) or the valence band
(holes).
In this way, all three can be calculated by running approximately seven calculations at different levels of strain. All of these can be obtained provided that the whole
structure is energetically relaxed to the ground state, both for atomic positions as
well as lattice constant(s).
6.2
Zr(HPO 3 )2 Mobility
In the previous section, the mobility was derived in such a way that it could be
calculated with DFT. In this section, the mobility of Zr(HPO 3 )2 will be analytically
calculated and will be shown to be much higher than that of traditional organic heterojunction materials; to increase the overall efficiency of the material, the phosphate
simply needs to have a mobility greater than 10 4 cm 2 /Vs, the order of charge mo-
bility in organic semiconductors. A literature review did not yield any information
about the mobility, or other electronic properties, of the transition metal phosphate
compounds studied in this work. This is principally due to the fact that these ma91
terials have such large bandgaps; thus this is the first study aiming to quantify the
charge carrier mobility within these materials.
In Equation 6.9, several values have to be calculated.
Namely
cq,
the electric
constant, E1 , the deformation potential, and m*, the effective mass. Each of these
values have a value for both the holes and electrons. All of these constants come from
the band diagram of the structure. For zirconium phosphate, this was calculated and
was found to be highly directionally dependent. The bandstructure was calculated
along, the reciprocal a*, b*, and c* directions. Along a* and b*, the band diagram
was plotted from F to M. Along the c* axis the band diagram was plotted from F to
A and was found to be completely flat. These points are the high symmetry points
inside the bruillion zone, and are generally accepted to be points of special interest.
These points were chosen because of their directionality relative to the real-space
directions as we primarily are interested in the mobility normal to the phosphate
sheet as well as along the sheet. These direction are shown in Figure 6-1. The band
diagram was then calculated along these directions and is shown in Figure 6-2. The
mobility of the structure is then only calculated in the a* and b* directions, however,
due to symmetry, these results are identical.
As Figure 6-2 shows, the curvature of the bands is zero along the c* direction
(the direction normal to the Zr(HPO 3 )2 layers, where van der Waals interactions
dominate), leading to an infinite effective mass and a mobility of exactly zero along
this direction. In contrast, the bands in the plane of the 2D sheets have a finite
curvature, and therefore a meaningful mobility along this direction.
By fitting a
curve to this raw data, a curvature can be found by taking the second-derivative of
the fitting function. By using Equation 6.12, this curvature then leads to the the
effective mass.
To calculate the elastic constant and the deformation potential and eleastic constant, the unit cell is now strained along the a* direction (only required along this
direction because of symmetry as well as the fact that the mobility along the c*
direction is identically zero). Increments of 1% are induced along this unit cell.
In this way, two curves are obtained, as dictated by Equations 6.10 and 6.11. For
92
k
M
b*
iA
j
.b
r
C*
-A
-M
(b) Reciprocal space showing the direction
(a) Reciprocal space showing the direction along c*
along a*
Figure 6-1: Directions of reciprocal space band structure
6
6
4
4
2
0
-2-
rI2 -2
-4
-4-
-6
-6
-M
-1
M
K [1/A]
(a) Band Diagram from F to
A
K [1/A]
(b) Band Diagram from F to
M
Figure 6-2: Band Diagram of Zr(HPO 3 ) 2 along reciprocal axes
93
A
0.015
I
'
U
-
Calculated Points
Fit(12.316x 0.0065x-3e-5)
0.010 --
W
0.005-
-0.04
-0.02
0
0.02
0.04
Al/ 0
Figure 6-3: Plot of curve required to fit the elastic constant
the elastic constnat, only one curve needs to be calculated, this is shown in Figure
It is worth noting that this is acutally plotting 2AE/Vo = (Al/1b)
6-3.
2
, to make
curve fitting easier. By fitting the curve with a parabola, the elastic constant can
be calculated as the coefficient to the parabolic curve that fits this data of the form
y
=
Ejx 2 +C 1 x+Cox0 . The additional constants are indicative of the error associated
with using the parabolic band approximation.
