effect of chemical impurities on the solid state physics of ......the impurity states in the band...
TRANSCRIPT
i
Effect of Chemical Impurities on the Solid State Physics of Polyethylene
by
Ahmed Ali Soliman Huzayyin
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy
The Edwards S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto
© Copyright by Ahmed Huzayyin 2011
ii
Effect of Chemical Impurities on the Solid State Physics of
Polyethylene
Ahmed Ali Soliman Huzayyin
Doctor of Philosophy
The Edwards S. Rogers Sr. Department of Electrical and Computer Engineering
University of Toronto
2011
Abstract
Computational quantum mechanics in the frame work of density functional theory (DFT) was
used to investigate the effect of chemical impurities on high field conduction in polyethylene
(PE). The impurity states in the band gap caused by common chemical impurities were
characterized in terms of their ―depth‖, i.e. energy relative to their relevant band edge (valence
band or conduction band), and in terms of the extent to which their wavefunctions were localized
to a single polymer chain or extended across chains. It was found that impurity states can affect
high field phenomena by providing ―traps‖ for carriers, the depths of which were computed from
first principle in agreement with estimates in literature. Since the square of the wavefunction is
proportional to the spatial electron probability density, transfer of charge between chains requires
wavefunctions which are extended across chains. Impurity states which are extended between
chains can facilitate the inherently limited interchain charge transfer in PE, as the DFT study of
iodine doped PE revealed.
The introduction of iodine into PE increases conductivity by several orders of magnitude,
increases hole mobility to a much greater extent than electron mobility, and decreases the
activation energy of conduction from about 1 eV to about 0.8 eV. These characteristics were
explained in terms of the impurity states introduced by iodine and wavefunctions of those states.
iii
Understanding the effect of iodine on conduction in PE provided a basis for understanding the
effect of common chemical impurities on conduction therein. In particular, carbonyl and vinyl
impurities create states which should promote hole mobility in a manner very similar to that
caused by iodine. It was demonstrated that in the context of high field conduction in PE, besides
the traditional focus on the depth of impurity states, it is important to study the spatial features of
the states wavefunctions which are neither discussed nor accounted for in present models.
iv
Acknowledgments
The author is deeply grateful to Prof. Reza Iravani, Professor of Electrical and Computer
Engineering at the University of Toronto, and Prof. Steven Boggs, Adjunct Professor at the ECE
Department, University of Toronto, and Professor of Material Science and Director of Electrical
Insulation Center at the University of Connecticut, who so wisely and patiently guided the
research work of the thesis.
The truthful support and unfailing guidance of Professor Iravani throughout the Ph.D. program
were most valuable indeed. Right from the first moment I arrived at U of T, as an international
student, I always felt that he is there whenever I needed academic supervision or personal advice.
I certainly learned a lot from him and, therefore, owe him a lot.
Working with Prof Boggs was one of the most wonderful coincidences in my life. He always put
what is the best for me, academically and life wise, as the utmost priority. In deed I have learnt
so much from his wide knowledge and long research expertise and clearly there was no limit to
how much he was always willing to provide. In addition, Professor Boggs was ever willing to
invest time to make a better researcher out of me. Furthermore, he was extremely kind and
supportive in his counsel at the personal level. My gratitude to Prof Boggs is far beyond what
these lines can convey.
Both Professor Iravani and Professor Boggs, at an early stage of the Ph.D. program when I had to
apply for a year of medical leave of absence, showed unlimited understanding and gave, the then
much needed, moral support and continuous encouragement.
Gratitude is due to Prof Ramamurthy Ramprasad, Associate Professor of Material Science,
University of Connecticut, for sharing his valuable knowledge in density functional theory
computations and keenness to advice and teach in the nicest manner possible. This work would
not have been possible without his support. The author is also thankful to colleagues and friends
at the University of Connecticut and University of Toronto who shared thoughts and useful
discussions during the course of the study.
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I am sincerely grateful to all my past teachers and professors throughout my schooling years in
Egypt, at schools and at the University of Cairo, who gave much without expecting any. In
particular, I mention Professor Amr Adly, Professor of Electrical Engineering and Vice Dean,
Faculty of Engineering, Cairo University, from whom I learnt a lot not only in science and
research but also at the personal level.
To my future wife, with whom I would have loved to share the PhD experience, I am already
grateful for all your future support. Last but not least, I owe my genuine love and gratitude to my
mother, father and sister, and lately my brother in law, all from the academia. Raised in such a
family is the best gift I have been awarded in life. Their support, compassion, and love are
beyond what words can express, and my admiration of their personalities is boundless. The more
I grow, the more I realize how valuable that has been to me. Any success I may achieve in life
and research would be a product of their support. I simply thank them for everything.
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Table of Contents
Acknowledgments.......................................................................................................................... iv
Table of Contents ........................................................................................................................... vi
List of Tables ...................................................................................................................................x
List of Figures ................................................................................................................................ xi
Chapter 1 Introduction .....................................................................................................................1
1 Introduction .................................................................................................................................1
1.1 Research Motivation ............................................................................................................1
1.2 The Need for Quantum Mechanical Models ........................................................................2
1.3 Major Uncertainties in the Understanding of High Field Conduction and the Role of
Chemical Impurities Therein ...............................................................................................3
1.3.1 Chemical Impurity States and the Mystery of Traps ...............................................3
1.3.2 Conduction Enhancement in Iodine Doped PE .......................................................4
1.3.3 Computational Quantum Mechanics Based Approach ............................................5
1.4 Density Functional Theory ..................................................................................................6
1.5 Thesis Objectives .................................................................................................................7
1.6 Thesis Organization .............................................................................................................7
Chapter 2 Brief Introduction to DFT ...............................................................................................8
2 Brief Introduction to DFT ...........................................................................................................8
2.1 Basics of Computational Quantum Mechanics ....................................................................8
2.1.1 Schrödinger Equation...............................................................................................8
2.1.2 Formulation of the Many Body Problem .................................................................9
2.1.3 Solving the Many Body Problem ...........................................................................10
2.2 Theoretical Basis of DFT ...................................................................................................12
vii
2.2.1 The Basic Idea of DFT ...........................................................................................12
2.2.2 Hohenberg and Kohn Theorems ............................................................................13
2.2.3 Kohn and Sham Method ........................................................................................14
2.3 Implementation of DFT .....................................................................................................15
2.3.1 Simplified Algorithm .............................................................................................15
2.3.2 Exchange-Correlation Approximation ...................................................................16
2.3.3 Basis Sets ...............................................................................................................18
2.3.4 Pseudopotentials ....................................................................................................19
2.3.5 Boundary Conditions .............................................................................................20
2.3.6 Moving the Nuclei; Geometry Optimization and Molecular Dynamics ................21
2.3.7 DFT in Brief ...........................................................................................................22
2.4 The SIESTA Code .............................................................................................................24
Chapter 3 Quantum Mechanical Modeling of Chemical Impurities in Polyethylene ....................27
3 Quantum Mechanical Modeling of Chemical Impurities in Polyethylene ...............................27
3.1 Review of DFT Studies of Polyethylene and Chemical Impurities Therein .....................27
3.2 Computational Details of the Present Work ......................................................................29
3.3 Modeling of Chemical Impurities in Polyethylene; From Single Chains to Crystalline
Bulk Models .......................................................................................................................30
3.3.1 Models of Pure Polyethylene .................................................................................30
3.3.2 Models of Chemical Impurities in Polyethylene ...................................................36
3.4 Core-Shell Model ...............................................................................................................41
Chapter 4 Effect of Chemical Impurities on Solid State Physics of Polyethylene ........................44
4 Effect of Chemical Impurities on Solid State Physics of Polyethylene ....................................44
4.1 Characterization of Chemical Impurities Studied ..............................................................44
4.2 Band Gap Impurity States ..................................................................................................46
4.3 Discussion of Chemical Impurity States in the Context of High Field Phenomena ..........51
viii
4.3.1 Impurity States as Trapping/Hopping Sites ...........................................................51
4.3.2 Comparison between Trap Depths Determined in the Present Thesis and
Estimates in Literature ...........................................................................................54
4.3.3 Interchain Extension of Impurity States ................................................................57
Chapter 5 Effect of Iodine on Conduction in Polyethylene ...........................................................59
5 Effect of Iodine on Conduction in Polyethylene .......................................................................59
5.1 Conduction in Iodine Doped Polyethylene ........................................................................59
5.2 Isolated Iodine Molecules ..................................................................................................60
5.3 Interaction between Iodine and Polyethylene ....................................................................62
5.4 Iodine Impurity States in Polyethylene ..............................................................................65
5.5 Effect of Iodine on Conduction..........................................................................................69
5.6 Effect of Bromine ..............................................................................................................71
5.7 Summary ............................................................................................................................72
5.8 Similarities between Iodine and Common Impurities .......................................................73
Chapter 6 Summary, Conclusions, and Future Work ....................................................................79
6 Summary, Conclusions, and Future Work ................................................................................79
6.1 Summary and Conclusions ................................................................................................79
6.1.1 DFT Models of Chemical Impurities in Polyethylene ...........................................79
6.1.2 Effect of Chemical Impurities on the solid State Physics of Polyethylene ............80
6.1.3 Effect of Iodine on Conduction in Polyethylene....................................................83
6.2 Original Contributions .......................................................................................................85
6.3 Future Work .......................................................................................................................85
6.3.1 Experimental Studies .............................................................................................85
6.3.2 Macroscopic Modeling ..........................................................................................86
6.3.3 DFT Studies ...........................................................................................................86
References ......................................................................................................................................88
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Appendix A ....................................................................................................................................95
A Determination of the Vacuum Level .........................................................................................95
A.1 Vacuum Level in System Finite in Two Dimensions ........................................................95
A.2 Vacuum Level in Bulk Periodic Systems ...........................................................................96
Appendix B ....................................................................................................................................98
B Pseudopotentials ........................................................................................................................98
B.1 Iodine Pseudopotential .......................................................................................................98
Appendix C ....................................................................................................................................99
C Main SIESTA Input Files .........................................................................................................99
C.1 Core-Shell Systems ............................................................................................................99
C.2 Bulkcell Crystalline System ...............................................................................................99
Appendix D ..................................................................................................................................100
D Computers Main Specifications ..............................................................................................100
x
List of Tables
Table 1.1 Various mobility expressions for bulk limited conduction in polymers. ...................... 4
Table 2.1 DFT Codes classified according to boundary conditions and basis set implemented .. 26
Table 4.1 Bond lengths and angles of impurities in Figure 4.1. ................................................... 45
Table 4.2 Impurity states depths and a description of their orbital formation. ............................. 48
Table 4.3 Various estimates of trap depths ................................................................................... 55
Table 4.4 A comparison between DFT estimates of electron trap depths in [5,64] based on
electron affinity computations and the estimates of shallow electron traps from the present work.
....................................................................................................................................................... 56
xi
List of Figures
Figure 2.1 Simplified flow chart of DFT computations (modified from [26]). ............................ 16
Figure 2.2 Pseudopotentials and Pseudo wavefunctions (slightly modified from [33]). .............. 20
Figure 2.3 Various systems created through periodic boundary conditions applied to a unit cell.
....................................................................................................................................................... 21
Figure 2.4 DFT and geometry optimization flowchart. ................................................................ 23
Figure3.1 The primitive unit cell of a PE chain. ........................................................................... 32
Figure 3.2 Unit cells of infinite and finite chains. ........................................................................ 32
Figure 3.3 The variation of the band gap of finite PE chains versus chain length. ...................... 34
Figure 3.4 DOS of a pure 40 C atom finite chain (C40H82). ......................................................... 34
Figure 3.5 Orthorhombic crystalline structure of PE. ................................................................... 35
Figure 3.6 Minimum energy structure of a 40 C atom chain with a carbonyl impurity (C40H80O).
....................................................................................................................................................... 36
Figure 3.7 DOS of a 40 C atom chain including a carbonyl group (C40H80O). ............................ 37
Figure 3.8 Energy separating impurity states of two carbonyl impurities in a 40 C atoms chain
versus distance separating the carbonyl impurities. ...................................................................... 38
Figure 3.9 Crystalline bulkcell of PE. ........................................................................................... 39
Figure 3.10 Initial structure of a Core-Shell model. ..................................................................... 41
Figure 3.11 The Core-Shell structure and crystalline bulkcell structure with carbonyl impurities.
....................................................................................................................................................... 42
Figure 4.1 Various common chemical impurities in the minimum energy structure of a Core-
Shell model. .................................................................................................................................. 45
xii
Figure 4.2 DOS of Core-Shell structure showing impurity states from a carbonyl impurity.. ..... 47
Figure 4.3 Square of the wavefunctions (probability density) of the unoccupied impurity states.
....................................................................................................................................................... 48
Figure 4.4 Square of the wavefunctions (probability density) of the occupied impurity states. .. 49
Figure 4.5 DOS of a Core-Shell and a crystalline bulkcell structures including a carbonyl
impurity. ........................................................................................................................................ 50
Figure 4.6 Impurity states introduced into the band gap of PE by various impurities. ................ 51
Figure 4.7 Overlap surface of impurity states and bands edge states. .......................................... 53
Figure 4.8 Energy diagram of the band gap of PE showing the energy depth of impurity states
created by carbonyl, vinyl, double bond, and conjugated double bond impurities. ..................... 54
Figure 4.8 The electron spatial probability density (square of the electron state wavefunction) of
impurity states extended towards neighboring chains. ................................................................. 58
Figure 5.1 Stability of various configurations of In (n = 3 to 5). Stability decreases from the Ina to
Ind................................................................................................................................................... 61
Figure 5.2 Binding energy per iodine atom (Eb[In]/n) for the most stable In configurations at each
n (n = 2 to 5). ................................................................................................................................ 61
Figure 5.3 Incremental binding energy (Einc[n]) for In from n = 2 to 5. ...................................... 62
Figure 5.4 Core-Shell model of PE with I2 molecule (PE-I2) shown from two perspectives. ..... 63
Figure 5.5 Binding energy of iodine molecules to PE chains in Core-Shell model per iodine
atom (Eb PE-In / n). .......................................................................................................................... 63
Figure 5.7 The difference electron charge density in PE-I2 with isosurfaces corresponding to
regions of charge depletion and accumulation.. ............................................................................ 65
Figure 5.9 DOS of PE-I2 showing the contribution of I2 orbitals in red. ..................................... 67
xiii
Figure 5.10 Contours of the square of the wavefunction of various iodine impurity states ........ 68
Figure 5.11 The electron spatial probability density surface plots of various iodine impurity
states. ............................................................................................................................................. 70
Figure 5.12 Various structures based on modifications to the Core-Shell model. ...................... 72
Figure 5.13. Electron densities in the vicinity of various impurities. .......................................... 74
Figure 5.14 The electron spatial probability density of various impurity states which are
extended between chains............................................................................................................... 75
Figure 5.15 The electron spatial probability density of iodine and carbonyl mixed impurity
states. ............................................................................................................................................. 77
Figure 5.16 Probability surface of impurity states of interacting impurities in adjacent chains. 78
Figure A.1 The planar average total local potential of a 40 C atoms isolated chain of which the
DOS plot is shown in figure 3.4.................................................................................................... 95
Figure A.2 The planar average total local potential of a Core-Shell model with a carbonyl
impurity in the Core-Shell structure which is shown in Figure 3.11. ........................................... 96
Figure A.3 The plots of the planar average total local potential (VKSav
) showing the ―bulk plus
band lineup‖ procedure applied to determine the electron affinity of PE (110). .......................... 97
Figure B.1 Input file to generate iodine pseudopotential using ATOM program. ........................ 98
1
Chapter 1 Introduction
1 Introduction
1.1 Research Motivation
POLYMERS are used widely as insulation in electrical power apparatus such as cables,
capacitors, protection devices, and electrical machines. Insulating polymers are easy to process,
relatively inexpensive, and have excellent dielectric properties [1]. Insulating polymers are
replacing other dielectrics as the material of choice in high voltage (HV) applications, and in
particular for cables and capacitors.
Bulk high voltage dielectrics operate at average geometric design fields up to 20 kV/mm at
power frequency, while laminar dielectrics, such as biaxially oriented polypropylene, operate at
DC fields well above 400 kV/mm in some capacitor designs. In the case of bulk dielectrics,
maximum fields in the dielectric are much greater than the design field as a result of ―defects‖ in
the insulation system due to chemical impurities in the dielectric and imperfections in the
manufacturing process. Engineering of HV insulation systems tends to be dominated by
management of ―defects‖ which contribute to local electric field enhancements and affect
significantly (if not dominate) high field aging of the dielectric. High field conduction in the
regions of field enhancement in insulating polymers contributes to the degradation of polymeric
insulating material as a result of ―hot‖ electrons and carrier recombination which results in UV
photons that can break chemical bonds. As a result, understanding of high field conduction in
insulating polymers and the role of chemical impurities therein is an important scientific
objective with significant industrial implications [1-2]. Such understanding is likely to contribute
to
More compact power system components.
Reliability of power systems.
Materials with reduced conduction and improved dielectric breakdown field.
Materials with tailored dielectric properties.
2
In spite of extensive research, understanding of the physical basis of high field conduction in
common insulating polymers such as polyethylene (PE), is incomplete [2-4] in that a thorough
atomic level explanation of the role of chemical impurities therein is lacking. Quantum
mechanical modeling of PE and chemical impurities therein, which only became possible
recently, can contribute significantly to a much needed rigorous understanding of the physical
basis of high field conduction in insulating polymers. PE, the material studied in the present
thesis, is among the most commonly used insulating materials [2,5,6] and is often used as a
model material for investigations of insulating polymers [2].
1.2 The Need for Quantum Mechanical Models
The present understanding of high field phenomena in insulating polymers and the effect of
chemical impurities therein is based on macroscopic models. Most of the uncertainties and gaps
in the present understanding can be traced down to the fact that quantum mechanical processes,
such as those responsible for the transfer of electronic carriers, are studied using macroscopic
models which often describe those processes using classical concepts. Macroscopic models
include parameters, such as ―trap‖ depths, which can only be understood at an atomic level using
computational quantum mechanical modeling. However in present models, the values of these
parameters are determined by adjusting them so that the models fit the experimental data.
Developing a thorough understanding of the physical significance of such parameters and
assessing the validity of the theory behind a macroscopic model are difficult when various
models based on differing theories can fit the same data ―well‖. Furthermore, macroscopic
models of high field conduction are often based on assumptions that do not represent the
microscopic features of the material properly. The assumption that initial and final states are
uniformly available throughout the material for transfer of carriers has no solid physical ground.
This assumption overlooks fundamental questions, such as what creates these states, how can
they be characterized, and what determines their spatial distribution throughout the material?
These questions are best addressed using computational quantum mechanics. Macroscopic
models also treat the polymer as a continuum neglecting local changes, as those in the vicinity of
impurities, which might be critical to charge transfer. Quantum mechanical modeling can
analyze microscopic changes to the material in the vicinity of impurities and, accordingly,
determines their effect on the material behavior. Treating the material as a continuum also
overlooks the fact that PE is formed of chains created by strong chemical bonds, and these chains
3
are held together with weaker van der Waal forces. As a consequence, the mechanism of transfer
of carriers along chains is expected to be very different from that across chains. This difference
is neither discussed nor accounted for in macroscopic models. All of the above leads to various
uncertainties in the understanding of high field phenomena in insulating polymers.
Computational quantum mechanics is suited naturally to explain many of such uncertainties. The
main advantage of a ―quantum mechanical microscopic‖ approach over the ―traditional
macroscopic‖ approach in understanding the effect of chemical impurities in high field
conduction in PE is that the former can describe atomic level features of the material, such as
local disturbance in the vicinity of a specific impurity.
All the above press the compelling argument that an untraditional quantum mechanical modeling
approach can contribute to a much needed rigorous understanding of the physical basis of high
field conduction in insulating polymers and the effect of chemical impurities therein, which is
the general goal of the present thesis. The thesis attempts to achieve this goal through a
computational quantum mechanics based investigation of two long standing problems in the field
(discussed in greater detail in section 1.3).
First, do chemical impurities create states in the band gap of PE, how to characterize these
states, what is the physical basis of the abstract concept of ―traps‖ which is central to
understanding high field phenomena, what is the correlation between traps and chemical
impurities?