In the same way, the deformation potential can be calculated. This, however,
requires knowledge of both the conduction band and valence band. Equation 6.10
dictates that the conduction band minimum is required for electrons and the valence
band maximum for holes. These curves are plotted in 6-4. By fitting a linear curve to
these curves, the deformation potential for both the holes and electrons can be calculated. The absolute value of this fitting will be used to guarantee a positive mobility
value. Now knowing the effective mass, the elastic constant, and the deformation
potential, the mobility can be calculated using Equation 6.9.
Each of these constants were then calculated for various "layer thicknesses." This
simply means that the bulk structure properties were calculated as well as a structure
with a vaccuum (20A) on either side of the slab.
94
A combination of bulk layers
0.3
N
-
0
-
0.2-
-
BM
VBM
CBM Fit (-0.9691x+0.0022)
VBM Fit (6.03x+0.0072)
0.1-
-
0
-0.1-
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
A1I1
Figure 6-4: Plot of curves required to calculate the deformation potential.
Table 6.1: Electron and hole transport properties of Zr(HPO3 )2
# Layers cq [eV/A2]
1
2
3
4
5
Bulk
13.18
12.32
12.32
12.35
13.06
12.87
E1 (e) [eV]
-2.92
-0.97
-0.79
1.11
0.52
3.32
E1 (h)
3.71
6.03
6.25
8.17
5.8
10.1251
m*' [mo]
0.95
0.81
0.77
0.78
0.83
0.75
m*' [mo]
1.43
1.49
1.05
1.45
1.34
1.45
were then combined, from a monolayer until five total repeats. The results of these
calculations are shown in Table 6.1.
The inverse-square relation of the effective mass of the electrons/holes are of the
most important property to the electronic property to dictate the mobility of the
holes and electrons in the material. The effective mass for each layer is plotted in
Figure 6-5, showing that the effective mass does not change significantly from a bulk
structure to a monolayer. Between a monolayer and a bulk-structure, the hole-mass
grew by 27.5% while the electron mass only shrunk by 1.78% between a bulk and
mololayer. This implies that a monolayer can be used without sacrificing a significant
95
1.6
1
-
Y-V Electron Mass
-V Hole Mass
1.2-
-
I
0.81
2
4
3
5
Bulk
# Layers
Figure 6-5: Effective Mass vs. Layers of Zr(HPO 3 )2
amount of performance over the bulk structure properties.
By combining the knowledge of Table 6.1 and Figure 6-5, it is then immediately
possible to use Equation 6.9 to calculate the electron and hole mobility in the plane of
the zirconium phosphate. Carrying this out it is found to be 107cm 2 /Vs for holes and
388cm 2 /Vs for electrons, a significant improvement over the total mobility of organic
semiconductors. The comparison between common semiconductors are shown in Table 6.2. It is important to note that the comparison should be between Zr(HPO 3 ) 2 and
traditional organic semiconductors. While traditional inorganic semiconductors such
as silicon and gallium-arsenide have much higher mobilities, they serve a different purpose than is desired of Zr(HPO 3 )2 . By having a mobility several orders of magnitude
greater than the organic semiconductors, the overall mobility of a traditional BHJ
material can be increased, allowing for an overall higher efficiency in the material.
96
Table 6.2: Comparison of mobilities between common semiconductors
Material
Electron Mobility (cm 2 /Vs)
Hole Mobility (cm 2 /Vs)
Monolayer Zr(HPO 3 )2
Silicon
Ga-As
P3HT-PEDOT
388
1400
8500
107
450
400
10 4
10-4
6.3
Conclusion
In this chapter, calculation of the mobility with existing DFT tools was derived. By
using this derivation, it was found that Zr(HPO 3 )2 has a significantly higher mobility
than organic semiconductors. By using this material with organic semiconductors, an
overall increase in the material mobility can be realized. This presents a key finding
to hybrid organic-inorganic BHJ operations and opens the door for higher efficiency
in such materials.
97
98
Chapter 7
Conclusions
In this thesis a model for a highly efficient solar material was presented. The functional groups that were explored were shown to both contribute to the overall voltage
drop across the organic molecule as well as shift the energetic levels of both the CB
and VB inside of Zr(HPO 3 )2 . Further, the high mobility in the Zr(HPO 3 )2 layer was
shown to be significantly higher than that of the organic layer.
7.1
Efficiency Estimate
The energetic barriers due to the band alignment will severely limit the overall efficiency of the material. Elimination of these barriers would make the system work in
an ideal way, but it would be prudent to estimate the overall efficiency of the devices
as they currently stand. There are three main ways in which the system will be
limited in terms of the overall efficiency: the Shockley-Queisser (SQ) limit, the area
fraction, and the efficiency associated with the energetic barrier.
The SQ barrier is the max efficiency associated with the bandgap.
This was
described in Section 1.1.2. This is a measure of the max efficiency that the material
can achieve with a given bandgap. In organic photovoltaics such as the materials
observed in this study, recombination would have to be taken into account to describe
the actual efficiency. The SQ limit was originally developed for inorganic material,
and hence no recombination is included in the use of it. In this way, the SQ limit is
99
Absorbs Light
Absorbs Light
Figure 7-1: Schematic representing the idea of the area fraction
an upper limit.
The area fraction needs to be taken into account because the phosphate materials
do not interact with the solar spectrum.
AF
=
Area interacting with solar spectrum
Total Area
This is shown schematically in Figure 7-1.
(7.1)
Because of the length of the organic
semiconductors used in this material, the area fraction will be quite high
(~ 90%).
The largest efficiency drop will be due to the energetic barriers at both the conduction and valence bands. In classical physics, energy barriers would prevent particles
from moving over this, however, electrons and holes behave quantum-mechanically.
One of the benefits of this is that the electrons can tunnel through an energetic barrier
that it does not have the energy to overcome. This is a well known effect and can
be found in any introductory physics book. Schematically, this tunneling is shown
in Figure 7-2. If an electron can tunnel through this barrier, it would maintain the
same energy that was incident on the barrier, but the wavelength of its wave-function
would be decreased. In this way, the transmission coefficient, T, describes the probability that the electron can tunnel through the barrier at either interface. T can be
described functionally as
T=
e- 2 k2L
where k 2 = f2m(Uo - E)/h and L is the length of the barrier. In k 2 ,
(7.2)
m
is the rest
mass of an electron, Uo is the energetic barrier, and E is the energy the electron
(hole) posses. The transmission coefficient is then used as a proxy to the efficiency
100
U Barrier
E Electron
EElectron
with probability T
j
Energy
X=0
x=L
Figure 7-2: Schematic for electron (hole) tunneling
of an electron leaving the organic region. The energy (E) of the electron is taken
to be and electron in the electric field created by the organic region and the length
of the barrier L is taken to be the length of a benzene ring, a characteristic length
in the system. This length was chosen because it is a good characteristic length of
the whole system. Because of the fact that there exists two energetic barriers in the
systems studies (a barrier for holes and a barrier for electrons), the larger of the two
barriers is taken as a Uo.
By using this model, the total efficiency of the system will be
-
WTotal ~~ TSQ * T7AF * 77Tunneling
(7-3)
Each of these values are calculated for the the materials listed is Table 5.3. The
results of this efficiency study is shown in Table 7.1. From this table is is apparent
that these devices are not the highly efficient materials that are to be desired. It can
101
Table 7.1: Max efficiencies of select systems examined
Molecule
MK2
MK2
MK2
BX
BX
PX
PX
TPD
TPD
FG1
-
Pyrimidine
Flourine
Benzene
Pyrimidine
Benzene
Pyrimidine
Benzene
Pyrimidine
Max-Barrier
2.83
1.89
1.92
3.27
3.25
3.45
3.49
2.66
2.83
Bandgap [eV]
1.90
2.54
2.99
2.20
2.17
2.65
1.80
2.89
1.90
77SQ
12.39%
11.59%
4.52%
18.51%
19.03%
22.35%
27.09%
6.06%
25.03%
Wunneling
7lTotal
1.81%
2.40%
2.25%
0.92%
0.20%
0.25%
0.09%
0.15%
0.92%
0.50%
0.53%
1.15%
0.95%
0.11%
0.10%
0.13%
0.06%
0.21%
be seen that this is due to the extremely low tunneling coefficient due to the large
energetic gap between the organic and inorganic material.