Second, what is the physical explanation of the effect of iodine on the conductivity of PE?
1.3 Major Uncertainties in the Understanding of High Field Conduction and the Role of Chemical Impurities Therein
1.3.1 Chemical Impurity States and the Mystery of Traps
Most existing models of conduction in polymers involve solving the Poisson equation along with
the Charge Transport and Continuity equations to form a ―transport model‖. Defining the types
and number of charge carriers and generation and transport mechanisms thereof is crucial in
solving any transport model [2]. The charge transport mechanism determines the carrier
mobility, which is a main parameter in any transport model. Various transport models have been
4
developed for insulating polymers, where each is based on a different charge transport
mechanism and hence a different mobility expression as given in Table 1.1.
Table 1.1 Various mobility expressions for bulk limited conduction in polymers. In the equations below, α and γ are constants, Et, F, T, k are trap depth, electric field, temperature, and Boltzmann constant, respectively. The table is based mainly on [2,7].
Conduction Mechanism Mobility Expression
Hopping Conduction µ = α e
-
sinh (
)
Poole Frenkel µ = α -
-
√
Space Charge Limited Conductivity (SCLC)
(modified by traps)
µ = α -
High field conduction is generally explained in terms of charge transfer between traps/hopping
sites due to the combined effect of thermal and electric fields [1,4]. Electrons/holes, rather than
ions, are the main carriers at high fields [1,4]. Chemical impurities affect, if not dominate, the
conduction process possibly by creating trapping sites for carriers, the energy depth (Et) of which
plays a major part in determining the carrier mobility as given in Table 1.1. The concept of traps
is fundamental to most high field conduction models and is essential to account for the effect of
chemical impurities in high field phenomena, such as space charge formation. However, a clear
identification of the physical basis of traps is lacking, in spite of extensive effort [2-4]. No
measurement can provide detailed and consistent information about trap depths and their sources
[2]. The possibility that traps do not exist has been proposed due to the lack of a convincing
physical basis for the concept [3]. Traps might be an abstraction of chemical impurity states
created in the band gap of PE. In such case, the characterization of these states only by an energy
depth, and ignoring the features of the states wavefunctions implies erroneously that the charge
carriers can be treated as classical particles.
1.3.2 Conduction Enhancement in Iodine Doped PE
Another long standing problem in the field of insulating polymers is the lack of an atomic level
explanation of the effect of iodine on conduction in PE. An iodine content of few percentages
increases the conductivity of PE by about 4 orders of magnitude [8-12]. Such effect is consistent
5
over many studies that involved various types of PE, electrode material, and methods by which
iodine is introduced into PE. Thus, the experimental observations can be attributed to the
interaction between iodine and the polymer chains and other factors can be excluded. The
present conduction models, and the transport mechanisms they imply, are not sufficient to
explain the increase of the conductivity of PE upon doping by iodine [8-12]. In addition, the
present picture, which explains the effect of chemical impurities on conduction through the
introduction of traps/hopping sites with certain depths and densities, cannot explain the effects of
iodine on PE. Understanding the physical basis of these effects at an atomic level is important, if
the large effects caused by iodine cannot be understood; there is little hope of understanding the
more subtle effects of common impurities such as carbonyls, double bonds in the backbone, etc.
1.3.3 Computational Quantum Mechanics Based Approach
Macroscopic models and existing experimental techniques have failed to provide clear answers
to the problems above. The more rigorous approach to these problems is to model PE at an
atomic level including various chemical impurities using computational quantum mechanics to
determine how these impurities change the electronic structure of the polymer, identify chemical
impurity states, and determine how to characterize them. A quantum mechanical characterization
of chemical impurity states using their energies and wavefunctions might explain the physical
significance of the abstract concept of ―traps‖ and their role in high field phenomena. Such an
approach can improve existing macroscopic models, and more importantly, lead to development
of new models which are better suited to describe high field conduction in insulating polymers.
In addition, it allows assessing the relative effects of various impurities which is otherwise
almost impossible experimentally.
The present thesis attempts to address the problems discussed in sections 1.3.1 and 1.3.2 through
atomic level modeling using computational quantum mechanics in the framework of Density
Functional Theory (DFT). Addressing the research problems above using DFT will help in
explaining the role of chemical impurities in charge transport in PE, which is necessary for a
rigorous understanding of the physical basis of high field conduction therein.
6
1.4 Density Functional Theory
DFT is a computational quantum mechanical method which allows practical and realistic
simulations of atomic behavior of matter and, accordingly, can offer better understanding of
various physical processes. Using DFT, the states of electrons and nuclei can be calculated, and
the atomic structure of material can be determined along with other ground state properties.
Accordingly, various material properties can be computed from first principles where no
adjustable parameters are involved and local properties (at an atomic level) of systems can be
investigated. Some physical properties simulated by DFT have led to valuable insight that was
otherwise almost impossible, as in the case of explaining the physical basis of ferroelectric
ferromagnetic property [13].
In the context of the present work, DFT is the most suitable computational quantum mechanics
approach for various reasons. In general, DFT is the most successful formalism within
computational quantum mechanics for systems of hundreds of atoms, as it provides the best
compromise between computational time and accuracy [14,15]. The limitations and capabilities
of DFT are reasonably well understood as a result of its widespread use to study various
properties of various types of materials. DFT has been employed successfully to study, liquids
and solids, insulators, semiconductors, metals, and superconductors, interfaces, surfaces, and
bulk material, and simple molecules up to complicated structures such as DNA [13,15]. In
addition, DFT has been applied successfully to determine, thermal, optical, electrical, and
magnetic properties among other features of matter [16].
The decision to use DFT in this thesis is also supported by the fact that it has been used
successfully in electrical engineering applications such as studying the band structure, band gap,
dielectric constant, and other parameters related to electrical conduction and solid state physics
of material [17]. DFT was recently applied in studies of the electronic properties of PE and the
effects of chemical impurities therein; a brief review of which is given in subsequent sections of
the thesis. Although the use of computational quantum mechanics in electrical engineering
applications, and particularly studies of insulating polymers, is scarce compared to other fields, it
is strongly believed that with the ever increasing computer power and the continuous refinement
of computational tools, DFT will find more applications and bring new insight into many
electrical engineering research problems.
7
1.5 Thesis Objectives
Through implementation of computational quantum mechanics, this thesis has the following
objectives:
Develop molecular models of bulk PE in which the effect of chemical impurities on electronic
properties of the polymer can be studied, as such models have not been reported.
Determine if common chemical impurities create impurity states in the band gap of PE and if
so, determine the nature of such states as characterized by their energy relative to the band
edges and the spatial features of the state wavefunction.
Provide a physical understanding of the concept of trapping and hopping sites which are
essential in understanding high field phenomena in insulating polymers.
Provide physical explanation of the effect of iodine on the conductivity of PE.
1.6 Thesis Organization
The thesis begins with an introduction which explains the research motivation, presents a brief
account of DFT, and states the thesis objectives. Chapter two presents the theoretical basis of
DFT and the details of its implementation. The main approximations required to implement DFT
are also discussed. The chapter also discusses briefly the main features of the SIESTA (Spanish
Initiative for Electronic Simulations with Thousands of Atoms) code which is used to implement
DFT computations in the present thesis. In Chapter 3, various models of PE through which the
effect of common chemical impurities can be studied are discussed. The effect of chemical
impurities on the solid state physics of PE is analyzed in Chapter 4. In addition, impurity states
in the band gap of PE are identified and characterized. In Chapter 4, the physical basis of the
concept of traps and their correlation with chemical impurities are explained. In Chapter 5, the
physical basis of the effect of iodine on conduction in PE is investigated. Based on the energy of
iodine impurity states introduced into the bandgap and spatial features of the impurity states
wavefunctions, a mechanism by which iodine increases the conductivity of PE is proposed. The
mechanism explains experimentally observed features of conduction in iodine doped PE. The
thesis ends with Chapter 6 which includes, summary, conclusions, the thesis contributions, and
proposed future work. The thesis has four appendices.
8
Chapter 2 Brief Introduction to DFT
2 Brief Introduction to DFT
2.1 Basics of Computational Quantum Mechanics
2.1.1 Schrödinger Equation
DFT is a quantum mechanical computational method which determines the ground state
configuration of a system of electrons (in general fermions) and nuclei forming a molecule or a
solid. Various ground state properties of matter can be determined from first principle using
DFT. The basic equation, which all computational quantum mechanical methods aspire to solve,
is the Schrödinger equation. The Schrödinger equation (given for a single particle in the time
independent form in 2.1 below) is a wave equation that describes the behavior of ―small‖
particles such as electrons, where the wave nature of the particle, which is dictated by the wave-
particle duality concept, becomes pronounced
22h
ψ(r) + V(r) ψ(r) = E ψ(r) 2 m
(2.1)
where, h is Planck’s constant h divided by 2 π, m is the particle mass, V(r) is the potential to
which the particle is subjected as function of the 3D space vector r, ψ is the wavefunction
describing the behavior of the particle, and E is the total energy of the particle. The physical
interpretation of the wavefunction ψ(r) is based on the square of its absolute value 2
ψ which
provides the probability of finding the particle at the location r.
Properties of the hydrogen atom can be determined from first principle by solving the
Schrödinger equation (2.1) analytically for the single electron of the hydrogen atom subjected to
the electrostatic field of the nucleus, which is represented through the potential term in (2.1). In
larger atoms, electrons interact together by contributing to the potential term in (2.1) which
prohibits analytical solutions. Solving a system of interacting quantum mechanical particles is
referred to as the ―Many Body problem‖, which in spite of its simple formulation, is among the
most complicated problems in computational physics.
9
2.1.2 Formulation of the Many Body Problem
The Many Body problem, as discussed in the context of the present work, is the problem of
determining the wavefunctions and energies of a system consisting of N electrons and M nuclei
which form a molecule or a solid. According to the Born-Oppenheimer approximation, the much
heavier nuclei can be considered fixed classical particles while studying electron dynamics [18].
Thus, the Many Body problem is reduced to determining the behavior of N electrons, i.e.,
determining their energies and wavefunctions. Each of the electrons is subjected to a potential
from M fixed nuclei as well as the other N-1 electrons. To determine the behavior of the N
interacting electrons, the many electron Schrödinger equation (2.2), must be solved [19].
ˆ1 2 N 1 2 NH Ψ(r , r ,............,r ) = E Ψ(r , r ,............,r ) (2.2)
where H is the system Hamiltonian, E is the energy of the electrons, and Ψ is the many electron
wavefunction. Together, these variables characterize the system of the N interacting electrons.
The position vectors 1 2 Nr , r ,....,r denote the location of each of the N electrons. The probability
of finding the N electrons in the locations 1 2 Nr , r ,....,r , is given by2
Ψ . The Hamiltonian H is
ˆn-e e-eH = T + V + V . (2.3)
The term T represents the kinetic energy of electrons, the term Vn-e is the potential acting on the
electrons from the nuclei and the term Ve-e is the potential to which an electron is subjected by
the other electrons. These terms can be elaborated as in the form
2
1
2 2N N M N Nk
n-e e-e ii i k i j , j ii k i j
Z e 1 eT + V + V = -
r - R 2 r - r
(2.4)
where, the kinetic energy term is given by the Laplacian operator2i , Zk is the charge of nucleus
k, Rk is the location of nucleus k, and e is the electron charge. The equation is written in atomic
units (e = me = h = 4 π ε0 = 1) [18].
10
2.1.3 Solving the Many Body Problem
2.1.3.1 Hartree-Fock
When the number of electrons N is greater than two, no method which solves (2.2) exactly has
been found. The Hartree-Fock (HF) method, which is among the earliest quantum mechanical
computational methods, provides an approximate solution to the Many Body problem formulated
in (2.2), (2.3), and (2.4). The HF method begins by assuming that Ψ is formed of an anti-
symmetric product of N single electron wavefunctions represented by the Slater determinant in
(2.5) [18]. The anti-symmetric nature of the Slater determinant in (2.5) accounts for the Pauli
Exclusion Principle which states that no two identical fermions, such as electrons, can occupy
the same quantum state simultaneously.
1 1 1 2 1 N
2 1 2 2 2 N
N 1 N 2 N N
ψ (r ) ψ (r ) . . . ψ (r )
ψ (r ) ψ (r ) . . . ψ (r )
. . .Ψ =
. . .
ψ (r ) ψ (r ) ψ (r )
(2.5)
By invoking a variational principle, the HF method can be formulated as a set of N single particle
like Schrödinger equations, such as that given in (2.6) for an electron ―i‖ [18].
-
2 ψ
(r) + - ψ
(r) + ∑ ∫ r |ψ (r
)|
|r-r | ψ
(r) - ∑ ( , )∫
r ψ *(r ) ψ (r
)
|r-r | ψ
(r) = ψ
(r) (2.6)
where, the Kronecker-Delta function i jδ(ζ ,ζ ) restricts the summation to electrons of same spin
[20]. Equation (2.6) must be solved self-consistently for N electrons in order to find the
wavefunction of each electron and, consequently, the many electron wavefunction through (2.5).
The HF method reduces the Many Body problem to solving N single particle like Schrödinger
equations.
The first and second terms in (2.6) represent the kinetic energy of the electron and its potential
energy due to the interaction with the nuclei, respectively. The third term in (2.6), which is
referred to as the Hartree potential, represents the potential energy due to the Columbic
interaction of electrons [19]. The Hartree potential, as given by the third term in (2.6), represents
the interaction between an electron ―i‖ and the average electron distribution. The forth term in
11
(2.6) is referred to as the exchange potential. The exchange potential is a direct reflection of
assuming Ψ to be anti-symmetric which in turn allows the inclusion of the Pauli Exclusion
Principle. According to the Pauli Exclusion Principle, electrons with the same spin cannot exist
at the same point in space. This implies a decrease in the potential energy, which is expressed by
the subtraction of the 4th
term in (2.6). Usually, (2.6) is written in a concise form analogous to
(2.1), which is given for an electron ―i‖ in (2.7) [20].
2
HF i HF i i i
-1H ψ (r) V (r) ψ (r) = E ψ (r)
2
(2.7)
where, HHF is the Hartree-Fock Hamiltonian and HFV (r) , given below in (2.8), is the Hartree-
Fock potential.
HF n-e Hartree xV (r) = V (r) + V (r)+V (r) (2.8)
where HartreeV (r) is the Hartree potential and xV (r) is the exchange potential. Further elaboration
on these terms will follow in subsequent sections.
The HF method starts with an assumption of the many electron wavefunction Ψ which renders
the method approximate in principle. The method proceeds through a variational principle to
reduce the Many Body problem to the set of equations given by (2.6), which is written concisely
in (2.7). The equations given by (2.6) are similar to the single electron Schrödinger equation with
an effective potential term (VHF in (2.7)) which couples the electrons. Solving the set of
equations given by (2.6) provides a solution to the Many Body problem given by (2.2). A major
shortcoming of HF method is that the assumed form of Ψ, given in (2.5), prevents equation (2.6)
from capturing the direct quantum mechanical electron-electron interaction properly [20]. In HF
method, an electron ―i‖ represented by the single particle like Schrodinger equation in (2.6), is
subjected to an average potential by the nuclei (Vn-e) and interacts with other electrons in the
system through the terms VHartree and Vx. The exchange term Vx represents a direct interaction
between the electrons of the system that accounts for repulsion of electrons with similar spin.
However, equation (2.6) overlooks the direct quantum mechanical interaction of individual
electrons due to electrostatic repulsion which is known as ―electron correlation‖ and replaces it
with the average term given by VHartree. The term VHartree represents a poor approximation of
12
electron correlation. The effect of electron correlation is expected to be accounted for in the
system Hamiltonian by a term referred to as the ―Correlation Potential‖ (Vc) which together with
the exchange potential Vx can represent the direct quantum mechanical interaction between
electrons properly [21]. The effect of electron correlation in solids is profound [21]. The lack of
proper representation of electron correlation and its large computation scaling with the system
size (α N4) renders HF a poor method for studying solids [18,21]. Accounting for the electron
correlation properly is a major challenge in computational quantum mechanics. Quoting [20]
―the role of correlation among electrons stands out as defining the great questions and challenges
of the field of electronic structure today‖.
2.1.3.2 Attempts to Account for Correlation; “Post Hartree-Fock Methods”
Other methods have been developed to solve the Many Body problem with a better
representation of electron correlation than that of the HF method. Some of these methods, such
as the Configuration Interaction (CI) and the Couple Cluster (CC) methods, are similar to HF in
starting with an assumption of Ψ and proceeding through a variational principle, while others are
based on perturbation theory such as Møller–Plesset (MP) method [18,22,23]. What is common
among the methods mentioned so far is that they solve for Ψ directly. The CI, CC, and MP
methods are more accurate than HF. However, the computational effort they require prohibits
their implementation as the number of electrons N increases to values which are necessary for
practical applications. Quoting [15] ―the problem with these methods is the great demand they
place on one’s computational resources: It is simply impossible to apply them efficiently to large
and complex systems‖.
2.2 Theoretical Basis of DFT
2.2.1 The Basic Idea of DFT
The basic idea of DFT is to solve the Many Body problem in terms of electron density n(r). A
main disadvantage of the above techniques is that the problem is formulated in terms of the
Many Body wavefunctionΨ , which is a function in 3N variables and is very challenging to
represent adequately for large values of N [15]. DFT gets around this disadvantage by
formulating the N interacting electron problem in terms of electron density n(r), which is a
function of three variables. The work of Hohenberg and Kohn (HK) in 1964 [24] and Kohn and
13
Sham (KS) in 1965 [25] establish the theoretical basis of DFT. The work of HK and KS maps
the problem of N interacting electrons (in general fermions) subjected to an external potential
into a problem of N non-interacting electrons subjected to an effective potential. Unlike HF
method, the effective potential in KS formulation accounts properly for all the physical aspects
of the problem, i.e., the kinetic energy, external field (for example due to nuclei) and, in
particular, both exchange and correlation effects between electrons. The HK and KS work, which
is the basis of DFT, presents, in principle, an exact reformulation of the Many body problem
[16].
2.2.2 Hohenberg and Kohn Theorems
Two theorems by HK lay the foundations of DFT [16]. The theorems relate to the problem of N
interacting electrons (applicable to all fermions) subjected to an external potential Vext, for
example that due to the nuclei of a solid, Vn-e in (2.3). The first HK theorem states that the
external potential Vext to which N interacting electrons are subjected, is (to within a constant) a
unique functional of the ground state electron density n0(r). In other words, n0(r) determines
uniquely, to within a constant, Vext [19]. Accordingly, the Hamiltonian (2.3) is determined to
within a constant, which in turn means that ground state Many Body wavefunction 0Ψ , the
ground state energy of the system E0, and other ground state properties are unique functionals of
n0(r) [16,20].
The second HK theorem states that a universal energy functional E[n] can be defined in terms of
the electron density such that the exact ground state is the global minimum value of this
functional. The functional F[n] (2.9) is universal and valid for any number of fermions and any
external potential such that the global minimum of F[n] coincides with the ground state value n0
[24]. The minimization should be subjected to the constraint given by (2.10).
extE[n] = V (r) n(r) dr + F[n] (2.9)
n(r) dr = N (2.10)
According to the two HK theorems, determining the ground state energy, ground state electron
density, and, consequently, the ground state properties of N interacting electrons subjected to an
external potential, is reduced to the problem of minimizing a functional E[n] in (2.9). If F[n] is
14
known and sufficiently simple, then minimizing F[n], which is a functional of the three
dimensional variable n(r), provides a simple [24] yet exact method of solving the ground state
Many Body problem of interacting electrons [20]. Now the question becomes how to find E[n].