7.2
Future Outlook
Even though the materials that have been calculated have been shown to have rather
low efficiencies, shown in Table 7.1, many positive outcomes can be gained from the
completion of this work. Primarily, the ability to control the energetic levels in the
phosphate group creates the ability to accurately control these levels, first computationally observed in this work. The phosphate groups themselves have much higher
mobilities than the organic materials, a property that has not been studied thus far.
The ability of Zr(HPO 3 ) 2 to form layered structures with organic molecules implies
that it can be used as a hybrid organic-inorganic BHJ material. The directionality
Zr(HPO 3 ) 2 creates for the organic molecules creates the ability the ability for traditional semiconducting organic materials to become highly ordered, utilizing their
intrinsic dipole to effectively separate the exciton. Several materials were shown to
have intrinsic dipoles great enough to effectively separate the generated exciton.
The main limiter in the estimated efficiencies of these materials is the energetic
barrier at each interface. The exact nature of this barrier is not precisely known,
but it could be due to charge transfer at the interface between the two materials,
102
or due to an interface dipole at this interface. However, the ability to control the
energetic properties of these phosphate layers via organic modification has many
implications to the organic BHJ designs as well other novel material designs. Further,
the order layered phosphate groups create a way to make highly ordered BHJ designs,
allowing for the organic molecules to use a designed dipole to overcome the exciton
binding energy. Further optoelectrical properties can be imagines from such functional
materials as optomechanical actuators as well as tunable-bandgap materials.
Utilizing this design and the results contained within this work, it will be possible
to engineer band alignment between the organic and inorganic material. The ability
to control the properties of the layered inorganic phosphates through organic functionalization could have wide implications. Fixing the energetic barriers present in
the material is the primary issue that needs to be addressed. Elimination of these
barriers would create a material that would be highly efficient photovoltaic material.
This would realize a cheap, solution-processable, organic photovoltaic material that
would open the world to abundant solar-electricity.
103
104
Appendix A
Molecule Figures
Because of the large number of combination that are present in this work, common
names were used extensively used throughout this work. In this appendix the chemical
structure of each molecule is represented in concise form. These figures represent the
organic materials that were used throughout this work to modify the bands of the
Zr(HPO 3 ) 2 layers. These molecules were placed between layers of Zr(HPO 3 ) 2 as well
as titanium-doped Zr(HPO 3 )2 and their electronic properties were measured.
105
HN
H 2N
0
0
Figure A-i: MK2 with Benzene interlinker
106
HN
HN
s
S
H 2N
H 2N
o
0
0
0
N
N
N
(b) MK2-Pyrimidine
(a) MK2-Pyridine
Figure A-2: Chemical structures of MK2- Pyridine and Pyrimidine
107
OH
OH
HN
HN
H2N
NH 2
S
S
H 2N
O
0
NN
0
N
(a) MK2-Pyridine-OH
(b) MK2-Pyrimidine-OH
Figure A-3: Chemical structures of MK2- Pyridine and Pyrimidine functionalized
with an OH group
108
OH
HO
OH
HO
HN
HN
S
S
NH 2
0
NH 2
0
0
N
N
(a) MK2-Pyridine-(OH) 2
0
N
(b) MK2-Pyrimidine-(OH)
2
Figure A-4: Chemical structures of MK2- Pyridine and Pyrimidine functionalized
with two OH groups
109
HN
HN
S
NH 2
0
H2B
NH 2
0
o
8H 2
F
(a) MK2 with Boron
0
F
(b) MK2 with Flourine
Figure A-5: Chemical structures of Boron and Flourine functionalized MK2
110
H 2B
BH2
HN
NH 2
0
0
F
F
Figure A-6: MK2 with both Fluorine and Boron
HN
HN
N
N
\I
N
0
N
N
-~N
(b) BX with Pyrimidine interlinker
(a) BX with Benzene interlinker
Figure A-7: BX shown with Benzene and Pyrimidine interlinkers
111
HN
HN
N)N
NMNN
NM.