2.2.3 Kohn and Sham Method
In 1965 Kohn and Sham (KS) formulated the energy functional E[n] given by HK as follows
ext 0 xc
1 n(r) n'(r)E[n] V (r) n(r) dr dr dr' T [n] E [n]
2 r - r' (2.11)
where the first term is the energy due to the external potential. The second term represents the
electron-electron Coulomb interaction energy, the third term T0[n] is the kinetic energy of a
system of non-interacting electrons of density n(r), and the fourth term Exc[n] represents the
exchange and correlation energy of a system of interacting electrons of density n(r) [25]. Up to
this point, the formulation is exact. Although the non-interacting electron kinetic energy T0 is
different from the true kinetic energy T, the difference can be accounted for within the exchange
correlation term Exc[n]. This means that Exc[n] is not simply the exchange and correlation
energies which result from Pauli Exclusion repulsion and quantum mechanical electrostatic
repulsion of electrons but also includes a correlation term which represents the residual between
T0 and T [16]. All terms of (2.11) can be evaluated exactly except for Exc[n], the approximation
of which plays a central role in DFT [19].
The work of KS proceeds by minimizing (2.11) subject to the constraint (2.10) using the method
of Lagrangian Multipliers, which leads to the N equations given by (2.12). DFT, as formulated
by KS, maps the problem of N interacting electrons subjected to an external potential (Vext) into a
system of N non-interacting electrons subjected to a fictitious effective potential as expressed in
(2.12) to (2.15). Equation (2.14) is similar to HF equation (2.7) with the major difference of the
presence of a potential term in the effective total potential which accounts for correlation exactly
in (2.12) which is not represented in (2.7).
2
KS i KS i i i
-1H ψ (r) V (r) ψ (r) E ψ (r)
2
(2.12)
15
where, HKS is KS Hamiltonian, KSV (r) is the KS effective potential given in (2.13), ψi(r) is the
wavefunction of the ith
electron, Ei is the total ground state energy of the ith
electron, and
KS ext xc
n(r')V (r) = V (r) + dr+V [n(r)]
r-r' (2.13)
where, Vxc is the exchange correlation potential given by (2.14).
xcxc
E [n]V (r)
n(r)
. (2.14)
The single particle wavefunctions ψi(r) should satisfy (2.15).
N 2
i
i
n(r) ψ (r) (2.15)
Solving (2.12) to (2.15) self consistently provides an exact solution of the Many Body problem
of N interacting electrons subject to an external potential [25]. Equations (2.12) to (2.15) provide
the theoretical basis of DFT, which is unlike the methods discussed previously is exact in
principle.
2.3 Implementation of DFT
2.3.1 Simplified Algorithm
The implementation of DFT requires solving equations (2.12) to (2.15) as demonstrated in the
simplified flow chart of Figure 2.1. To date, no exact representation of the exchange-correlation
potential Vxc[n(r)] in (2.13) exists. Thus, as presently implemented, DFT ceases to be exact. The
numerical implementation of the algorithm shown in Figure 2.1 involves many approximations.
Some of these approximations are in evitable, such as in the case of the Vxc[n(r)], whereas others
are necessary to render the computational time reasonable, such as the pseudopotentials
approximation (discussed later). Some approximations are dictated by the discrete nature of
numerical methods such as the need for a basis set to represent the wavefunctions, and some are
related to the numerical techniques used to solve the system of equations (2.12) to (2.15). Below
is a list of three of the main conceptual approximations used in the implementation of DFT.
These approximations will be discussed briefly in subsequent sections.
16
The exchange-correlation approximation (to represent the Vxc[n(r)]).
Basis Set approximation (to represent the wavefunctions).
Pseudopotentials approximation (to represent the interaction of the nuclei and core electrons
with valence electrons).
A range of techniques is available to handle each approximation. Determining the best technique
depends on the physical system investigated and requires understanding of the nature of each
approximation in terms of its advantages and limitations.
Figure 2.1 Simplified flow chart of DFT computations (modified from [26]).
2.3.2 Exchange-Correlation Approximation
All the terms which contribute to VKS (2.13) can be evaluated exactly, except for Vxc, the
exchange correlation potential [19]. No exact explicit form of the exchange correlation energy
Yes
Assume an Initial Value of nin(r)
Calculate KS Effective Potential (2.13)
Solve KS Single Electron Equations for electrons i = 1 to N (2.12)
Calculate final electron density nf(r) (2.15)
No
Self Consistent?
nf ≈ nin
Calculate Final Values; Energies, Wavefunctions and other ground state properties
17
Exc is available, and consequently none is available for Vxc. Moreover, no clear guidelines are
available for a systematic approximation of Exc [16]. The approximation of Exc is central to how
well DFT performs. Two main classes of approximations of Exc exist, the local density
approximation (LDA), which is the approximation used in the present work, and the generalized
gradient approximation (GGA). The Local Density approximation (LDA) is surprisingly
successful for many systems [21], in spite of being very simple in principle. The approximation
was first proposed by Kohn and Sham [25]. According to the LDA Exc[n] is defined as
LDA
xc xcE [n] n(r) ε ( n(r) ) dr (2.16)
where xcε ( n(r) ) is the exchange correlation energy of an electron at point r in a homogeneous
electron gas of density n(r). The term εxc in (2.16) can be split into two parts, one which accounts
for exchange (εx) and the other which accounts for correlation (εc) [20]. Explicit exact analytical
expressions for εx exist, but none are available for εc [16]. Accurate numerical data for
representing correlation energy (εc) are available. Analytical expressions of εc are produced by
fitting to such data [13]. By comparing various material properties as computed using DFT and
the LDA with experimental values the range of validity of LDA became well established [18].
An extensive survey of LDA accuracy is beyond the scope of this document. In general, the use
of LDA within DFT results in prediction of energies, such as cohesive energy and ionization
energy of molecules, within 10% to 20% of experimental values. The LDA is usually accurate to
within 2 % of experimental values in determining the molecular structure of solids [27]. The
success or failure of LDA depends on the material studied. The LDA performs well in structural
studies of isolated chains and crystals of polymers such as PE [28,29]. For strongly correlated
materials, the deficiency of the LDA is most evident [30]. Although a theoretical basis for the
success of LDA is lacking [16,19], LDA reproduces many experimentally determined physical
properties of materials to a useful level of accuracy and is successful for most applications [27].
The LDA is an important part of the success of DFT [18-21]. The performance of DFT based on
the use of LDA deteriorates as the variation of n(r) becomes more rapid. The Generalized
Gradient Approximation (GGA) improves the performance of DFT for systems with more
rapidly varying n(r) [15,21] by making Exc[n(r)] an explicit function of both n(r) and its
gradient, n(r) [15,21]. Various mathematical forms of GGA are based on what achieves
numerically reasonable results for the physical phenomena studied rather than being derived
18
from a fundamental theory related to the material under investigation [16]. The choice of a
specific form of GGA depends on experience, as no single form is considered optimum [18,24].
In general, GGA provides more accurate results than LDA, especially for energies [31]. Methods
other than LDA and GGA are used to approximate Exc such as meta-GGA. However, LDA and
GGA continue to be the most common approximations used in DFT. While research continues to
improve LDA and GGA performance, they are regarded as reliable approximations for
representing Exc in DFT.
2.3.3 Basis Sets
The set of N single electron like equations given in (2.12) can be written concisely as
KS i i iH ψ (r) = E ψ (r) . (2.17)
The set of the coupled nonlinear integro-differential equations given in (2.17) is complicated to
solve [16]. A challenging aspect of the solution is the representation of the wavefunctions ψi
[21]. In DFT codes, ψi are usually represented as a linear combination of a predefined set of basis
functions Φk(r), as given in (2.18) below,
ψ (r) ∑ (r)
. (2.18)
The coefficients Ck are constants determined through solving (2.17). Starting with a ―suitable‖
set of basis functions, the substitution of (2.18) in (2.17) results in a set of linear algebraic
equations. Solving these equations determines the coefficients Ck, and, consequently, the
wavefunctions ψi through (2.18). Representing ψi as a combination of basis functions can, in
principle, be exact if the set of Φk is complete, which requires the number of functions Φk to be
infinite. Since the basis sets have to be finite, the representation of the wavefunctions in DFT is
approximate. The quality of the approximation depends on the ability of the basis sets to capture
the physical features of the wavefunctions, which, in turn, depends on the type and number of
basis functions Φk. The number of basis functions required for a good approximation depends
largely on the type of the basis function Φk [16]. If the function is chosen to capture the features
of ψi, fewer functions (Φk) are required which saves computational effort. Various basis set
functions have been used, the most common of which are Linear Combination of Atomic
Orbitals (LCAO), Plane Waves (PW), Atomic Sphere (AS), and Gaussian [20]. The LCAO used
19
are typically those of a hydrogen-like atom [21]. Hybrids of these basis sets can be used as well.
No basis function is generally superior. Each type of function is suitable to a range of
applications. In general, PW functions remain the simplest type and are very successful with
various materials [21]. On the other hand, the LCAO functions are more efficient and allow
handling larger systems than PW functions [32]. The SIESTA code uses LCAO functions.
2.3.4 Pseudopotentials
The pseudopotential approximation refers to representing the interaction of valence electrons of
an atom with the potential created by the atom core electrons and nucleus through the use of a
fictitious slowly varying potential [21]. The pseudopotential differs from the true potential of the
core electrons and the nucleus of an atom only up to a certain distance from the nucleus, known
as the cutoff radius (rc), beyond which the pseudopotential coincides with the true potential as
shown in Figure 2.2. In solids and molecules, core electrons of neighboring atoms do not usually
interact. Hence, the assumption that the orbitals of core electrons in atoms of solids or molecules
remain unchanged from those in isolated atoms is valid [15]. Solving the KS equations for
molecules or solids, representing the potential of the nucleus of an atom by a pseudopotential is a
reasonable approximation [21]. Based on the pseudopotentials approximation, the KS equations
need only be solved for the valence electrons under the pseudopotential effect of the ionic core
(nucleus and core electrons) [33].
The use of a pseudopotential decreases the number of electrons for which KS equations must be
solved, which allows handling larger numbers of atoms with less computation effort. In addition,
the use of a pseudopotential increases the stability of the numerical solution of KS equations by
avoiding calculating energies of core electrons, which are three orders of magnitude greater than
those of valence electrons. The pseudopotential approximation allows DFT computations to be
focused on calculating the energies and states of the valence electrons which ultimately
determine the material behavior [33]. Finally, the pseudo wavefunctions, which coincide with the
true wave functions beyond the cutoff radius, are slowly varying in space, easier to represent
numerically, and can be characterized with a smaller basis set [21]. Efficient pseudopotentials for
all elements in the periodic table have been developed [21] and various commercial and free
programs are available to generate pseudopotentials used in DFT codes.
20
Figure 2.2 Pseudopotentials and Pseudo wavefunctions (modified from [33]).
2.3.5 Boundary Conditions
DFT computations are applied to an atomic structure in a unit cell. Either zero boundary
conditions or periodic boundary conditions are applied at the cell boundaries. The effect of zero
boundary conditions is to create a finite structure that is isolated from the outside world. The
periodic boundary conditions create identical replicas of the unit cell in 3D, which creates an
infinite extension of the structure in the unit cell in 3D. Zero boundary conditions DFT codes are
more suited to treating finite structures (isolated molecules) and charged systems, while periodic
boundary conditions codes are more suited to treat bulk materials. The two boundary conditions
can be mixed to create structures such as wires (periodic in one dimension) or surfaces (periodic
in two dimensions). In SIESTA, where periodic boundary conditions are applied, the proper
choice of the unit cell shape and dimensions relative to the structure within it allows for the
creation of finite structures, surfaces and slabs (infinite in 2D and finite in the third dimension),
in addition to bulk structures (infinite in 3D) as shown in Figure 2.3. If the dimensions of the unit
cell are sufficiently larger than the atomic structure in the cell, the interaction between the
structure and its neighboring replicas vanishes (Figure 2.3a), and the structure can be regarded as
isolated or finite. In such case the unit cell is referred to as a supercell.
21
Figure 2.3 Various systems created through periodic boundary conditions applied to a unit cell. The unit cell, which includes the structure to be studied, is in the center of each system and is outlined by a black line. The periodic replicas created by the periodic boundary conditions are outlined by the dotted line. a) Isolated system where the structure and its periodic replicas are far enough apart that they do not interact (supercell) b) Slab of a material with an infinite extension in 2D and a finite extension in the third dimension creating a surface c) Slabs of two differing materials creating an interface d) Bulk structure of orthorhombic crystalline PE with infinite extension in 3D.
2.3.6 Moving the Nuclei; Geometry Optimization and Molecular Dynamics
Up to this point, the atomic nuclei are assumed to be fixed. In DFT computations, energy
minimization (geometry optimization) or molecular dynamics (MD) are the classical theories
commonly used to move the nuclei towards their final positions. As the nuclei advance towards
their final positions, the electrons, which are treated quantum mechanically, follow their motion
[18,20,21]. Ideally, the wavefunctions which describe the nuclei should be determined through
quantum mechanical computations similar to those applied for valence electrons. Treating the
nuclei quantum mechanically increases computational time and complicates the problem
substantially. Treating the motion of the nuclei classically is a very reasonable approximation
[21] that is justified by the fact that the wavelength of the nuclei wavefunctions is very small
compared to the distances between the atoms. The Born-Oppenheimer approximation justifies
treating the nuclei of an atomic system classically, while treating its electrons quantum
mechanically [18].
a)
b)
c)
d)
a)
b)
c)
22
Energy Minimization (Geometry Optimization or structure relaxation) The total energy of an
atomic system is a function of the states of electrons and the location of the nuclei. To reach the
ground state energy of an atomic system in a DFT calculation, the nuclei must be moved from
their initial positions to the positions which minimize the system total energy. The electron states
are decoupled from the nuclei states (positions) [26] which are allowed to vary based solely on
energy minimization. This process, through which the ground state (absolute zero temperature)
of an atomic system can be determined, is also referred to as structure relaxation or geometry
optimization. The basic method for performing structure relaxation is to represent the energy
Lagrangian as a function of nuclei positions and then minimizing the Lagrangian, subject to
other constraints, to find the minimum energy position of the nuclei and thus the whole system
[26]. Typical minimization techniques used with DFT computations are conjugate gradient (CG),
steepest descent, and Verlet algorithm. The CG method is the most successful, stable, and widely
used [26].
Molecular Dynamics While geometry optimization determines the absolute zero temperature
ground state of an atomic system, molecular dynamics (MD) can be used with DFT to impose
certain temperature, and/or pressure on the system. In molecular dynamics, the nuclei motion is
based on satisfying various canonical ensembles. This opens the possibility of studying the
system under various conditions beyond zero temperature conditions provided by DFT and
geometry optimization. For example satisfying a constant NVT (N stands for number of
particles, V for volume, and T for temperature) ensemble in MD allows the temperature of the
system to be controlled, and satisfying a constant NPT ensemble (P stands for pressure) in MD
allows the temperature and pressure of the system to be controlled. Among the common MD
methods which are used with DFT are Nose Thermostat for constant temperature simulations
[34] and Nose-Parrinello-Rahman for constant temperature and pressure simulations [35]. Shown
in Figure 2.4 is a flowchart which demonstrates how DFT computations are typically combined
with geometry optimization (structure relaxation) or molecular dynamics to study a system of
electrons and nuclei which form a molecule or a solid.
2.3.7 DFT in Brief
Density functional theory is a computational quantum mechanical method in which the Many
Body problem of interacting electrons is reformulated in terms of the electron density n(r) rather
23
than the many electron wavefunction Ψ. Unlike other computational quantum mechanics
methods such as Hartree-Fock or Configuration Interaction, DFT is exact in principle. However,
upon implementation, DFT becomes approximate, mainly, due to the lack of an exact
representation of the electron correlation potential which is necessary to represent the quantum
mechanical electron-electron interaction. Applying DFT involves a number of approximations
that have been tested successfully for various materials and in numerous applications. Using the
appropriate approximations requires a thorough understanding of the material and the properties
studied.
Figure 2.4 DFT and geometry optimization/molecular dynamics flowchart. The flowchart demonstrates how DFT combined with geometry optimization or molecular dynamics is applied to study a system of electrons and nuclei forming a molecule or a solid.
The ground state configuration of a system of electrons and nuclei which form a molecule, bulk
material, surface, or interface can be determined through a combination of DFT (for the
electronic structure) and optimization of the nuclear positions based on energy minimization.
Within that framework, DFT provides the solution of the eigenvalue problem given by the
system of equations given by (2.19) for the electronic states of the system.
24
[- + (r)] ψ (r) ψ
(r) (2.19)
where is the Laplacian operator. The first term in the brackets represents the kinetic energy
and the second term, VKS, represents the effective potential energy seen by an electron, which is
composed of the electron-electron interaction, the electron-nuclear interaction, as well as the
potential from any external electric field. The terms Ei and ψi are the eigenvalues and eigenstates
which represent the electronic energy states and the wavefunctions of the system, respectively
[36]. Once the electronic eigenvalues (i.e., energy states) and wavefunctions (i.e., eigenstates) of
the system are determined, various properties of the system, such as the electron density and
density of states (DOS), can be studied. DFT typically predicts the structural details of materials
within 2% of experimental values. In general, relative energies determined in DFT are more
accurate than absolute energies. Relative energies, such as electron affinity and work functions,
are determined within 2%, and dielectric constants of insulators within 5% of experiment. DFT
tends to underestimate the band gap of insulators by about 30%; however, changes in band gaps
and in the energy of impurity states relative to the band edge to which they are related are
rendered accurately [14,20,36]. In general, relative electronic energies (e.g., work functions,
band offsets, etc.) and relative changes in electronic structure are well represented by DFT [14].
2.4 The SIESTA Code
SIESTA (Spanish Initiative for Electronic Simulations with Thousands of Atoms) is both a
method and a computer code implementation thereof to perform electronic structure calculations
using DFT and MD simulations for atomic systems. SIESTA was developed with the goal of
handling large number of atoms with reasonable computational effort. SIESTA is an open source
code written in Fortran 90 which was developed over years by researchers from various
Universities mainly Spanish, British, and American [32]. DFT codes can be categorized based on
the boundary conditions and the basis set used to represent the wavefunctions. In that sense,
SIESTA can be characterized as a periodic boundary condition code which implements an
atomic orbitals basis set. Compared to other common DFT codes (such as those in Table 2.1), the
main advantage of SIESTA is its ability to handle large systems [32]. The theoretical details of
the SIESTA code are described in [32]. SIESTA is freely available to the academic community.
SIESTA has been applied to a large variety of systems including surfaces, interfaces, and bulk
material. SIESTA has been used to study adsorbates, nanotubes, nanoclusters, and biological
25
molecules. SIESTA has also been applied to study amorphous and crystalline materials, metals,
semiconductors, dielectrics, polymers, ferroelectric, and magnetic materials [13]. SIESTA has
been used successfully to study a variety of material properties such as dielectric constants, band
structures, and defect energies. SIESTA, along with other post processing codes can mainly
provide [37];
Electronic structure (atoms coordinates).
Energy eigenvalues.
Wavefunctions (using Denchar post processing code [37])
The system total energy.
Atomic forces.
Energy decomposition into; kinetic, hartree, external field, exchange-correlation, ion-electron,
ion-ion.
The inputs to a SIESTA simulation includes the usual parameters required for any numerical
solver such as convergence criteria, initial guesses, and rules for weighted iterations. The main
input parameters are those related to the conceptual approximations discussed in section 2.2. The
choice of such input parameters depends on the system studied and the goals of the study. Input
pseudopotential files must be available for each type of atom of the system under investigation.
SIESTA provides pseudopotential files for most elements. Pseudopotential files for SIESTA can
be generated by the user using the ATOM program which is provided by the developers of
SIESTA [37]. Following is a brief list of the main input parameters in a SIESTA computation.
1. Exchange Correlation approximation.
a. LDA (Two different types)
b. GGA (Three different types)
c. Hybrid (Weighted Mixture of the above)
2. Eigenvalue problem solution method.
a. Diagonal (5 different methods of diagonalization)
b. Order N (13 different adjustable parameters)
3. Orbital Basis Type.
a. User Defined
b. Finite Range Pseudo atomic orbitals (PAO) (Four different types)
26
4. Mesh Cut Off (fitness of real space grid).
5. K-grid Cut Off (fineness of the k-grid used for Brillouin zone sampling).
6. Parameters for self consistent solution of DFT equations.
7. Geometry optimization and MD techniques and parameters.
Various other parameters are required for running a successful SIESTA computation. The
SIESTA input parameters used in the present work will be discussed in subsequent sections as
well as appendices.