N
N
NN
N
(a) PX with Benzene interlinker
(b) PX with Pyrimidine interlinker
Figure A-8: PX shown with Benzene and Pyrimidine interlinkers
HN
HMN
0
0
N
-
N
-
0
0
NH2
o
NH2
0
0
0
N
(a) TPD with Benzene interlinker
N
(b) TPD with Pyrimidine interlinker
Figure A-9: TPD shown with Benzene and Pyrimidine interlinkers
112
Bibliography
[1] Giulio Alberti, Mario Casciola, Umberto Costantino, and Riccardo Vivani. Layered and pillared metal(iv) phosphates and phosphonates. Advanced Materials,
8(4):291-303, 1996.
[2] Alfonso Baldereschi, Stefano Baroni, and Raffaele Resta. Band offsets in latticematched heterojunctions: A model and first-principles calculations for gaas/alas.
Phys. Rev. Lett., 61:734-737, Aug 1988.
[3] Stefano Baroni and Raffaele Resta. ii ab initioi/iZ calculation of the macroscopic
dielectric constant in silicon. Phys. Rev. B, 33:7017-7021, May 1986.
[4] Serge Beaupre and Mario Leclerc. Pcdtbt: en route for low cost plastic solar
cells. J. Mater. Chem. A, 1:11097-11105, 2013.
[5] J. L. Bredas, R. Silbey, D. S. Boudreaux, and R. R. Chance. Chain-length
dependence of electronic and electrochemical properties of conjugated systems:
polyacetylene, polyphenylene, polythiophene, and polypyrrole. Journal of the
American Chemical Society, 105(22):6555-6559, 1983.
[6] Jr. Chase, M.W. NIST-JANAF Themochemical Tables, Fourth Edition J. Phys.
Chem. Ref. Data,. Monograph 9, 1998.
[7] William Chockley and Hans J. Queisser. Detailed balance limit of efficiency of
p-n junction solar cells. Journal of Applied Physics, 32:510-519, 2002.
[8] A. Clearfield and J.A. Stynes. The preparation of crystalline zirconium phosphate
and some observations on its ion exchange behaviour. Journal of Inorganic and
Nuclear Chemistry, 26(1):117 - 129, 1964.
[9] Abraham Clearfield and Zhike Wang. Organically pillared microporous zirconium
phosphonates. J. Chem. Soc., Dalton Trans., pages 2937-2947, 2002.
[10] Albert Einstein. Concerning an heuristic point of view toward the emission and
transformation of light. Annals of Physics, 17:132-148, 1905.
[11] D.E. Elliot. Density Functional Theory of Molecules, Clusters, and Solids.
Springer, 1995.
113
[12] J. D. Fast. The allotropic transformation of hafnium and a tentative equilibrium
diagram of the system zirconiumhafnium. Journal of Applied Physics, 23(3):350351, 1952.
[13] J. Frenkel. On the transformation of light into heat in solids. i. Phys. Rev.,
37:17-44, Jan 1931.
[14] Paolo Giannozzi, Stefano Baroni, Nicola Bonini, Matteo Calandra, Roberto Car,
Carlo Cavazzoni, Davide Ceresoli, Guido L Chiarotti, Matteo Cococcioni, Ismaila
Dabo, Andrea Dal Corso, Stefano de Gironcoli, Stefano Fabris, Guido Fratesi,
Ralph Gebauer, Uwe Gerstmann, Christos Gougoussis, Anton Kokalj, Michele
Lazzeri, Layla Martin-Samos, Nicola Marzari, Francesco Mauri, Riccardo -Mazzarello, Stefano Paolini, Alfredo Pasquarello, Lorenzo Paulatto, Carlo Sbraccia,
Sandro Scandolo, Gabriele Sclauzero, Ari P Seitsonen, Alexander Smogunov,
Paolo Umari, and Renata M Wentzcovitch. Quantum espresso: a modular and
open-source software project for quantum simulations of materials. Journal of
Physics: Condensed Matter, 21(39):395502 (19pp), 2009.
[15] Stefan Grimme. Accurate description of van der waals complexes by density functional theory including empirical corrections. Journal of Computational Chemistry, 25(12):1463-1473, 2004.