Table 2.1 DFT Codes classified according to boundary conditions and basis set implemented
Code Boundary Conditions Basis Sets License
SIESTA Periodic Atomic Orbitals Free Academic
VASP Periodic Plane Waves Commercial
ABINIT Periodic Plane Waves Free
Gaussian Zero Gaussian Type Commercial
DMOL Zero Atomic Orbitals Commercial
27
Chapter 3 Quantum Mechanical Modeling of Chemical Impurities in
Polyethylene
3 Quantum Mechanical Modeling of Chemical Impurities in Polyethylene
3.1 Review of DFT Studies of Polyethylene and Chemical Impurities Therein
Pure PE
In 1996, the LDA was used to calculate the structure of a single PE chain with helical symmetry
over a range of carbon backbone dihedral angles [29]. The minimum energy experimental and
theoretical structure was reproduced. In 1997, bond lengths and angles were computed for a
single PE chain and for the orthorhombic crystalline structure using the LDA approximation to
within 5% of experimental values. However, the error in the lattice constant Lb (7.41 Å in
experiment), which is perpendicular to the chains, was up to 9%. The GGA did not provide any
significant improvement in results when compared to LDA [38]. In 2000, the PE structure was
calculated using LDA with similar errors to [38]. However, all lattice constants were determined
using GGA to within 3% of experimental values [39].
DFT studies of PE band structure are more relevant to high field phenomena. In the early DFT
work of [29] and [38] (1996-1997), the band structure of a single PE chain was determined using
the LDA, and was in qualitative agreement with experimental and theoretical results [29,38]. In
2000, the band gap of PE was calculated as 6.7 eV and 6.2 eV using the LDA and GGA [39],
respectively, and the band structure of crystalline orthorhombic PE was determined along all
principle directions of the Brillouin zone [39]. In the band diagram determined using the LDA
and GGA, the profile of the valence band was in qualitative agreement with experiment [39]. In
2003 [40], the band gap of crystalline PE was calculated as 6.4 eV using LDA, which is within
range of other DFT studies [39,41,42]. The DFT computations underestimate the band gap of PE
by about 25% relative to the experimental value of 8.8 eV, which is typical of DFT studies of
dielectrics. The LDA rendered better estimates of the band gap of PE than the GGA in [40,43].
28
In 1997, the electron affinity of alkane chains (CnH2n+2) was determined starting from n = 1
(methane) to n = 36 (hexa-triacontane C36H74). The gradual increase in the negative electron
affinity appears to converge by n = 36 to about -0.75 eV, which is consistent with the trends in
the experimental data for alkane chains [44]. The electron affinity was defined as the difference
of the ground state energy of the neutral and the -1 charged molecule.
In 2000, the spatial features of the wavefunctions of the CBM (conduction band minimum) and
VBM (valence band maximum) states of PE were studied using DFT [41]. The CBM states show
an interchain character (interchain peaks), while the VBM states show an intrachain character
(extended along the chain backbone) [41]. The interchain character of the CBM states was
correlated with the well established negative electron affinity of PE [41]. In 2001, DFT was used
to investigate the electronic properties of PE surfaces [42]. The computed electron affinities of -
0.17 and -0.1 for surfaces which are perpendicular and parallel to the chains were within range of
the experimental electron affinity of -0.5±0.5 eV [45]. Surface states were identified at -1.2±0.5
eV relative to CBM. Other parameters of PE, such as cohesive energy, elastic constant [46], and
Youngs’ modulus [39], were computed using DFT in agreement with experimental values.
Chemical Impurities in PE
Few DFT studies have considered the effect of chemical impurities in PE, especially in the
context of HV insulation. Trap depths of chemical impurities which are commonly found in HV
cables insulation were determined using the DFT code DMOL [5,44,47,48]. Chemical impurities
which can be represented as a defect on the polymer chain, such as carbonyl groups, vinyl
groups, double bonds, and conjugated double bonds, were modeled as a chemical modification of
a single alkane chain [5]. Chains of 10 (decane) [5] 13 (tridecane) [35] and 15 carbon atoms [47]
were used as a representative of PE in various studies. Other chemical impurities which consist
of byproducts and large molecules, such as acetone, butanol, and water, were modeled in the
center of a triangle of three short linear chains [49]. The difference between the electron affinity
of pure ―short‖ PE chains and that of chains with impurities was determined using DFT to
provide an estimate of trap depths. The electron affinity was defined as the difference between
the energy of a neutral PE chain and a chain with a charge of -1. The carbonyl and conjugated
double bond impurities were responsible for the deepest traps. The DFT estimates of trap depths
were included in a DC conduction model in which the electron mobility was determined from a
29
multiple trapping electron transport model. The model provided realistic estimates only for low
fields from 10 kV/mm to 60 kV/mm [50].
3.2 Computational Details of the Present Work
In the present work, the ground state configuration of all structures, such as bulk PE and
chemical impurities therein, are determined through a combination of DFT (for the electronic
states) and optimization of the nuclear positions, which is based on energy minimization using
the conjugate gradient (CG) method. The computations can be represented by the flow charts of
Figures 2.1 and 2.4. The electron correlation potential is approximated using the LDA according
to the Ceperly Alder method (CA) [51]. The SIESTA code supports both LDA and GGA.
Although GGA is generally superior to LDA, this is not true in the case of PE [38,39]. In studies
of structural and electronic properties of PE [38,39] as well as other polymers [52,53], LDA and
GGA have their respective shortcomings and merits; however, their performance is generally
similar.
Pseudopotentials of the Troullier-Martins type [54] are used to eliminate the core electrons and
represent the nuclei potentials. The pseudopotentials provided by SIESTA were used for all
atoms studied in the present work (C, H, O, Br) except iodine (I). No pseudopotential was
provided by SIESTA for iodine, and it was generated as discussed in Appendix B.1 using the
ATOM program [37]. The SIESTA basis functions are finite-range pseudo-atomic orbitals [32].
The wavefunctions are expanded using a double zeta plus polarization (DZP) basis set. The DZP,
which is the richest basis set within SIESTA, employs more than twice the basis functions in a
minimal basis set. Additional flexibility is available through the inclusion of higher angular
momentum basis functions which allow capture of atomic polarization. Within this basis set, the
hydrogen and carbon atoms are described by 5 and 13 orbitals, respectively. The DZP basis sets
are sufficient to describe states above the vacuum level and to treat the CBM and states close
thereto.
The CG geometry optimization is based on requiring each component of the force on each atom,
which are calculated according to the Hellman-Feynman theorem [55-56], to be less than 0.05
eV/Å. This convergence criterion is typical of DFT studies [57]. The convergence of the self
consistent field (SCF) computations of DFT equations 2.12 to 2.15 (as demonstrated by the
flowchart of Figure 2.1) is achieved when the difference between input and output values at each
30
point in the grid in an SCF cycle is < 10-4
eV. A 3rd
order Pulay convergence accelerator with 0.2
mixing weight [37-58] is used in the SCF computations unless otherwise mentioned. Details of
other parameters which are related to SCF and energy minimization computations are in
Appendix C. Such parameters were tuned based on suggestions from the SIESTA user manual
[37-59], previous SIESTA studies [59], and trial and error. During the SCF computations in
SIESTA, the Hamiltonian matrix elements are partially computed on a real space grid, the
fineness of which is controlled by a parameter referred to as the ―Mesh Cutoff‖ [37]. For
periodic structures, Brillouin zone sampling is applied. The parameter that controls the fineness
of the grid which is used for Brillouin zone sampling is referred to as the ―Kgrid Cutoff‖ [37].
The values of Mesh Cutoff and Kgrid Cutoff were chosen as the minimum values at which a
further increase in Mesh Cutoff and Kgrid Cutoff will result in a change of the system total
energy of less than 0.01 eV. Mesh Cutoff and Kgrid Cutoff values are reported in Appendix C.
Since SIESTA is a periodic boundary condition code, the dimensions of the unit cell can be
chosen to model infinite (periodically repeating) structures or finite structures. In the latter case a
supercell which is much larger than the system under investigation is used so that the atoms of
adjacent cells do not interact. The SIESTA code is used to determine the electronic energy states
(i.e., eigenvalues), wavefunctions (i.e., eigenstates), potentials, and electron density, from which
various properties of the system can be determined, including, band structure, binding energies,
density of states (DOS), and the projected density of states (PDOS). The DOS is the number of
electronic states per unit energy and unit cell, as a function of energy. The PDOS is the
contribution of any orbital, atom, or group of atoms to the DOS of the system. Certain material
properties, such as the electron affinity, are determined through complementing computational
methods which are applied to SIESTA outputs, as will be explained in details later.
3.3 Modeling of Chemical Impurities in Polyethylene; From Single Chains to Crystalline Bulk Models
3.3.1 Models of Pure Polyethylene
The main goal of the present chapter is to develop suitable models of bulk PE in which the effect
of chemical impurities on the electronic properties of the polymer can be studied. The
application of DFT to complex systems requires a systematic, incremental approach which starts
from simple systems and moves toward more complex systems while establishing agreement
31
with experimental data at each step along the way. The following sections begin by modeling
impurity free (pure) single PE chains, both finite and infinite, and proceeds to modeling pure
crystalline PE. Chemical impurities are modeled in single chains as well as in a crystalline
environment. Finally, bulk models, which include disorder (Core-Shell models), are developed to
study chemical impurities in an environment that captures features of amorphous regions in
which chemical impurities are likely to exist. In addition, this chapter investigates whether
single, short PE chains, which have been used in previous DFT studies [5,6,47,48,49,50], are
sufficient to understand the effect of chemical impurities on the electronic properties of bulk PE.
Throughout the chapter, the DFT computational scheme and approximation parameters which
are used in the remainder of the thesis are validated.
Pure Single PE Chains
Using SIESTA, models of finite and infinite PE chains can be created with the proper choice of
the size and shape of the unit cell which is subjected to periodic boundary conditions.
Rectangular unit cells are used unless otherwise mentioned. The ―minimum unit cell‖ which is
required to create an infinite chain is the primitive unit cell of a PE chain. Such unit cell consists
of the two ethylene groups (C2H4) which are shown in Figure 3.1. The length of the unit cell in Z
direction matches the lattice constant of the primitive unit cell in Z direction (Lc). The lengths of
the minimum unit cell in X and Y directions are large enough (allow at least 10 Å separation
between atoms of the unit cell and the closest atom in their periodic replicas unless otherwise
mentioned) to avoid interaction between the chain and its neighboring replicas which are created
by the effect periodic boundary conditions. Infinite chains can also be created using multiples of
the minimum unit cell in Z direction as shown in Figure 3.2a, in which the unit cell consists of 5
multiples of the minimum unit cell (C10H20).
A finite PE chain must be terminated by a methyl group to saturate the carbon atoms at the ends
of the chain. Thus, the chemical structure of a finite PE chain in a unit cell is CnH2n+2 rather than
CnH2n as in the case of an infinite chain, where ―n‖ is the number of carbon atoms in the chain.
To create a finite PE chain, the length of the unit cell in Z direction must be large enough to
ensure no interaction between the chain and its neighboring replicas along that direction, as
shown in Figure 3.2b. In that sense, an isolated CnH2n+2 molecule is being modeled in a supercell.
32
Figure 3.1 The primitive unit cell of a PE chain. The unit cell in ZY plane is shown to the left. The atoms in the unit cell are shown in XY plane to the right.
Figure 3.2 Unit cells of infinite and finite chains. a) Unit cell of an infinite chain (C10H20) using 5 multiples of the primitive cell of a PE chain (C2H4). The unit cell dimension along the direction of the chain backbone (Z direction) creates an infinite extension of the chain, while the unit cell dimensions in X and Y directions are large enough to prevent interaction of the chain and its periodic replicas in X and Y directions. b) The unit cell of a finite PE chain consisting of 10 carbon atoms and terminated by methyl groups (C10H12). The unit cell dimensions in X, Y, and Z directions are large enough to prevent interactions with the neighboring periodic replicas in 3D.
The minimum energy structure of an infinite PE chain is determined by optimization of the
geometry (positions of nuclei) and the lattice constants, while in the case of a finite chain, only
the geometry is optimized. For an infinite PE chain, the calculated C-C and C-H bond lengths
b) a)
33
were 1.51 and 1.12 Å, which is in agreement with other DFT studies and within 2% of X-ray
diffraction data in [38,39]. The computed band gap of 8.5 eV is in agreement with previous DFT
estimates [38,39]. Taking into consideration the limitations of DFT, the accuracy of the results
for infinite PE chains is acceptable. The above band gap, average bond lengths, and angles were
also reproduced using integer multiples of the primitive unit cell (up to 20 multiples with a
C40H80 chain in the unit cell).
The variation of the band gap of finite PE chains of varying chain lengths from 10 to 60 carbons
is shown in Figure 3.3. The band gap of finite chains approaches that of an infinite chain as the
number of carbons increases, which is in agreement with trends in experimental and theoretical
work [2] as well as previous DFT studies [44,60]. The energetic location of highest occupied
molecular orbital (HOMO) and least unoccupied molecular orbital (LUMO) plays a significant
role in determining the electronic properties of any system. Hence, it is important to understand
how the orbitals of the terminal methyl groups affect the HOMO and LUMO of a finite PE chain
and, consequently, the VBM and CBM states. The DOS of C40H82 and the contribution of the
terminal methyl groups to the DOS as determined using PDOS analysis are shown in Figure 3.4.
The DOS plots are generated by fitting Gaussian functions of 0.1 eV width to the energy eigen-
values which are determined using DFT. The reference energy in the DOS plots of the present
work is chosen to be the vacuum level. The vacuum level for systems, which are finite in two
dimensions, is determined as discussed in Appendix A.1. Traditionally, the Fermi energy level
(Ef) is indicated in DOS plots; however, the Ef which is determined by SIESTA for a bulk
dielectric has no physical significance except to indicate the occupancy of states. The ―true‖ Ef of
an insulator can only be determined in the presence of an interface with a metal through the
―bulk plus band lineup‖ procedure [14]. As shown in Figure 3.4, the orbitals of the terminal
methyl group do not contribute to the HOMO and LUMO and, accordingly, not to the CBM and
VBM states of a system created by short PE chains. The average bond lengths of a C40H82 chain
are identical to those of an infinite chain. Past experimental and theoretical results [2], and the
present DFT computations, indicate that a 40 carbon atom chain (C40H82) provides a good
approximation of the electronic properties of an infinite PE chain. Finite chains with various
lengths (CnH2n+2, n = 10 to 15) were used in previous DFT studies of PE [5,6,47,48,50]. The
present work suggests that such chains are too short to represent infinite PE chains adequately
[5,6,47,48,50].
34
Figure 3.3 The variation of the band gap of finite PE chains versus chain length. The chain length is indicated by the number of carbon atoms in the chain (n) (CnH2n+2). The band gap of infinite PE chain is indicated by the gray dashed
line.
Figure 3.4 DOS of a pure 40 C atom finite chain (C40H82). The contribution of the orbitals of terminal methyl groups is shown in gray. The vacuum level is taken as reference energy. The CBM and VBM energies are 1.51 and -6.31 eV.
Band gap of an
infinite chain = 8.5 eV
Number of carbon atoms in a finite chain CnH2n+2
Ener
gy (
eV)
DO
S (
arb. U
nit
s)
Energy (eV)
35
Pure Crystalline PE
The primitive unit cell of crystalline orthorhombic PE consists of two chains (Chains ―a‖ and ―b‖
in Figure 3.5). Each of these chains is formed of two ethylene groups [39]. Using the
approximations and DFT parameters which are described in section 3.2, the lattice constants of
bulk PE, La, Lb, and Lc, are calculated as 6.63 Å, 4.54 Å, and 2.53 Å, respectively. Compared to
measurements in [39], the largest error in the lattice constants is in the value of La (10% and 6%
for X-ray and Neutron beam measurements, respectively) which is similar to errors in other LDA
studies of crystalline PE [38,39]. The C-C and C-H bond lengths are calculated as 1.51 Å and
1.12 Å, and the C-C-C and the H-C-H bond angles are 113.7 degrees and 105.13 degrees, which
are all within 2% of X-ray diffraction measurements in [38,39]. The computed data are also in
excellent agreement with prior DFT work [38,39]. The computed band gap of 6.39 eV is within
1% of other LDA estimates [39-42]. Taking into consideration the limitations of DFT, the
present estimates of the structure and band gap of crystalline orthorhombic PE are satisfactory.
Figure 3.5 Structure of orthorhombic crystalline PE. The primitive unit cell, formed of chains (a) and (b), is shown in XY plane with lattice constant La and Lb in X and Y directions, respectively. Z direction is in the paper with a lattice constant Lc = 2.54 Å (Modified from [38]).
36
3.3.2 Models of Chemical Impurities in Polyethylene
Chemical Impurities in single PE chains
PE structures with chemical impurities can be modeled by modifying the initial structure of PE
to include the impurity and then optimizing the modified structure. Figure 3.6 shows the
minimum energy structure of a 40 carbon atom (40 C) PE chain which includes a carbonyl group
(C40H80O). Two hydrogen atoms were removed from an ethylene group and an oxygen atom was
introduced in the vicinity of the carbon atom from which the hydrogen atoms had been removed.
In the minimum energy structure (Figure 3.6), a π planar double bond is formed between the
oxygen and the carbon atom (C=O), as confirmed by the bond length, angles, and the PDOS, all
of which indicate the formation of a carbonyl group. The effect of the impurity on the electronic
properties of the chain can be investigated through the DOS and PDOS analysis. The DOS of the
40 C chain with a carbonyl impurity (C40H80O) and the contribution of the orbitals of the
impurity atoms are shown in Figure 3.7. Two states appear in the band gap of the chain, an
occupied state 1.63 eV above the VBM and an unoccupied state 2.91 eV below the CBM. No
such states exist in the band gap of a pure chain (Figure 3.4). The PDOS analysis shows that the
band gap states are formed mainly by the orbitals of the impurity atoms (the carbon and the
oxygen atoms of the carbonyl group), in particular the 2p orbitals of the oxygen and carbon
atoms which form the carbonyl group. Thus the new states in the bandgap can be classified as
impurity states. The energies of the unoccupied impurity state below the CBM and of the
occupied impurity state above the VBM are referred to as the ―depths‖ of the impurity states.
The above approach will be used to model various impurities, confirm their formation, and
analyze their effect on the electronic properties of PE.
Figure 3.6 Minimum energy structure of a 40 C atoms chain with a carbonyl impurity (C40H80O).
37
Figure 3.7 DOS of a 40 C atoms chain including a carbonyl group (C40H80O). The contribution of the orbitals of the carbonyl group to each state is shown in gray.
Interaction of Chemical Impurities
Multiple impurities can be studied simultaneously in the same chain. In that case, one should
determine whether the impurities are interacting or sufficiently separated such that they do not
interact. If two carbonyl impurities are separated sufficiently (usually with a separation of about
10 Å or 8 carbons along the backbone), the impurity states they introduce appear as degenerate
states in the DOS of the polymer chain. As the carbonyl impurities approach each other on the
polymer backbone, their impurity states split (i.e., become non degenerate). Figure 3.8 shows the
energy which separates the two unoccupied and the two occupied band gap impurity states of
two carbonyl impurities in a 40 C chain (C40H78O2) versus the distance which separates the
carbonyl groups. The separation sufficient to prevent interaction is 7.6 Å.
Chemical Impurities in Crystalline PE
If carbonyl impurities are modeled in a crystalline structure which is created by the primitive unit
cell in Figure 3.5, the neighboring carbonyl replicas are separated by 6.63, 4.54, and 2.53 Å in X,
Y, and Z directions, respectively. Such separations are less than the separation necessary to
prevent interaction between the impurity atoms. Using the primitive cell also implies an
Energy (eV)
DO
S (
arb. U
nit
s)
38
unrealistic ratio of carbonyl impurities to ethylene groups. The disturbance that a carbonyl
impurity causes to the primitive unit cell of PE will be substantial, such that the system will not
correspond to bulk PE with an impurity [61]. Therefore, impurities in crystalline PE are studied
in a unit cell which is formed of multiples of the primitive unit cell. Such an enlarged unit cell
ensures that impurities do not interact appreciably with their periodic images. In the present
work, an enlarged 192-atom unit cell (or bulkcell) which consists of eight PE chains was used to
study impurities in crystalline environment of PE (Figure 3.9). Each of the eight chains consists
of eight ethylene groups. DFT computation time increases proportional to N3
, which imposes a
constraint on the number of atoms studied. Thus the size of the crystalline bulkcell is a
compromise between decreasing computation time, preventing interaction between neighboring
replicas of impurities, and minimizing the effect of carbonyl on the overall crystal structure.
Figure 3.8 Energy separating impurity states of two carbonyl impurities in a 40 C atoms chain versus distance separating the carbonyl impurities.
The minimum energy structure of the crystalline bulkcell without impurities is determined
through DFT and optimization of nuclei positions and lattice constant, after which the optimized
structure of the unit cell is modified to include the impurity. Finally, DFT and geometry
optimization of the modified structure determine the minimum energy structure of the crystal
Distance separating carbonyl groups in Å
Ener
gy s
epar
atin
g i
mpuri
ty s
tate
s (e
V)
39
bulk system including the impurity. In the latter step when the impurity is included, the lattice
constants are fixed such that the computations correspond to a single isolated impurity in an
infinite solid [61]. In the crystalline bulkcell, the interaction between impurities and their
neighboring replicas is negligible. The minimum energy structure of the pure bulkcell of
crystalline PE has bond lengths, and angles within 0.6% of values calculated using the primitive
unit cell (i.e. within 3% of experimental values in [38,39]). The computed band gap of 6.39 eV is
identical to that determined using the primitive unit cell. The vacuum level in such a 3D infinite
bulk is determined using the ―bulk plus band lineup‖ procedure which is described in [14,62]
(Appendix A.2), on the basis of which the electron affinity is -0.17 eV, in excellent agreement
with prior DFT work [42] and within the range of the experimental value of -0.5±0.5 eV [50].
Based on computed structure, band gap, and electron affinity, and taking into consideration the
limitations of DFT, the crystalline bulkcell model is satisfactory in the context of the present
work which suggests that the approximations and parameters used in the present DFT approach
are appropriate.
Figure 3.9 Crystalline bulkcell of PE. The bulkcell which creates an infinite bulk through boundary replicas is in the black frame. Each chain in the supercell has 8 ethylene groups in the Z direction (into the paper). The neighboring replicas, shown in the dashed gray frames, are created by the effect of the periodic boundary conditions.
To create a carbonyl impurity in a crystalline PE environment, two hydrogen atoms of an
ethylene group are replaced by an oxygen atom in the optimized crystalline bulkcell. The
minimum energy structure of the bulkcell with the impurity is determined through DFT and
geometry optimization. The resulting O=C bond is a π planar double bond, as confirmed by the
bond length, angles, and PDOS, all of which indicate the formation of a carbonyl group. The
40
carbonyl impurity introduces local disorder in the PE structure, as the ethylene groups close to
the carbonyl impurity have shorter carbon-carbon bond lengths than those which are further
away. The CBM and VBM energies are within 0.7% of those of the structure without the
impurity. The perturbation of the overall PE structure and band gap caused by the impurity is
less than that in [63] as a result of a model which is closer to a single, isolated defect in an
infinite solid [61]. The band gap of the crystalline bulkcell with the carbonyl impurity has two
impurity states, an occupied state 0.68 eV above the VBM and an unoccupied state 1.81 eV
below the CBM.
Although single chain models have been used to study the effect of chemical impurities in PE
[5,6,47,48,49,50], results based on such models differ significantly from those based on a
crystalline bulkcell model. The difference between the band gap of a single PE chain (~ 8.5 eV)
and crystalline PE (~ 6.4) as determined by DFT is 2.1 eV (~ 33% referred to the experimental
crystalline band gap). In the case of carbonyl impurities, the depths of the impurity states in a
single chain (40 C atoms) and in a crystalline bulkcell differ by 0.95 eV and 1.11 eV for the
occupied and unoccupied impurity states, respectively. The depths of the impurity states are
important to conduction processes (as demonstrated later), and differences in the range of 1 eV
are significant given the typical activation energies in the conduction process of about 1 eV. In
addition, the interaction of an impurity with neighboring chains, which is undoubtedly important
to the conduction processes, can only be studied in a bulk model. Thus single chain models of
chemical impurities should not be used to study the effect of chemical impurities on the
electronic properties of bulk PE.
The crystalline bulkcell model is an important step towards studying chemical impurity states in
semi-crystalline PE and is a reasonable representation of crystalline regions therein. However, in
the context of amorphous regions of semi-crystalline PE in which impurities are likely to exist,
the crystalline bulkcell model is over constrained with artificially imposed periodicity. When
impurities are included in a crystalline bulkcell model, symmetry is preserved, and the
backbones of the PE chains remain parallel with almost fixed separations. Such features are not
characteristic of amorphous regions in which impurities are more likely to exist. In the next
section, a ―Core-Shell‖ model, which is developed in the present work to capture more features
of the amorphous state of PE, is discussed.
41
3.4 Core-Shell Model
The rationale behind the Core-Shell model is to study a chemical impurity in a PE chain which is
sufficiently long to behave as infinite, while including the effect of neighboring chains in an
amorphous like environment. The initial structure of the Core-Shell model (Figure 3.10) consists
of a ―Core‖ chain, to which the chemical impurity is added, surrounded by six ―Shell‖ chains
with crystalline PE spacing and orientation. The seven chains are in a unit cell which is large
enough to prevent interaction with chains in neighboring replicas (supercell). Each of the seven
chains consists of 40 carbon atoms and is terminated by a methyl group (C40H82). As discussed
above, the C40H82 chain is a good approximation of the electronic properties of infinite PE
chains. The Core-Shell model has around 900 atoms, which is large in the context of
computational quantum mechanics. The Core chain is surrounded by what would be the first
layer of neighboring chains in a crystalline environment. The Shell chains provide the Core chain
and the region in its vicinity with a reasonable approximation of the surrounding amorphous bulk
environment in that the chains are free to distort around the impurity without the constraint of a
crystalline environment.
Figure 3.10 Initial structure of a Core-Shell model. The Core-Shell initial structure in XY plane is shown above and in ZY plane is shown below.
When an impurity is added to the Core chain, the minimum energy structure is distorted
significantly from crystalline periodicity in the vicinity of the impurity. The minimum energy
structure of a Core-Shell model with a carbonyl impurity in the Core chain is shown in Figure
3.11 along with the minimum energy structure of a similar part of the crystalline bulkcell model
42
which includes a carbonyl impurity. Figure 3.11 demonstrates the ability of the Core-Shell model
to capture physical disorder, which is common in amorphous parts of PE.
Figure 3.11 The Core-Shell structure and crystalline bulkcell structure with carbonyl impurities. The optimized Core-Shell structure including a carbonyl impurity in the Core chain is shown to the right, and an equivalent part of the crystalline bulkcell structure is shown to the left. Bending of the Shell chains in the vicinity of the carbonyl group is evident as compared to the crystal supercell model or the initial Core-Shell structure without impurities in Figure 3.10.
The DOS of the Core-Shell model with a carbonyl impurity has a band gap of 6.78 eV, which is
close to other DFT estimates of the band gap and slightly higher than that of the crystalline
bulkcell model. The vacuum level is identified according to the procedure described in Appendix
A.1. The electron affinity of the Core-Shell model is -0.36 eV, which is within range of the
experimental value of -0.5±0.5 eV [45]. In the Core-Shell model, wavefunctions of VBM states
have an intrachain character (interchain peaks), while wavefunctions of CBM states have an
interchain character (localized maxima between the chains). The spatial features of VBM and
CBM states in the Core-Shell model are in agreement with previous DFT work [41] (discussed in
greater detail in Chapter 5). The depths of the carbonyl impurity states in the Core-Shell
environment are very close to those in a crystalline environment (differs by 0.27 eV for the
occupied state and by 0.15 eV for the unoccupied impurity state). The similar effect of the
carbonyl impurity on the DOS of both the over constrained crystalline bulkcell model and the
under constrained Core-Shell model suggests that the DOS of both models provides reasonably
accurate data for studying chemical impurities in semi-crystalline PE.
43
The Core-Shell model represents an acceptable approximation to amorphous PE which can be
implemented with a reasonable number of atoms and periodic boundary conditions. The Core-
Shell model has various appealing features in that the length of the Core-Shell chains allows
incorporation of multiple impurities simultaneously without interaction between impurities or
their neighboring replicas. The ratio of impurities to ethylene groups in the Core-Shell model is
more realistic than what could be achieved in a crystalline bulkcell model in the same
computation time. The Core-Shell model can accommodate large impurity atoms that reside
between the chains and cause significant deformation of the structure or large atoms which can
only be located in highly distorted amorphous regions (e.g., iodine impurities discussed in
Chapter 5). In such cases, the periodicity which is enforced in a crystalline bulkcell model
represents a much less accurate representation of the morphology when compared to the Core-
Shell model. As demonstrated later, the Core-Shell model reveals details which are missed by a
crystalline bulkcell model (discussed in Chapter 4). Based on the computed bond lengths and
angles, band gap, electron affinity, and the features of the wavefunctions of CBM and VBM
states, and given the limitations of DFT, the Core-Shell model is acceptable for studying the
effect of chemical impurities in bulk PE, and especially in the amorphous regions therein. Since
impurities are more likely to exist in the amorphous regions, the following discussion will be
based on the Core-Shell model. A typical Core-Shell model simulation with an impurity requires
about 7 weeks in a parallel SIESTA computation using an 8 Core computer with the
specifications in Appendix D. Two of such computers were dedicated for the thesis work.
44
Chapter 4 Effect of Chemical Impurities on Solid State Physics of
Polyethylene
4 Effect of Chemical Impurities on Solid State Physics of Polyethylene
4.1 Characterization of Chemical Impurities Studied
The chemical impurities considered in this chapter include carbonyl, vinyl, double bond, and
conjugated double bond impurities. These impurities, shown in Figure 4.1, have been identified
experimentally in PE [5,64] and are among the most common chemical impurities in XLPE cable
dielectric [5,64]. The procedures employed in this chapter to study the effect chemical impurities
on the solid state physics of PE, including the use of the ―Core-Shell‖ model, can be applied to a
wide range of impurities. The impurities are studied in the Core chain of a Core-Shell model.
Carbonyl is discussed in greater detail than other impurities since it is probably the most
common impurity in PE and increases conductivity thereof [65,66]. The molecular structures of
the impurities studied in the present chapter are described below. The labeling of atoms in the
description refers to the labeling in Figure 4.1. The calculated bond lengths and angles of the
impurities are in Table 4.1. All bond lengths are within 2.5% of X-ray and neutron beam
diffraction measurements in [67].
A carbonyl impurity is composed of a carbon atom in a PE chain, which is double bonded to an
oxygen atom. The carbon atom of the carbonyl impurity (C) is single bonded to the adjacent
carbon atoms (Ca and Cb). The atoms C, O, Ca, and Cb are in one plane. A vinyl impurity consists
of a carbon atom in a PE chain (C1) which is double bonded to a side carbon atom (C2) instead of
two hydrogen atoms as in other ethylene groups. The side carbon atom (C2) is single bonded to
two hydrogen atoms (H1 and H2). The atoms C1, C2, Ca, Cb, H1, and H2 are in one plane. A
double bond impurity consists of two carbon atoms in a PE chain (C1 and C2), which are double
bonded together. Each of these atoms is single bonded to one hydrogen atom (H1 and H2) and is
single bonded to the carbon atoms of neighboring ethylene groups (Ca and Cb). The atoms C1, C2,
H1, and H2 are in one plane. A conjugated double bond impurity consists of 4 carbon atoms (C1,
C2, C3, and C4), each of which is single bonded to one hydrogen atom (H1, H2, H3, and H4),
single bonded to one of its two neighboring carbon atoms, and double bonded to the other. The
45
conjugated double bond impurity includes two double carbon-carbon bonds (C1=C2 and C3=C4)
and one single carbon-carbon bond (C2-C3). The atoms C1, C2, C3, and C4 and H1, H2, H3, and H4
are in one plane.
Figure 4.1 Various common chemical impurities in the minimum energy structure of a Core-Shell model, a) carbonyl, b) conjugated double bond, c) vinyl, and d) double bond.
Table 4.1 Bond lengths and angles of impurities in Figure 4.1.
Impurity Bond Lengths in Å Bond Angles in Degrees
Carbonyl C=O
1.23
OCCa, OCCb
121.27, 120.8
Vinyl C1=C2, C2-H1, C2-H2
1.35, 1.11, 1.11
C2C1Ca, C2C1Cb,
121.77, 121.71
Double Bond C1=C2, C1-H1, C2-H2
1.35, 1.12, 1.12
CaC1H1, CbC2H2,
118.51, 118.31
Conj. Double
Bond
Ca-C1, C1=C2, C2-C3, C3=C4, C4-Cb,
1.49, 1.36, 1.44, 1.36, 1.49
C1C2C3, C2C3C4, C3C4Cb,
124.41, 124.3, 124.72
The formation energy Ef of an impurity ―X‖ in bulk PE is defined as
a) b)
c) d)
46
[ ] = [ + ] - [ ] - ∑ (4.1)
where, E[PE+X] is the total energy of bulk PE including the impurity X, E[PE] is the total
energy of bulk PE without the impurity, ni indicates the number of atoms of type i that were
added (ni positive) or removed (ni negative) from bulk PE to create the impurity, and i are the
corresponding chemical potentials of the added or removed species [61]. Based on (4.1), the for-
mation energies of carbonyl, vinyl, double bond, and conjugated double bond impurities are 1.63
eV, -1.19 eV, -2.47 eV, and -4.41 eV, respectively. The formation energy of carbonyl in the
crystalline bulkcell model is 1.77 eV, which is slightly higher than in the Core-Shell
environment, as would be expected, since the Core-Shell system has more opportunity to
decrease its energy through relaxation. The above formation energies depend on the choice of
chemical potentials, i, of species which are added or removed from the bulk to form the
impurity. The chemical potentials used in the present work are, μO as half the energy of O2
molecule, μH as half the energy of H2 molecule, and μCH2 as half the energy of C2H4 molecule.
The chemical reactions implied in forming the impurities are
X = Carbonyl: PE + ½ O2 → (PE+X) + H2
X = Vinyl: PE + ½ C2H4 → (PE+X) + H2
X = Double bond: PE → (PE+X) + H2
X = Conj. Double bond: PE → (PE+X) + 2 H2
4.2 Band Gap Impurity States
All the impurities introduce local disorder in the PE structure. The ethylene groups close to the
impurities have shorter carbon-carbon bond lengths than those further away. The disorder caused
by introduction of impurities in their vicinity is more pronounced in the Core-Shell model than
the crystalline bulkcell model. The band gap of the Core-Shell structure with any of the
impurities is 6.78 eV, which is in agreement with DFT estimates of the band gap of PE [39,40].
The band gap is underestimated by about 2 eV compared to the experimental band gap. Such
underestimation of the band gap with respect to experiments, occasionally by as much as 50%, is
a well-known deficiency of DFT. However, changes in band gaps and the energy of impurity
states relative to the band edge to which they are related are rendered accurately [14,20,36]. Thus
the band gap impurity states, such as those in Figure 4.2, will be referred to by their energies
relative to the adjacent band edge, which are relevant to the conduction process and rendered
with reasonable accuracy by DFT [14,20].
47
Figure 4.2 DOS of Core-Shell structure showing impurity states from a carbonyl impurity. The vacuum level is taken as reference. The occupied and unoccupied impurity states are formed mainly by the carbonyl group orbitals. The shallow impurity state is created indirectly by the carbonyl through the physical disorder it causes to adjacent ethylene groups.
All the chemical impurities studied introduce an occupied state above the VBM and an
unoccupied state below the CBM in the band gap of PE, which are similar to those shown for a
carbonyl impurity in Figure 4.2; however, the energy of the state relative to the adjacent band
edge varies with the impurity. A detailed description of the impurity states is provided in Table
4.2, including the depth of the state (energy relative to the adjacent band edge) and the type of
orbitals which forms the state, based on the PDOS analysis and wavefunction plots. The
occupancy of impurity states (as indicated in Table 4.2) is at the ground state of the system
(absolute zero temperature), and charge injection processes, which might change the occupancy
of the impurity states, are not taken into consideration. The wavefunctions were generated from
SIESTA outputs using a FORTRAN code based on DENCHAR program [37]. DFT
computations allow the inspection of the spatial features of impurity states. The square of the
unoccupied impurity state wavefunction, which indicates the spatial probability distribution of
electrons in that state, is shown for each impurity in Figure 4.3. The square of the wavefunction
of each occupied impurity state, which indicates the spatial probability distribution of
electrons/holes in that state, is shown for each impurity in Figure 4.4. The SIESTA
Carbonyl
occupied state
Carbonyl
unoccupied state
Shallow impurity state D
OS
(ar
b. unit
s)
Energy (eV)
48
wavefunctions of Figure 4.3 must be normalized to provide the probability density distribution
data shown in Figure 4.4. A FORTRAN code was written to normalize SIESTA wavefunctions.
All the electron probability density plots of impurity states in the present thesis have the same
scale.
Table 4.2 Impurity states depths and a description of their orbital formation.
Impurity Occupied state, Energy
above VBM (eV)
Unoccupied state, Energy
below CBM (eV)
Carbonyl π2p , 0.95 π*2p,, 1.96
Vinyl π2p , 1.0 π*2p, 0.97
Carbon-carbon double bond π2p , 1.06 π*2p, 1.06
Carbon-carbon conjugated
double bond π2p , 1.53 π*2p, 1.85
Figure 4.3 Square of the wavefunctions (probability density) of the unoccupied impurity states. Carbonyl impurity state (top left), double bond impurity state (top right), side vinyl impurity state (bottom left), and conjugated double bond impurity state (lower right) in the Core-Shell structure plotted in the XY plane. The plots are at planes which include the highest probability value of each impurity state.
Ele
ctro
n p
robab
ilit
y d
ensi
ty
49
Figure 4.4 Square of the wavefunctions (probability density) of the occupied impurity states. Carbonyl impurity state (top left), double bond impurity state (top right), side vinyl impurity state (bottom left), conjugated double bond impurity state (lower right) in the Core-Shell structure plotted in the XY plane. The plots are at planes which include the highest probability value of each impurity state. The probability scale in the plot is the same as that in Figure 4.3 and will be used in the rest of the thesis for electron probability density contour plots.
In addition to the impurity states discussed above, which are deep in the band gap, the carbonyl,
vinyl, and conjugated double bond impurities introduce ―shallow‖ unoccupied states slightly
below the CBM. The carbonyl and vinyl impurities introduce the states indirectly through the
physical disorder they cause in their vicinity, as shown by PDOS analysis and wavefunction
plots. The neighboring ethylene groups of the carbonyl and vinyl impurities have bond lengths
which are shorter than those of ethylene groups further from the impurity by about 1%. These
shallow states could not be identified when carbonyl was studied in the crystalline bulkcell
model. Figure 4.5 shows the CB edge of the Core-Shell model with a carbonyl impurity and of
the crystalline bulkcell model with carbonyl impurity. Shallow impurity states which appear as a
perturbation to the CB edge in Figure 4.5 are only evident in the case of the Core-Shell model.
On the other hand, the shallow impurity state which is introduced by the conjugated double bond
Ele
ctro
n p
robab
ilit
y d
ensi
ty
50
is caused mainly by the orbitals of the impurity atoms. The shallow impurity states due to
carbonyl, vinyl, and conjugated double bond are 0.22 eV, 0.21 eV, and 0.32 eV below the CBM,
respectively. The CBM states, shallow traps, and hopping sites have presence in the interchain
vacuum. Although a local orbital basis set has been used, the CBM and the shallow impurity
states are treated adequately (although they may suffer from the inherent deficiencies of the LDA
within DFT). The basis set used (DZP) includes a generous number of unoccupied states which
tend to be ―diffuse‖ or ―delocalized‖, i.e., they extend into the region away from the
corresponding atoms. Thus linear combinations of these functions have significant ―presence‖ in
the region between chains (e.g., between the Core and Shell chains) so that states localized in
such regions can be captured adequately by the present treatment.