[16] Stefan Grimme. Semiempirical gga-type density functional constructed with
a long-range dispersion correction.
Journal of Computational Chemistry,
27(15):1787-1799, 2006.
[17] Lars Hedin. New method for calculating the one-particle green's function with
application to the electron-gas problem. Phys. Rev., 139:A796-A823, Aug 1965.
[18] Gerhard Herzberg. Molecular Spectra and Molecular Structure: Electronic Spectra and Electronic Structure of Polyatomic Molecules. Krieger Pub Co, 1991.
[19] P. Hohenberg and W. Kohn. Inhomogeneous electron gas. Phys. Rev., 136:B864B871, Nov 1964.
[20] K.P. Huber and G. Herzberg. Molecular Spectra and Molecular Structure. IV.
Constants of Diatomic Molecules. Van Nostrand Reinhold Company, 1979.
[21] K. Hudon, T. Merrigan, J. Burch, and J. Maguire. Cost solar water heating
research and development roadmap. NREL Report (NREL/TP-5500-54793),
2012.
[22] R. 0. Jones and 0. Gunnarsson. The density functional formalism, its applications and prospects. Rev. Mod. Phys., 61:689-746, Jul 1989.
[23] E. A. Katz, D. Faiman, S. M. Tuladhar, J. M. Kroon, M. M. Wienk, T. Fromherz,
F. Padinger, C. J. Brabec, and N. S. Sariciftci. Temperature dependence for the
photovoltaic device parameters of polymer-fullerene solar cells under operating
conditions. Journal of Applied Physics, 90(10):5343-5350, 2001.
114
[24] Aung Ko Ko Kyaw, Dong Hwan Wang, Hsin-Rong Tseng, Jie Zhang,
Guillermo C. Bazan, and Alan J. Heeger. Electron and hole mobility in solutionprocessed small molecule-fullerene blend: Dependence on the fullerene content.
Applied Physics Letters, 102(16):-, 2013.
[25] W. Kohn and L. J. Sham. Self-consistent equations including exchange and
correlation effects. Phys. Rev., 140:A1133-A1138, Nov 1965.
[26] G. Kresse and J. Furthmiiller. Efficient iterative schemes for ab initio totalenergy calculations using a plane-wave basis set. Phys. Rev. B, 54:11169-11186,
Oct 1996.
[27] G. Kresse and J. Furthmller. Efficiency of ab-initio total energy calculations
for metals and semiconductors using a plane-wave basis set. Computational
Materials Science, 6(1):15 - 50, 1996.
[28] G. Kresse and J. Hafner. Ab initio molecular dynamics for liquid metals. Phys.
Rev. B, 47:558-561, Jan 1993.
[29] G. Kresse and J. Hafner. Ab initio molecular-dynamics simulation of the
liquid-metalamorphous-semiconductor transition in germanium. Phys. Rev. B,
49:14251-14269, May 1994.
[30] Muhammet E. Kse. Evaluation of acceptor strength in thiophene coupled donoracceptor chromophores for optimal design of organic photovoltaic materials. The
Journal of Physical Chemistry A, 116(51):12503-12509, 2012.
[31] David C. Langreth and M. J. Mehl. Beyond the local-density approximation in
calculations of ground-state electronic properties. Phys. Rev. B, 28:1809-1834,
Aug 1983.
[32] David C. Langreth and John P. Perdew. Theory of nonuniform electronic systems. i. analysis of the gradient approximation and a generalization that works.
Phys. Rev. B, 21:5469-5493, Jun 1980.
[33] W Y Liang. Excitons. Physics Education, 5(4):226, 1970.
[34] Mengqiu Long, Ling Tang, Dong Wang, Yuliang Li, and Zhigang Shuai. Electronic structure and carrier mobility in graphdiyne sheet and nanoribbons: Theoretical predictions. ACS Nano, 5(4):2593-2600, 2011.
[35] Anthony Lopez, Billy Roberts, Donna Heimiller, Nate Blair, and Gian Porro.
U.s. renewable energy technical potentials: A gis-based analysis. NREL Report,
2002.