Figure 4.5 DOS of a Core-Shell and a crystalline bulkcell structures including a carbonyl impurity. The shallow impurity state indirectly introduced by carbonyl through the physical distortion of its neighboring ethylene groups is only evident in the Core-Shell model. The vacuum level is taken as reference energy.
Computations of the Core-Shell model with more than one impurity in the Core chain
demonstrate that as long as the impurities are separated by more than five carbon atoms along the
polymer backbone, the states introduced into the band gap are independent of the other
impurities. The DOS of a Core-Shell model which includes various impurities, as that shown in
Figure 4.6, can be reproduced by superimposing the DOS plots of Core-Shell models which
DO
S (
arb. unit
s)
Energy (eV)
51
include each of the impurities separately while aligning the vacuum levels of the plots. The
ability of DFT to determine the source of each impurity state in the system, as shown in Figure
4.6, provides a basis for assessing the impact of each impurity independently, which is almost
impossible experimentally.
Figure 4.6 Impurity states introduced into the band gap of PE by various impurities. The arrows indicate which impurity creates which states. The dotted arrows refer to the creation of the impurity state indirectly due to physical distortion of ethylene groups adjacent to the impurity. A hydroxyl impurity has been included in the Core-Shell model of the figure. Hydroxyl impurities have been identified experimentally in PE [68]; however, they are not as common as the other impurities in the figure.
4.3 Discussion of Chemical Impurity States in the Context of High Field Phenomena
4.3.1 Impurity States as Trapping/Hopping Sites
The minimum energy required to elevate a carrier from the occupied VBM states to the
unoccupied CBM states in the impurity free PE is the band gap energy. In the presence of the
impurity states, this energy is reduced to the energy separation between the highest occupied
impurity state and the lowest unoccupied impurity state; thus the effective band gap of the
system is reduced. However, the 20% to 30% reduction of the effective band gap due to the
impurities studied in the present work (referred to the experimental band gap of 8.8 eV) is
unlikely to allow the activation of carriers from the VB to the CB in typical operating electric
fields and is unlikely to have a significant impact on high field conduction.
DO
S (
arb.u
nit
s)
Energy (eV)
52
The effect of chemical impurities on high field conduction is generally explained in terms of
traps which provide trapping/ hopping sites for carriers [3,4]. In conduction models, traps are
usually characterized through their energy depth (Table 1.1) and average separation. While
general estimates of trap depths are possible experimentally, the trap depth of specific chemical
impurities is difficult to determine. Although the concept of traps is central to high field
phenomena in dielectrics, such as conduction and space charge formation, a clear explanation of
the physical basis of traps and their correlation with chemical impurities is lacking, in spite of
extensive efforts [2-4]. In addition, unambiguous experimental determination of trap depths
remains a challenge.
The impurity states which are identified in the present work are localized in space since they are
mostly created by the orbitals of the impurity atoms. The wavefunctions of the impurity states
overlap with the states of the adjacent band (VB for occupied impurity states and CB for
unoccupied states). This provides the quantum mechanical basis for the exchange of carriers
between the impurity states and the states of the adjacent band. Figure 4.7 shows the spatial
features of overlap surface of the double bond unoccupied impurity state (1.06 eV deep) and a
CBM state and overlap surface of the carbonyl occupied impurity state (0.95 eV deep) and a
VBM state. Carriers which occupy the impurity states will be localized in space, i.e. trapped,
unlike carriers in the CB and VB states which are created by orbitals of multiple ethylene groups.
In contrast to the impurity states, the extension of CB and VB states in space and the small
energy separation of the states within each band prevents the localization of carriers therein. The
characteristics of states introduced by chemical impurities allow them to play the role of traps.
Accordingly, the effect of chemical impurities on high field conduction and space charge
formation is related to the energies of the impurity states they introduce relative to the adjacent
band edge and the nature of the impurity state wavefunctions. Such impurity states provide
trapping/hopping sites for electrons and holes. The energy difference between the unoccupied
impurity states and CBM provide a reasonable estimate of electrons trap depths (Table 4.2,
Column 2). Similarly, the energy difference between the occupied impurity states and VBM
provides a reasonable estimate of holes trap depth (Table 4.2, Column 1).
53
Figure 4.7 Overlap surface of impurity states and bands edge states. To the left, overlap surface between double bond unoccupied state and a CBM state is shown. To the right, overlap surface between carbonyl occupied impurity state and a VBM state is shown. The difference in the spatial features of the overlap in both cases is due to the difference in the spatial features of CBM and VBM states in addition to difference of the spatial features of the carbonyl and vinyl impurity states.
Electron traps created by chemical impurities can be defined as the unoccupied orbitals of the
impurity atoms which are associated spatially with the location of the impurity and have energies
below the energy of the CBM of PE. This energy difference is referred to as the electron trap
depth. Similarly hole traps created by chemical impurities can be defined as the occupied orbitals
of the impurity atoms which are associated spatially with the location of the impurity and have
energies greater than the energy of the VBM of PE. This energy difference is referred to as the
hole trap depth. Shallow traps, such as those created at the edge of the CB (Figures 4.5 and 4.6),
act as ―hopping sites‖. Hopping sites and trapping sites differ mainly by the ease of exchanging
carriers between them and the states of the adjacent PE band. The residence time of a carrier in a
trapping site is proportional to the exponential of the trap depth [5]. Due to the small depth of
hopping sites, carriers spend less time in them (hop rather than being trapped and later
detrapped) than in deeper impurity states (traps). Hopping sites are more likely to be associated
with the physical distortion created by chemical impurities to their neighboring ethylene groups
than with the impurity atoms. Figure 4.8 shows the depths of various holes and electrons
54
trapping sites and hopping site of carbonyl, vinyl, double bond, and conjugated double bond
impurities.
Figure 4.8 Energy diagram of the band gap of PE showing the depth of impurity states. The states are created by carbonyl, vinyl, double bond, and conjugated double bond impurities. Unoccupied impurity states are in green and occupied states are in red.
4.3.2 Comparison between Trap Depths Determined in the Present Thesis and Estimates in Literature
4.3.2.1 Comparison with Estimates Based on Macroscopic Modeling and Measurements
Literature estimates of trapping/hopping sites for electronic carriers are in the range 0.1 to 2.0 eV
[2,3,4,69]. Estimates of trap depths from various studies and the approach used in each study are
provided in Table 4.3. Some of these estimates are based on experimental approaches such as X-
ray induced thermally stimulated current (TSC) measurements [69]. Other estimates are based on
fitting conduction models, which include trap depths as an adjustable parameter, to current vs
electric field (I-F) measurements or space charge (SC) measurements. Most experimental
techniques and models (table 4.3) do not differentiate between trap depths for holes and
electrons. In general, traps caused by chemical impurities are deeper than those caused by
Ener
gy (
eV)
55
physical disorder [2,69,70] which agrees with the present predictions based on DFT. In the
present work, the trap depths for hole and electrons (given in table 4.2) are between 0.95 eV and
1.96 eV which agrees with estimates in literature of deep traps caused by chemical impurities
which are in the range of 0.8 to 2.0 eV [2,3,4,69,70,75]). The shallow traps (hopping sites)
determined within 0.32 eV of the CBM are also within the range of estimates in literature for
shallow traps caused by physical disorder (0.1 to 0.3 eV) [2,70].
Table 4.3 Various estimates of trap depths
Ref Method Details of Trapping States Estimate
2,67 Estimate Due to physical disorder from 0.1 eV to 0.3 eV.
Due to Chemical impurities at deeper levels.
0.1 to 2 eV
66,68 Experimental Various discrete states in the range indicated.
Deep traps are attributed to chemical impurities.
From 0.1 eV to 2 eV
70,72,
73,76
Fitting model to I-F
measurements.
Single trap level
1 eV - 1.15 eV - 0.85
eV - 1.08 eV.
71 Fitting model to I-F
measurements.
Continues parabolic distribution.
0.01 to 1 eV
69 Fitting computations to
mobility measurements
Two distributions Shallow in the range
of 0.3-0.6
Distribution of deep
traps around 1.5 eV
80 Fitting model to I-F
and Space charge (SC)
measurements.
Single trap level for holes and single trap level
for electrons
Holes 1eV
Electrons 1 eV
79 Fitting model to SC
measurements. Exponential distribution with deepest level
indicated for electrons and holes
Electrons 0.85 eV
Holes 0.79 eV
77 Fitting model to SC
measurements. Exponential distribution with deepest level
indicated for electrons and holes
Electrons 0.6 eV to
1eV for various data
sets
Holes 0.53 eV to 0.83
eV for various data
sets
81 Fitting model to SC
measurements. Single trap level for holes and single trap level
for electrons
Electrons 0.9 eV
Holes 0.8 eV
56
4.3.2.2 Comparison with Estimates from Previous DFT Work
The estimates of trap depths from the present work differ from previous DFT studies. The only
previous DFT estimates of electron trap depths of chemical impurities were based on electron
affinity computations [5,6,47,48,49,50]. No estimates of hole trap depths have been provided in
previous DFT studies. The impurities were studied in single short PE chains, and an electron trap
depth was defined as the difference between the electron affinity of a pure PE chain and the
electron affinity of the PE chain with the impurity. Such an approach can only identify the
shallowest trap created by the impurity. This limitation, in addition to the fact that the DFT
studies of trap depths in literature were based on single short PE chains, are likely to cause
substantial differences from values determined from Core-Shell models or crystalline bulkcell
models of PE used in the present work. Table 4.4 shows DFT estimates of trap depths in the case
of 10 C atoms chain in [5], which is too short to represent an infinite chain as discussed in
Chapter 3, and the shallow trap depths calculated in the present work.
Table 4.4 A comparison between DFT estimates of electron trap depths in [5,64] based on electron affinity computations and the estimates of shallow electron traps from the present work.
Method Estimates in [5,64] for 10 C chain
(eV)
Present work (eV)
Carbonyl 0.45,0.49 0.22
Vinyl 0.16,0.23 0.21
Double bond 0.04,0.16 1.06
Conjugated double bond
0.44,0.51
0.32
4.3.2.3 Carbonyl Trapping/Hopping Sites
The present work suggests that carbonyl impurities are responsible for both deep traps and
hopping sites. The calculated depths of hole traps, electrons traps, and hopping sites of carbonyl
impurities are 0.95, 1.96, and 0.22 eV, respectively, which are in general agreement with
estimates in literature. Experimental estimates of trap depths of carbonyl impurities are 1.5 eV in
[77] and 1.4 eV in [69]. Models that fit well to measurements of activation energies and I-F
(current vs electric field) characteristics of carbonyl doped PE were based on ~ 0.3 eV deep
hopping sites which were attributed to carbonyl impurities [3,65] and a trap depth of ~ 0.8 eV
(total activation energy ~ 1.1 eV). The carbonyl electrons trap depths based on electron affinity
57
computations were 0.49 eV and 0.45 eV in [5,64] which may correspond to the carbonyl hopping
site. The present work provides a clear association of carbonyl impurities with both trapping and
hopping sites, as has been proposed previously with no formal justification [3,65].
4.3.2.4 Summary
The conclusion that the band gap impurity states are the traps and hopping sites is supported by
the physical features of their wavefunction and the general agreement of their depths with
estimates of previous experimental and theoretical work. The impurity state wavefunctions are
localized in space, and their overlap with CBM and VBM states, as shown in Figure 4.7,
provides the quantum mechanical basis for the processes of trapping, detrapping and hopping. In
general, the combined effect of ~1eV deep holes traps and the ~1eV deep electrons traps
identified in the present work can explain the experimentally determined activation energy of
conduction in PE which is about ~ 1 eV, and is in general agreement with the ―effective trap
depth‖ in various conduction models (Table 4.2) which is in the range of 0.8 to 1.2 eV
[65,71,73,82]. Hopping sites are usually associated with physical disorder in the amorphous
regions and trapping sites with chemical impurities, both of which are necessary to account for
the dielectric properties of PE. However, the shallow impurity states/hopping sites due to
chemical impurities identified above suggests that the dielectric properties of PE, and in
particular high field conduction, may be entirely dominated by such impurities as they can
account for the observed solid state features of the material, although the degree to which
disorder in the amorphous region contributes similar shallow states even without chemical
impurities, is not yet known. The clear identification of trapping and hopping sites is believed to
be one of the main concluded contributions of the present work.
4.3.3 Interchain Extension of Impurity States
While the impurity states in the band gap are caused mainly by impurity orbitals, PDOS analysis
shows minor contributions from orbitals associated with the atoms of neighboring chains.
Accordingly, the wavefunctions of some impurity states have low amplitude peaks around
neighboring chains in spite of being localized mainly around the impurity atoms. This is
reflected on the spatial probability distribution (square of states wavefunctions) of electron/holes
occupying such states. The impurity states of the carbonyl occupied state and vinyl occupied and
unoccupied states have significant presence at neighboring chains compared to other impurity
58
states identified above as the probability surface plots in Figure 4.8 demonstrate (all electron
probability density isosurface plots in the rest of the thesis have the same value of probability
surface). The planar average of the square of the wavefunctions also shows that, in general,
carbonyl and vinyl impurity states are more extended in space towards neighboring chains than
double bond and conjugated double bonds impurity states. The conjugated double bond shallow
impurity state extends towards neighboring chains more than the conjugated double bond deep
trap states. The extension of the impurity states of carbonyl and vinyl towards neighboring
chains suggest that they can play a role in enhancing interchain charge transfer and thereby
increase conduction. Such role will be discussed in greater detail in the next chapter. The above
analysis demonstrates that although the energies of states from various impurities might be
similar, the states effect on conduction can differ based on the shape of the impurity states
wavefunctions, which is not usually discussed in literature.
Figure 4.8 The electron probability density of impurity states extended towards neighboring chain. a) carbonyl occupied impurity state, b) the double bond occupied band gap impurity state, c) vinyl occupied impurity state, and d) vinyl unoccupied impurity state. The isosurfaces in the various figures have the same value of probability. The carbonyl and vinyl impurity states are extended towards neighboring chains compared double bond and conjugated double bond states which are more localized around the impurity atoms. The double bond occupied impurity state is shown in b) for comparison.
a) b)
c) d)
59
Chapter 5 Effect of Iodine on Conduction in Polyethylene
5 Effect of Iodine on Conduction in Polyethylene
5.1 Conduction in Iodine Doped Polyethylene
Iodine, which can be diffused into PE from an aqueous electrode [8,9] or vapor [10,11],
increases the electrical conductivity of PE by about four orders of magnitude [8-12] and
decreases the thermal activation energy of conduction (determined from Arrhenius type curves)
from ~ 1 eV to about 0.8 eV [10]. Iodine also increases hole mobility and, to a much lesser
extent, electron mobility in PE [12]. A long standing problem in the field of insulating polymers
is the lack of a complete physical explanation of such effects. The effect of iodine on the
conductivity of PE is a good place to start an investigation of the effect of chemical impurities on
conduction in PE for various reasons. The effects of iodine are large compared to the more subtle
effects of common chemical impurities such as carbonyls, double bonds, etc. The effects of
iodine on conduction in PE are consistent over many investigations which involved various types
of PE, electrode material, and methods by which the iodine is introduced into PE. Thus the
experimental observations can be attributed to the interaction between iodine and the polymer
chains, while other factors, such as the concentration of other impurities or electrode material,
can be excluded. Understanding the effects of iodine at an atomic level and developing a
mechanism which explains the experimental observations discussed above would be a significant
step toward understanding the more subtle effects of common impurities such as carbonyl, vinyl,
double bonds, and conjugated double bonds.
In this chapter, a DFT analysis of the atomic level interaction between iodine and PE is
presented, on the basis of which a mechanism that explains the physical basis of observed effects
of iodine on conduction in PE is proposed. The interaction between iodine and PE is
characterized in terms of quantities such as binding energy, bond lengths, electron charge
density, changes in electron charge density, and mixing of atomic orbitals. Since studying
isolated In molecules is essential in identifying the interaction of In and PE, various stable
molecular configurations of In (n=2 to 5) are determined. The work proceeds by studying the
interaction of the most stable configurations of In with PE in Core-Shell models.
60
5.2 Isolated Iodine Molecules
The literature suggests that iodine diffuses in PE in the form of neutral I2 [8-12] with the
possibility of molecular aggregates In [9]. Accordingly, In molecules are considered with a more
detailed discussion of I2 molecule as it is expected to be the most abundant form of In in PE [8-
12]. The pseudopotential for iodine was generated using the ATOM program according to the
parameters given in Appendix B.1 and taking into consideration basic relativistic effects as they
are important to consider in case of heavy elements such as iodine. The calculated bond length of
I2 is 2.66 Å, which is within 1% of the experimental value [83]. The HOMO in I2 is a 2p π anti-
bonding (π*2p) orbital, and the LUMO is a 2p σ anti-bonding (σ*2p) orbital, both in agreement
with the accepted molecular orbital energy diagram of I2 [84]. The binding energy of an isolated
In molecule per atom is defined in 5.1
b n nE I / n = E I / n - E I (5.1)
where, E[In] is the ground state energy of the In molecule, E[I] is the ground state energy of an
isolated iodine atom, and n is the number of atoms in the iodine molecule. The calculated I2
binding energy per atom (Eb[I2]/2) of 1.58 eV is within 1% of the experimental value [85],
which is better than the anticipated accuracy for this type of DFT computation. The above
agreement with experimental data for isolated I2 molecule is satisfactory and validates the
various DFT approximations used in modeling iodine. Similar DFT computations for In>2 should
provide reliable data where, unlike the case of I2, experimental data may not be available for
verification.
As ―n‖ increases, the number of possible configurations of In increases rapidly, and exploring all
possible configurations is impractical. Thus symmetric configurations for each n value up to 5
were studied, as shown in Figure 5.1. The most stable configuration at each ―n‖ is that with the
greatest absolute value of Eb[n] [86]. Based on present DFT computations, linear configurations
and zigzag configurations are more favorable energetically than configurations with higher
degrees of symmetry such as triangular, tetrahedral, etc. Although other stable configurations of
In may exist, the discussion is limited to the most stable configurations indicated in Figure 5.1.
61
Figure 5.1. Stability of various configurations of In (n = 3 to 5). Stability decreases from the Ina to In
d.
A plot of the binding energy per atom (Eb[In]/n) as a function of ―n‖ for the most stable
configurations in Figure 5.1 is shown in Figure 5.2. The binding energy per atom does not
change monotonically with n, which indicates that the stability of In relative to In-1 and In+1 might
not vary monotonically with molecular size (n). The incremental binding energy, Einc[In], as
defined in (5.2), determines the stability of an In molecule relative to In-1 and In+1 [87,88]. The
analysis of Einc[In] (Figure 5.3) indicates that I2 and I4 are more stable than I3 and I5 [87,88].
inc n n-1 nE I = E I + E I - E I (5.2)
Figure 5.2 Binding energy per iodine atom (Eb[In]/n) for the most stable In configurations at each n (n = 2 to 5).
I3a I3
b I3
c I4
a I4
b
I4c
I5a I5
b
I4d
Most stable Stable
Unstable
relaxes to I3a
Most stable
Stable
Unstable relaxes to I4a Most stable Stable
Stable Stable
Stable
I3a I3
b I3c
I4a I4
b
I4c I4
d I5a I5
b
Number of atoms (n) in
B
indin
g e
ner
gy p
er a
tom
(eV
)
Number of atoms (n) in In
62
Figure 5.3 Incremental binding energy (Einc[n]) for In from n = 2 to 5.
5.3 Interaction between Iodine and Polyethylene
The interaction between the most stable configurations of In for n = 2 to 5 and PE is studied in
Core-Shell models (PE-In models) similar to that shown in Figure 5.4. The chain deformation in
the vicinity of the iodine molecules, shown in Figure 5.4, allows the model to capture features of
the amorphous regions of PE in which iodine molecules are most likely to be present [8-12]. The
bond length of iodine molecules in PE-In increases by about 1% compared to isolated In. The
binding energy of the In molecules per iodine atom to the PE chains in the Core-Shell model Eb
PE-In / n is defined as
b PE-In n nE / n = E PE+I - E PE - E[I ] / n (5.3)
where, E[PE+In] is the total energy of the Core-Shell model with In and E[PE] is the total
energy of the Core-Shell model without In. As shown in Figure 5.5, the minimum Eb PE-In per
iodine atom is 0.32 eV, which is about an order of magnitude less than the binding energy per
atom of isolated iodine molecules (Eb[In]/n shown in Figure 5.1). The In-PE binding energy per
iodine atom, which is an order of magnitude greater than the thermal energy, is large enough to
ensure that iodine forms stable structures in PE.
Number of atoms (n) in
Incr
emen
tal
bin
din
g e
ner
gy (
eV)
Number of atoms (n) in In
63
Figure 5.4. Core-Shell model of PE with I2 molecule (PE-I2) shown from two perspectives. The I2 molecule resides between deformed PE chains in an amorphous like environment.
Figure 5.5. Binding energy of iodine molecules to PE chains in Core-Shell model per iodine atom (Eb PE-In / n).
The electron charge density of the PE-I2 model is shown in Figure 5.6 for two planes, one
through the I2 molecule and the other away from it. As shown in Figure 5.6, the electron charge
density in the vicinity of the I2 molecule is extended between chains, which is evidence of the
interaction between the I2 and PE chains. The electron density is localized to the polymer chains
away from the I2. Analysis of the change in electron charge density in the vicinity of the I2
Number of atoms (n) in In
PE
-I n
bin
din
g e
ner
gy p
er I
ato
m (
eV)
64
molecule in the Core-Shell model provides insight into the charge redistribution which takes
place with introduction of I2.
Figure 5.6. Electron charge density contours in PE-I2. Shown to the left is a plane through the I2 molecule, and to
the right, a plane far from the I2. Charge density contour lines extend between PE chains only in the vicinity of the I2
molecule. The plots are in the XY plane where the Z direction is along the chains backbones and into the paper. The
scale units are in electron/Å3
The ―difference electron charge density‖, which is the PE-I2 electron charge density after
subtracting the electron charge density of an isolated I2 molecule and of the PE chains without
the I2 molecule followed by relaxation of the electronic states only [89], shows a region of
charge accumulation around the I2 molecule and a region of charge depletion closer to the PE
chains as shown in Figure 5.7. This is consistent with the greater electronegativity of I2 relative
to carbon and hydrogen. The electron charge density of PE-In systems (n=3 to 5) has similar
features to that of PE-I2. The electron charge density distribution is determined by the occupied
states and their wavefunctions. The interaction between iodine and PE that creates the charge
density features in Figure 5.6 must be reflected on the occupied states of the system, which can
be investigated through DOS and PDOS analysis.
Electron charge density (electron/Å3)
65
Figure 5.7. The difference electron charge density in PE-I2 with isosurfaces corresponding to regions of charge depletion and accumulation. A region of charge depletion of 0.08 electron/Å
3is shown in blue of and a region of
charge accumulation of 1.08 electron/ Å3 is shown in red.
5.4 Iodine Impurity States in Polyethylene
The iodine impurity states in the PE-I2 Core-Shell model, shown in Figure 5.4, along with their
energies relative to the CBM and the VBM, are shown in an energy diagram in Figure 5.8. The
band gap is 6.78 eV in agreement with DFT estimates of PE band gap [90]. The iodine
introduces four impurity states, three of which are in the band gap, two of which are occupied
while the third is unoccupied. The fourth state is an occupied impurity state just at the edge of
the valence band (VB) and is considered a ―mixed impurity state‖ for reasons explained below.
The process of hole and electron injection at the electrode, which might change the occupancy of
the impurity states, is not considered.
Conduction can take place through transfer of electrons between unoccupied impurity states and
the conduction band (CB) of PE, whereas holes can be transferred between occupied impurity
states and the VB of PE. An energy equal to the impurity state depth must be supplied to, or
released by, the carrier in order to move between an impurity state and a state at the PE band
edge. Carriers can move to a higher energy state by gaining energy from the field and/or through
thermal energy. Alternatively, a carrier can move between two impurity states, which could be
an energy conserving process in the absence of energy absorption or release. In general, the
smaller the energy difference between two states the more likely the transfer of carriers between
66
them. Accordingly, the four impurity states can be divided into three groups based on their
depths. First, the unoccupied state (electron trap ~4 eV below the CBM) is too deep to have
significant effect. Second, the occupied impurity states (hole traps) at 0.7 eV and 0.9 eV above
the VB are within the typical range of PE activation energies of conduction and are likely to play
a role in hole conduction processes. Third, the mixed impurity state at 0.03 eV (within thermal
energy of VBM given the accuracy of the present work) above the VBM, is likely to play a
major role in hole conduction processes, as it is extended between polymer chains, which
provides a mechanism for interchain transfer of holes, as described below.
Figure 5.8. Energy diagram of the PE-I2 band gap showing iodine impurity states. The energies of the impurity states relative to the VBM and CBM, which result from the introduction of I2, are indicated. Unoccupied states are in green and the occupied states are in red.
PDOS analysis determines the contribution of the orbitals of each atom to each state in the
system. The DOS of PE-I2, and the contribution to the impurity states from orbitals of I2 and PE,
is shown in Figure 5.9. The I2 orbitals that form the LUMO and the two degenerate highest
occupied molecular orbitals of the isolated I2 molecule are the main contributors to the band gap
impurity states. As a result, these states are localized around the I2 molecule (Figure 5.10a and
5.10b). The band gap impurity states differ from the LUMO and HOMO of the isolated I2
molecule through the small contribution of PE orbitals as shown in Figure 5.9, which breaks the
degeneracy of the occupied impurity states (compared to the degenerate highest occupied
molecular orbitals of the isolated I2 molecule) to create distinct states at 0.7 and 0.9 eV above the
67
VBM. Such differences reflect the interaction between I2 and PE. The impurity state at the VB
edge has almost equal contribution from PE orbitals and I2 orbitals, hence its designation as a
―mixed impurity state‖. The PE orbitals which contribute to the mixed impurity state belong to
chains 1 and 7 in Figure 5.4. The hybrid nature of this mixed impurity state sets it apart from the
band gap impurity states and contribute significantly to the extension the electron charge density
between the polymer chains (Figure 5.6).
Figure 5.9 DOS of PE-I2 showing the contribution of I2 orbitals in red. Band gap impurity states are formed mainly by I2 orbitals, while the mixed impurity state at the VB edge has an almost equal contribution from I2 and PE orbitals.
Other mixed impurity states were identified further below the VBM. The creation of states with
mixed I2 and PE orbitals binds the I2 to PE and creates the extension of electron charge density
between polymer chains in the vicinity of the I2 molecule (Figure 5.6). Mixed impurity states
have been identified in all PE-In systems (n=3 to 5) with a varying contribution percentage from
In orbitals and at varying depths below the VBM.
The orbital composition of a state determines the spatial features of its wavefunction. The square
of the wavefunction of an electron state represents the probability of finding an electron/hole
occupying that state at a given point in space (spatial electron probability density). Figure 5.10
shows the spatial electron probability densities of an occupied band gap impurity state above the
VBM, the unoccupied band gap impurity state below the CBM, and the mixed impurity state.
The probability density of the band gap impurity states is localized around the I2 molecule as a
result of the large contribution of its orbitals to these states. The low amplitude probability
Energy
DO
S
DO
S (
arb.u
nit
s)
Energy (eV)
68
density peaks at the PE chains in Figures 5.10a and 5.10b result from deformations of the LUMO
and HOMO of isolated I2 caused by the interaction between I2 and PE. The PE orbitals contribute
to the mixed impurity state near the VBM more than to the band gap impurity states shown in
Figure 5.9. Orbitals from two chains contribute significantly to the mixed impurity state, unlike
the band gap impurity states where the contribution of the PE orbitals is largely from a single
chain. As a result, the electron probability density of the mixed impurity state is extended
between chains to a much greater degree than the probability density of the band gap impurity
states, as shown in Figures 5.10c and 5.10d. The exchange of carriers between the iodine
impurity states and the PE states at the edges of the VB and/or CB should be responsible for the
increase in conduction. The role each impurity state plays in the conduction process is
determined by the impurity state energy and the spatial features of its wavefunction. Analysis of
the energies of the impurity states shows that the main influence of iodine takes place through
the occupied impurity states created above the VB rather than the unoccupied state created below
the CB. This is in general agreement with the experimental observation that iodine increases
hole mobility in PE to a much greater degree than electron mobility [12].
Figure 5.10 Contours of the electron probability density of various iodine impurity states. a) the unoccupied band gap impurity state, b) the occupied band gap impurity state 0.9 eV above the VBM which is very similar to that at 0.7 eV above the VBM, c) and d) the mixed impurity states in two XY planes at locations along the Z axis with relatively large values of electron probability at two adjacent PE chains. The probability scale was selected to demonstrate differ-ences among various impurity states.
a) b)
c) d)
Ele
ctro
n p
robab
ilit
y d
ensi
ty
69
5.5 Effect of Iodine on Conduction
The morphology of PE is related closely to charge transport therein. PE chains are held together
through van der Waals forces which are much weaker than those of a chemical bond, as is
reflected in the charge density contour lines being continuous along chains and not across chains.
The difference in the interactions along chains and between chains results in greatly differing
carrier mobilities along chains and between chains. The minimum mobility along PE chains
which is required by band theory is 1.2x10-5
m2V
-1s
-1 [91]. Experimental estimates of carrier
mobility in bulk PE are in the range of 10-10
to 10-14
m2 V
-1 s
-1 [91], which suggests that the
conduction process is limited by interchain rather than intrachain carrier mobility [91]. The
probability densities of a VBM state and a CBM state in the Core-Shell model are shown in
Figures 5.11c and 5.11d. The VBM state has an intrachain character (extended along chains),
unlike the CBM state which has an interchain character (interchain peaks). The spatial features
of VBM and CBM states of PE are in agreement with previous DFT work [41]. Although the
literature value of measured carrier mobilities in polyethylene vary, the hole mobility is
consistently less than the electron mobility [12,92,93] The limited mobility of holes in PE may
be related to the intrachain character of the VBM states which are better suited to transport holes
along chains. Some features of PE which impede interchain charge transfer are altered by the
interaction between PE and iodine. In the Core-Shell model, the charge density contour lines
become extended across chains in the vicinity of the I2 molecule (Figure 5.6) which increases the
interaction between chains. Analysis of the DFT total local potential (VKS in equation 2.12) in the
Core-Shell model demonstrates that the potential barriers, which are higher between chains than
along chains, are lowered in the vicinity of the I2 molecule. Further, the iodine occupied
impurity states and mixed impurity state at the valence band edge facilitate a mechanism which
increases hole transfer between chains and reduces the activation energy. The unoccupied
impurity state is unlikely to play a role in the charge transport mechanism since it is too deep
below the CBM (~ 4eV).
70
Figure 5.11. The electron probability density surface plots of various iodine impurity states. a) The mixed impurity state at the VBM edge, b) the occupied band gap impurity state 0.7 eV (hole trap), c) VBM state, and d) CBM state. The isosurfaces in the various figures have the same value of probability. The mixed impurity state in a) is extended across chains compared to the localized band gap impurity in b) and to the VBM in c) which is extended along chains unlike the CBM in d) which has an interchain character.
A plausible mechanism through which iodine may increase the hole mobility of PE is presented.
The mechanism involves the mixed impurity state and the occupied impurity states above the
VBM. The occupied impurity states above the VBM act as hole traps with depths 0.7 and 0.9 eV,
and holes injected from the electrodes will tend to concentrate in these states, as electrons will
tend to occupy the lowest energy orbitals. A hole can be introduced into the valence band by an
electron being excited (activated) from the VBM into a ―hole trap‖ state at 0.7 or 0.9 eV above
the VBM. If the hole is created in the mixed impurity state which is extended between chains
(Figure 5.11a), it can move to another chain, where an electron can drop from the impurity state
into the hole, thereby completing the charge transfer between chains and nearly conserving
a) b)
c) d)
71
energy in the process. Thus, the mixed impurity state helps to overcome the inherent difficulty of
transferring holes between chains in PE, which causes the experimentally observed increase in
hole mobility in iodine doped PE [12]. The above process requires an energy of ~ 0.8 eV (either
0.7 eV or 0.9 eV depending on to which iodine impurity state the electron is transferred to create
the hole in the VB). The iodine hole trap depths of 0.7 and 0.9 eV are less than the depths of
hole traps created by common chemical impurities such as carbonyl, double bonds, and vinyl (~1
eV, Chapter 4). The experimental data indicate that iodine lowers the activation energy of PE
from the range of 1 eV to about 0.8 eV [10] which correlates well with the introduction of iodine
impurity states at 0.7 and 0.9 eV above the VBM of PE. This suggests that the 1eV activation
energy of PE is correlated with impurity states created by common chemical impurities which
concentrate around 1eV above the VBM and around 1eV below the CBM (Figures 4.6, Chapter
4). Detrapping of carriers from the 2eV deep electron traps to the CB can occur indirectly
through the trap levels at 1eV in a sequential activation process that would require energy of 1eV
at a time; however, such compound processes should have low probabilities.
The proposed mechanism through which iodine increases the conductivity of PE is based on the
occupied impurity states above the VBM and the existence of a mixed impurity state which is
extended across chains. Similar states were identified when In molecules (n=2 to 5) were studied
in the Core-Shell model. The I2 molecule was also studied in an extended Core-Shell model
(Figure 5.12a) and in a model simulating a void or an interstitial layer where charge transfer is
expected to be limited (Figure 5.12b). The interaction between I2 and PE in such systems is
similar to the interaction of the In molecules and PE in Core-Shell models. The I2 molecule
extends the charge density between chains and creates hole trap states and mixed impurity states
in both systems shown in Figure 5.12.
5.6 Effect of Bromine
Experimental results show that bromine has the same qualitative but less quantitative effect on
conduction in PE as compared to iodine [12]. A Br2 molecule was studied in a Core-Shell model
similar to that in Figure 5.4. The DFT computations for the isolated Br2 reproduced the bond
length and binding energy within 1% and 10% of experimental values, respectively [85].
Bromine shows similar effect to that of iodine on the charge density. Bromine introduces hole
trap states above the VBM and a mixed impurity state 0.23 eV below the VBM. Thus, bromine
72
can provide a similar mechanism to increase PE conduction as that proposed for iodine. The less
quantitative effect of bromine on the conductivity of PE compared to iodine might be related to
the depth of the bromine mixed impurity state or the fact that bromine molecules, which are
smaller than those of iodine, would not be able to facilitate interchain charge transfer at larger
chain separations at which the iodine molecules are still effective.
Figure 5.12. Various structures based on modifications to the Core-Shell model. a) An I2 molecule in an “extended” Core-Shell model, and b) A model simulating a void or an interstitial layer where charge transfer is expected to be limited. In the top of the figure the structures are shown in the XY plane.
5.7 Summary
In summary, DFT was used to identify various stable configurations of In molecules (n=2 to 5).
The In molecules tend to exist in a linear or a zigzag configuration rather than configurations
with higher degrees of symmetry such as triangular, square, tetrahedral, or pyramid
configurations. According to the incremental binding energy analysis [87,88], I2 and I4 are more
stable than I3 and I5. Iodine interaction with PE was studied in the amorphous like Core-Shell
model (PE-In). The interaction between In and PE was identified through the change in In bond
length, In binding energy to PE chains, and electron charge distribution in the vicinity of In. The
binding energy of PE and In in PE-In models indicates that iodine forms stable structures in PE.
b) a)
73
The binding could be traced to the formation of electronic occupied states consisting of mixed In
and PE orbitals. The mixed states are also responsible for extending the electron charge density
distribution between chains in the vicinity of In molecules. A mechanism through which iodine
can increase hole mobility was proposed based on iodine hole trap states and the mixed impurity
state. The iodine hole trap states at 0.7 and 0.9 eV above the VBM account for the
experimentally observed decrease in activation energy when iodine is introduced into PE, and the
mixed impurity state facilitates interchain hole transfer, which accounts for the experimentally
observed increase in hole mobility [12]. All of the above observations are consistent for In
molecules (n =2 to 5) in Core-Shell models, I2 molecule in other models such as those in Figure
5.12, and for Br2 in a Core-Shell model, which increases confidence in the conclusions. The
present work points to the importance of studying the orbital composition and the spatial features
of impurity states, along with their energies relative to adjacent band edges, yet none of the
existing conduction models can account for such features of impurity states, and most of the
published work discusses impurity states solely based on their energies.
5.8 Similarities between Iodine and Common Impurities
Iodine facilitates interchain hole transfer and, thus, increases hole mobility (i.e., conduction) in
PE which is otherwise limited by the intrachain nature of the VBM states. At an atomic level, the
effect of iodine can be characterized through two features. The first feature is the extension of
charge density contour lines between chains in the vicinity of iodine as shown in Figure 5.6. The
second feature is the creation of an occupied impurity state to which orbitals from two differing
chains contribute (mixed impurity state) and is close in energy to the VBM. Impurities which
exhibit these two features should increase conduction in PE through increasing hole mobility.
These two features are investigated for carbonyl, vinyl, double bond, and conjugated double
bonds to determine their potential effect on interchain hole transfer. As the analysis for iodine
reveals the importance of studying the spatial features of impurity states, such analysis for the
states created by common chemical impurities is conducted.
The electron charge density plots in the vicinity of the common chemical impurities studied in
Chapter 4 are shown in Figure 5.13. The charge density contour lines are extended significantly
across the chains only in the vicinity of the carbonyl and vinyl group. The highest uninterrupted
charge density contour lines spanning two chains in the vicinity of carbonyl and vinyl are 0.108
74
and 0.0945 electron/A3, respectively. The effect is most significant in carbonyl followed by
vinyl, then double bond and conjugated double bond.
Figure 5.13 Contour lines of electron charge densities in the vicinity of various impurities. a) carbonyl, b) vinyl, c) double bond, and d) conjugated double bond. Electron charge density scale is in electron
/A
3 and is chosen to
demonstrate the relative effect of different impurities. The plots are in the XY plane where the Z direction is along the chain backbone and into the paper. The highest uninterrupted charge density contour lines spanning two chains in the vicinity of carbonyl and vinyl impurities are 0.108 and 0.0945 e/A
3, respectively.
While in the case of common chemical impurities, the band gap impurity states are caused
mainly by the orbitals of the impurity atoms, in some cases the PDOS analysis shows
considerable contributions from orbitals associated with carbon and hydrogen atoms of
neighboring chains. The extension of the occupied impurity states between chains will have more
significant effect on conduction than the extension of the unoccupied states because the CBM
states of PE have an interchain character, while the VBM states have an intrachain character.
Figure 5.14 shows the probability density of the carbonyl and vinyl band gap occupied impurity
states (hole trap states), an iodine hole trap state, and the iodine mixed impurity state. The
carbonyl and vinyl impurity states show an extension towards neighboring chains which is
comparable to that of the iodine mixed impurity state, while no such extension exist in the case
a) b)
c) d)
Ele
ctro
n c
har
ge
den
sity
(el
ectr
on/Å
3)
75
of double bond and conjugated double bond states on the scale adopted in Figure 5.14. This is
expected in light of the charge density profiles in Figure 5.13. The carbonyl and vinyl occupied
band gap impurity states can facilitate interchain hole transfer.
Figure 5.14 The electron probability density of various impurity states extended between chains. a) The iodine mixed impurity state at the VBM edge, b) the iodine occupied band gap impurity state 0.7 eV (hole trap), c) the carbonyl occupied band gap impurity state 0.95 eV (hole trap), and d) the vinyl occupied band gap impurity state 1.06 eV (hole trap). The isosurfaces in the various figures have the same value of probability. The carbonyl and vinyl impurity states show an extension towards neighboring chains that is comparable to that of the iodine mixed impurity state, thus, they can facilitate interchain charge transfer.