[36] Roberto Millini, Giovanni Perego, Umberto Costantino, and Fabio Marmottini.
Structure of layered -zirconium phosphite and zirconium phosphatephosphites
from x-ray powder diffraction data. Microporous Materials, 2(1):41 - 54, 1993.
115
[37] S. 0. Morgan and W. A. Yager. Dielectric properties of organic components relation to chemical composition and physical structure. Industrial & Engineering
Chemistry, 32(11):1519-1528, 1940.
[38] Paula Mori-Sinchez, Aron J. Cohen, and Weitao Yang. Localization and delocalization errors in density functional theory and implications for band-gap
prediction. Phys. Rev. Lett., 100:146401, Apr 2008.
[39] John P. Perdew. Density-functional approximation for the correlation energy of
the inhomogeneous electron gas. Phys. Rev. B, 33:8822-8824, Jun 1986.
[40] John P. Perdew, Kieron Burke, and Matthias Ernzerhof. Generalized gradient
approximation made simple. Phys. Rev. Lett., 77:3865-3868, Oct 1996.
[41] John P. Perdew and Yue Wang. Accurate and simple analytic representation of
the electron-gas correlation energy. Phys. Rev. B, 45:13244-13249, Jun 1992.
[42] John P. Perdew and Wang Yue. Accurate and simple density functional for the
electronic exchange energy: Generalized gradient approximation. Phys. Rev. B,
33:8800-8802, Jun 1986.
[43] Andrew M. Rappe, Karin M. Rabe, Efthimios Kaxiras, and J. D. Joannopoulos.
Optimized pseudopotentials. Phys. Rev. B, 41:1227-1230, Jan 1990.
[44] Steven K. Rishi, Benson M. Kariuki, Neal J. Checker, John Godber, and
Adrian J. Wright. Synthesis and crystal structure of alh2p3ol0[middle dot]2h2o;
a new structure-type for layered acid phosphates. Chem. Commun., pages 747749, 2006.
[45] M. A. Salvad6, P. Pertierra, S. Garcia-Granda, J. R. Garcia, J. Rodriguez, and
M. T. Fernandez-Diaz. Neutron powder diffraction study of a-Ti(HPO 4 )2 .H 2 0
and a-Hf(HPO 4 ) 2 .H 2 0; H-atom positions. Acta CrystallographicaSection B,
52(5):896-898, Oct 1996.
[46] M.C. Scharber and N.S. Sariciftei. Efficiency of bulk-heterojunction organic solar
cells. Progress in Polymer Science, 38(12):1929 - 1940, 2013. Topical issue on
Conductive Polymers.
[47] Erwin Schr6dinger.
79:361-376, 1926.
Quantisierung als eigenwertproblem.
Annals of Physics,
[48] Zhigang Shuai, Linjun Wang, and Chenchen Song. Theory of Charge Transport
in Carbon Electronic Materials. Springer Berlin Heidelberg, 2012.
[49] S.M. Sze and J.C. Irvin. Resistivity, mobility and impurity levels in gaas, ge,
and si at 300k. Solid-State Electronics, 11(6):599 - 602, 1968.
[50] M. van Schilfgaarde, Takao Kotani, and S. Faleev. Quasiparticle self-consistent
gw theory. Phys. Rev. Lett., 96:226402, Jun 2006.
116
[51] Zhong-Sheng Wang, Nagatoshi Koumura, Yan Cui, Masabumi Takahashi, Hiroki Sekiguchi, Atsunori Mori, Toshitaka Kubo, Akihiro Furube, and Kohjiro
Hara. Hexylthiophene-functionalized carbazole dyes for efficient molecular photovoltaics: Tuning of solar-cell performance by structural modification. Chemistry of Materials, 20(12):3993-4003, 2008.
[52] Masatomo Yashima, Teruo Hirose, Susumu Katano, Yasuo Suzuki, Masato Kakihana, and Masahiro Yoshimura. Structural changes of zrO 2 -ceO 2 solid solutions
around the monoclinic-tetragonal phase boundary. Phys. Rev. B, 51:8018-8025,
Apr 1995.
117