In the case of carbonyl and vinyl, the extension of charge density is created by the combined
effect of various impurity states. Many of these states do not bridge neighboring chains
completely, as the iodine mixed impurity state does. In case of iodine, the extension of charge
density between chains is created by fewer states which bridge neighboring chains than in the
case of the carbonyl and vinyl. However mixed impurity states, similar to that of iodine which
bridges neighboring chains completely, have been identified in the case of carbonyl. The nearest
of such states to the VBM in the case of carbonyl is 0.2 eV below VBM, while the iodine mixed
a) b)
c) d)
76
impurity state is at the VB edge (0.03 eV above VBM). The carbonyl mixed impurity state is less
extended between chains compared to the iodine mixed impurity state. Figure 5.15 shows the
extension of the probability density of the mixed impurity states of iodine and carbonyl. The
probability scale shown in Figure 5.15 is the same as that in Figures 4.3 and 4.4 which show the
probability density of all bandgap impurity states of carbonyl, vinyl, double bond, and
conjugated double bond impurities. The carbonyl mixed impurity state will have less effect on
increasing interchain hole transfer and mobility than that of iodine since the former is less
extended in space and is further in energy from the VBM than the later. In the case of carbonyl
and vinyl impurities, the band gap hole traps which are extended towards neighboring chains can
also facilitate interchain charge transfer. The atomic level features through which iodine
increases hole mobility and thus conduction in PE are only manifested in the case of carbonyl
and vinyl. The extension of charge density and the presence of a carbonyl mixed impurity state
which is close to VBM indicate that carbonyl will increase conduction more than vinyl. This is in
agreement with the experimental observation that carbonyl impurities increase conduction in low
density PE, for which no rigorous physical explanation has been presented [65,66,94,95]. In the
same context, the double bond and conjugated double bond impurities are not expected to
increase conduction in PE as carbonyl and vinyl would do. In general, a comprehensive
understanding of the effect of chemical impurities on hole mobility, and thus conduction, should
include an analysis of charge density and states which are spatially extended between chains, in
addition to the traditional analysis of trap depths of impurities.
Impurities might play a role in facilitating interchain electron transfer which would increase
conduction. However it is a topic which requires further research beyond the following brief
discussion. The carbonyl and vinyl hopping states are extended between chains and thus can
facilitate interchain electron transfer. However, their extension between chains is only slightly
higher than that of CBM states which have an interchain nature with peaks at adjacent chains.
This is different from the situation at the VBM where the extension of mixed impurity states, as
those due to carbonyl and iodine is significant compared to the intrachain VBM states. The
advantage of the carbonyl and vinyl hopping states in facilitating interchain charge transfer over
CBM states might come from the fact that their energy is lower than that of the CBM by about ~
0.2 eV. This means that electrons are activated easier into the carbonyl and vinyl states than into
the CBM states from electron trap states.
77
Figure 5.15. The electron spatial probability density of iodine and carbonyl mixed impurity states. a) The mixed impurity state of a carbonyl impurity 0.2 eV below the VBM edge and b) the mixed impurity state of iodine impurity at the VB edge.
A situation through which impurities might play a significant role in interchain electron transfer
can be created by the presence of certain impurities close to one another in adjacent chains.
Figure 5.16 shows a probability density surface of a CBM state and two unoccupied impurity
states (1.22 and 0.86 eV below CBM) created by a vinyl in one chain and a double bond 3.5 Å
away in an adjacent chain. The binding energy of this combination is two orders of magnitude
greater than thermal energy which indicates the stability of such combination once it is created.
The impurity states in this case are significantly extended between chains compared to CBM
states. This combination would have a great impact on conductivity due to the creation of
occupied and unoccupied impurity states which are significantly extended between chains and
have an effect that is almost similar to that of cross linking. This is further supported by the value
of the binding energy of these impurities, which is an order of magnitude larger than that of the
interchain binding energy.
b) a)
Electron probability density
78
Figure 5.16 Electron density probability surface of impurity states of interacting impurities in adjacent chains. The electron spatial probability density (square of the electron state wavefunction) of impurity states created by a vinyl and a double bond impurity in adjacent chains and 3.5 Å apart is shown a) an unoccupied impurity state 1.22 eV deep, b) an unoccupied impurity state 0.86 eV deep, and d) CBM state. The isosurfaces in the various figures have the same value of probability and is similar to that used in all probability surface plots in the whole thesis.
b) a) c)
79
Chapter 6 Summary, Conclusions and Future Work
6 Summary, Conclusions, and Future Work
6.1 Summary and Conclusions
6.1.1 DFT Models of Chemical Impurities in Polyethylene
Using DFT, molecular models of single PE chains and bulk PE were created to study the effect
of chemical impurities therein in the context of high field conduction. Two models of bulk PE
were employed, the crystalline bulkcell model and the Core-Shell model. The crystalline bulkcell
model is created using multiples of the primitive unit cell of orthorhombic crystalline PE. The
Core-Shell model, unlike crystalline models, can capture features of the amorphous state of PE.
The Core-Shell model is developed in the present work to study the effect of chemical impurities
which are more likely to exist in amorphous regions of semi-crystalline PE. The computed bond
lengths and angles, lattice constants, and electron affinity in all models are within reasonable
agreement with experimental values in [38,39,45]. The experimental band gap of 8.8 eV is
underestimated by about 25%. Underestimation of the band gap of insulators, occasionally by as
much as 50%, is a well-known deficiency of DFT. Based on the computed bond lengths and
angles, band gap, electron affinity, and the features of the wavefunctions of CBM and VBM
states [41], and given the limitations of DFT, the accuracy of the implemented DFT
approximations and developed models of PE is satisfactory.
The effect of carbonyl, vinyl, double bond, conjugated double bonds, iodine, and bromine on the
solid states physics and high field conduction in PE was investigated using mainly the Core-Shell
model. Although single chain models of 10 to 15 C atoms have been used to study the effect of
chemical impurities in PE [5,6,47,48,49,50], such chains are too short to represent infinite
chains. The present work demonstrates that the effects of chemical impurities on the electronic
properties of PE based on single chain models differs significantly from those based on bulk
models (crystalline and Core-Shell). Moreover, single chain models do not allow investigating
the interaction of impurities with neighboring PE chains, which is important to high field
conduction. Thus, single polymer chains should not be used to study the effect of chemical
impurities on the electronic properties of bulk PE.
80
The present work indicates that the crystalline bulkcell model and the Core-Shell model can be
used to study the effect of chemical impurities on the solid states physics of bulk PE. However,
in the context of semi-crystalline PE, and in particular amorphous regions therein, the crystalline
bulkcell model is over constrained with artificially imposed periodicity. When impurities are
included in a crystalline bulkcell model, symmetry is preserved, and the backbones of the PE
chains remain parallel with almost fixed separations. Such features are not characteristic of
amorphous regions in which impurities are more likely to exist. On the other hand, the ―Core-
Shell‖ model captures more features of the amorphous state of PE in that the chains are allowed
to deform as a result of the introduction of impurities. The Core-Shell model represents a
reasonable approximation to amorphous PE which can be implemented with a reasonable
number of atoms and periodic boundary conditions. The Core-Shell model with 40 C atoms long
chains which is employed in this work allows incorporation of multiple impurities
simultaneously without interaction between impurities or their neighboring replicas. The ratio of
impurities to ethylene groups in the Core-Shell model is more realistic than what could be
achieved in a crystalline bulkcell model in the same computation time. The Core-Shell model
accommodates large impurity atoms that reside between the chains and cause significant
deformation of the structure or large atoms which can only be located in highly distorted
amorphous regions (e.g., iodine impurities discussed in Chapter 5). In such cases, the periodicity,
which is enforced in a crystalline bulkcell model, provides a much less accurate representation of
the morphology when compared to the Core-Shell model. The Core-Shell model reveals details
that are missed from a crystalline bulkcell model, such as the creation of shallow impurity
states/hopping sites caused by distortion of bonds adjacent to impurities. The development and
demonstration of the Core-shell model is one of the original contributions of this thesis.
6.1.2 Effect of Chemical Impurities on the solid State Physics of Polyethylene
The effect of several common chemical impurities on the solid state physics of PE was studied in
crystalline and Core-Shell models of bulk PE. The investigation focuses on chemical impurities
studied in the Core-Shell model which provides an acceptable representation of the amorphous
regions of PE in which impurities are more likely to be found. All the chemical impurities
studied introduce occupied states above the VBM and unoccupied states below the CBM in the
band gap of PE. A quantum mechanical based characterization of the impurity states is presented
81
including, their energies relative to CBM or VBM, the type of bonds and orbitals which form the
impurity states, and the spatial features of the impurity states wavefunctions. In addition to the
impurity states deep in the band gap (depths between 0.95 to 1.96 eV), carbonyl, vinyl, and
conjugated double bond impurities introduce ―shallow‖ unoccupied states slightly below the
CBM (depths between 0.2 and 0.33 eV). The carbonyl and vinyl impurities introduce such states
indirectly through the physical disorder they cause in their vicinity. On the other hand, the
shallow impurity state introduced by a conjugated double bond is caused mainly by the orbitals
of the impurity atoms. The carbonyl and vinyl shallow impurity states, which appear as a
perturbation to the CB edge, are only evident when impurities are studied in the Core-Shell
model as a crystalline model constrains the atoms to the point that physical disorder introduced
by a chemical impurity is very limited.
The effect of chemical impurities on high field conduction is generally explained in terms of
traps which provide trapping/hopping sites for carriers [3,4]. Although the concept of traps is
central in explaining various high field phenomena in dielectrics, such as conduction and space
charge formation, a clear explanation of the physical basis of traps and their correlation with
chemical impurities is lacking, in spite of extensive effort [2-4]. In addition, unambiguous
experimental determination of trap depths and their sources remains a challenge. The
characteristics of states introduced by chemical impurities in the band gap of PE allow them to
play the role of trapping/hopping sites. The impurity state wavefunctions are localized in space,
and their overlap with CBM and VBM states provides the quantum mechanical basis for the
processes of trapping, detrapping, and hopping. Since the energy of impurity states in the band
gap near the CBM or VBM relative to the respective band edge are rendered with reasonable
accuracy by DFT [14,20], the energy differences between the unoccupied impurity states and the
CBM provide a reasonable estimate of electrons trap depths. Similarly, the energy differences
between the occupied impurity states and the VBM provide a reasonable estimate of holes trap
depths. Shallow traps, as those created at the edge of the CB, can act as ―hopping sites‖.
The conclusion that the band gap impurity states are the traps and hopping sites is supported by
the physical features of their wavefunctions and the general agreement of their depths with
estimates of previous experimental and theoretical work. In general, traps caused by chemical
impurities are deeper than those caused by physical disorder [2,69,70] which agrees with the
present predictions based on DFT. The deep traps identified in the present thesis have depths
82
between 0.95 eV and 1.96 eV and shallow traps/hopping sites have depths between 0.2 eV and
0.33 eV, which are in agreement with estimates in literature [2,66,67,68,69]. In general, the
combined effect of ~1eV deep holes traps and the ~1eV deep electrons traps can explain the
experimentally determined activation energy of conduction in PE which is about ~ 1 eV, and is
in general agreement with the ―effective trap depth‖ in various conduction models which is in the
range of 0.8 to 1.2 eV [65,71,73,82]. The present work provides a clear association of carbonyl
impurities with both trapping and hopping sites, as has been proposed previously with no formal
justification [3,65].
Shallow traps/hopping sites are usually associated with physical disorder in the amorphous
regions and trapping sites with chemical impurities, both of which are necessary to account for
the dielectric properties of PE. However, the shallow impurity states/hopping sites due to
chemical impurities identified above suggests that the dielectric properties of PE may be entirely
dominated by such impurities as they can account for the observed solid state features of the
material, although the degree to which disorder in the amorphous region contributes similar
shallow states even without chemical impurities, is not yet known. The clear identification of
trapping and hopping sites is believed to be one of the main contributions of the present work.
The procedure employed in chapter 4 to study the effect of chemical impurities on the solid state
physics of PE can be applied to a wide range of impurities.
DFT allows investigating the spatial features of impurity states which are as important as their
energies in understanding their effect on high field conduction. Impurity states which are
extended between chains can facilitate interchain charge transfer and, accordingly, conduction as
the iodine problem demonstrates (chapter 5). The carbonyl occupied state and the vinyl impurity
states are the most significantly extended between chains among the impurity states deep in the
band gap. The carbonyl and vinyl impurities also introduce mixed occupied impurity states
below the VBM which are expected to play a major role in increasing hole conduction in analogy
with the iodine mixed impurity state (chapter 5). The spatial features of the shallow unoccupied
impurity states/hopping sites which are introduced by carbonyl, vinyl, and conjugated double
bond at 0.22, 0.21 and 0.32 eV below the CBM differ significantly in spite being close in energy.
The carbonyl and vinyl states are much more extended in space than the conjugated double bond
state which is more localized around the impurity atoms. While the common impurities studied
in the present work introduce impurity states which are similar in energies the spatial features of
83
these states differ from one another. In general, the carbonyl and vinyl states are more extended
between chains than the double bond and conjugated double bond states. The analysis of the
spatial features of the impurity states concludes that among the common impurities studied in
Chapter 4 carbonyl is expected to increase conduction the most followed by vinyl.
6.1.3 Effect of Iodine on Conduction in Polyethylene
Iodine increases the electrical conductivity of PE by about four orders of magnitude [8-12] and
decreases the thermal activation energy of conduction from ~ 1 eV to about 0.8 eV [10]. Iodine
also increases hole mobility in PE to a much greater extent than electron mobility [12]. These
effects are consistent over many investigations which involved various types of PE, electrode
material, and methods by which the iodine is introduced into the PE. Thus the experimental
observations can be attributed to the interaction between iodine and the polymer chains.
Understanding the effects of iodine on conduction in PE at an atomic level can provide a basis
for understanding the more subtle effects of common chemical impurities.
DFT was used to identify various stable configurations of In molecules (n = 2 to 5). Iodine
interaction with PE is studied in the amorphous like PE Core-Shell model (PE-In model). The
interaction between In and PE is identified through the change in In bond length, In binding
energy to PE chains, and electron charge distribution in the vicinity of In. The binding energy of
PE and In in PE-In models (at a minimum of 0.32 eV/iodine atom) suggests that iodine forms
stable structures in PE. The binding was traced to the formation of electronic occupied states
consisting of mixed In and PE orbitals. The mixed states are also responsible for extending the
electron charge density distribution between PE chains in the vicinity of In molecules. All of the
above observations are consistent for In molecules (n = 2 to 5). A mechanism through which
iodine can increase hole mobility was proposed, which involves the iodine hole trap states at 0.7
and 0.9 eV above the valence band and the mixed impurity state at the valence band edge. A hole
can be introduced into the valence band by an electron being excited (activated) from the VBM
into a ―hole trap‖ state at 0.7 or 0.9 eV above the VBM. If the hole is created in the mixed
impurity state which is extended between chains, the hole can move to another chain, where an
electron can drop from an impurity state into the hole, thereby completing the charge transfer
between chains and (nearly) conserving energy in the process. Thus, the mixed impurity state
helps in overcoming the inherent difficulty of transferring holes between chains in PE.
84
Accordingly, the hole mobility will increase which agrees with the experimental observation that
iodine increases hole mobility to a much greater degree than electron mobility in PE. The iodine
holes trap states at 0.7 and 0.9 eV account for the experimentally observed decrease in activation
energy, and the mixed impurity state facilitates interchain hole transfer, which accounts for the
experimentally observed increase in hole mobility [12]. Correlating the decrease in activation
energy in iodine doped PE and the depth of iodine hole traps is supported by the similarity
between typical activation energies in PE and the depths of band gap impurity states identified
for common chemical impurities. Most common chemical impurities trap states are around ~ 1
eV deep (0.95, 1.0, 1.06 eV for carbonyl, vinyl, and double bond occupied states, respectively,
and 0.97 and 1.06 eV for vinyl and double bond unoccupied states, respectively), which is in
agreement with the experimental activation energy of conduction in PE of about ~ 1eV. Among
the various interpretations of the effect of iodine on the conductivity of PE [8-12], our proposed
mechanism is closest to that in [9].
On an atomic level, the effect of iodine can be characterized through two features. The first
feature is the extension of charge density between polymer chains in the vicinity of iodine. The
second feature is the creation of the mixed impurity state ―close‖ to the VBM. Chemical
impurities which exhibit these two features are expected to increase conduction in PE through
increasing hole mobility. Among the common chemical impurities studied, carbonyl and to a
lesser extent vinyl exhibit these features and as a result, are expected to increase interchain hole
transfer and conductivity. This agrees well with the experimental observation that carbonyl
impurities increase conduction in low density PE for which no rigorous physical explanation has
been presented [65,66,94,95].
The present work demonstrates the importance of studying the orbital composition and the
spatial features of the impurity state wavefunctions along with their energies relative to adjacent
band edges. None of the previous conduction models includes the spatial features of impurity
state wavefunctions, and most of the published work discusses impurity states solely based on
their energies.
85
6.2 Original Contributions
The major original contributions of the thesis are
Developing the Core-Shell model which can be used to study the effect of various impurities
on the solid state physics of PE and high field conduction therein while capturing features of
the amorphous state.
Providing a physical atomic level explanation of the concept of traps which plays a central
role in high field phenomena, identifying traps due to common chemical impurities, and
providing a first principle estimate of their depths.
Providing a physical atomic level explanation of effects of iodine on the conductivity of PE.
Proposing the concept of mixed impurity states and explaining their role in interchain charge
transfer, and, accordingly, conduction.
Demonstrating the importance of studying the spatial features of impurity states in addition
to their depths.
Providing a procedure through which the effect of chemical impurities on conduction in PE
can be studied using DFT.
6.3 Future Work
6.3.1 Experimental Studies
Many experimental studies of the effect of iodine and bromine on conduction in PE have been
published. However none of these studies compares the effect of iodine and bromine on PE
samples from the same manufacturer. The present work suggests that atomic level parameters,
such as the depths of hole traps and the depths of mixed impurity states which are created by
iodine or bromine, can be correlated with the decrease in activation energy and the increase in
conductivity of PE, respectively. A comparison between the effect of doping similar samples of
PE with various percentage of iodine and bromine on activation energy and conductivity of PE
will help establishing a quantitative correlation between the atomic level parameters and the
experimental observations.
86
The thesis concludes that carbonyl increases hole mobility to a greater extent than electron
mobility in PE. Experimental work should be carried out to validate this conclusion.
Measurements of activation energy, hole mobility, and electron mobility of PE samples with
differing percentages of carbonyl impurities should be carried out.
6.3.2 Macroscopic Modeling
The present thesis demonstrates that the main flaw in existing high field conduction models in
PE is that they can only account for the depth of impurity states, and they lack parameters which
can account for the spatial features of such states. Two impurity states with the same energy, one
of which is extended along the polymer chains and the other extended across chains will have a
significantly differing effect on conduction. In general, for better modeling of high field
conduction in insulating polymers a conduction model which can account for the spatial features
of impurity states should be developed. Such a model should also account for the differing ease
of charge transport along chains and across chains.
6.3.3 DFT Studies
Mixed occupied impurity states which extend between chains increase conduction. The closer
the mixed impurity state is to the VBM, the greater the increase in conductivity. On the other
hand, impurity states which are far from the band edge and are localized around the impurity will
not increase conduction. Impurity states can be mapped on a 2D space in which one dimension
indicates the extension of the state towards neighboring chains and the other indicates the depth
of the state. Such a map can identify impurities which are likely to increase conduction. To
create such a map, a parameter that quantifies the extension of impurity states between chains
should be selected carefully. Such a study requires data from manufacturers for important
chemical impurities in PE.
Although iodine in PE exists mostly in a neutral form and in particular I2, the presence of iodine
charged molecules is possible [8-12], for example in the form of I3 and I5 which have stable -1
structures [84]. Charged iodine species have not been addressed in the present work (as such
charged calculations pose formal difficulties in a periodic supercell approach) and should be a
topic of future research.
87
The increase of hole mobility plays a greater role in the increase of conduction than the increase
in electron mobility, since the increase in the former is more significant as experiments indicate
[12]. Yet, it is important to understand the mechanism by which iodine affects the electron
mobility. This represents a topic of further research.
The key to a rigorous understanding of the effect of chemical impurities on high field conduction
is an atomic level understanding of their effect on a) charge transport in the bulk and b) charge
transfer from the electrodes to the bulk through the interface region. The present work addresses
the role of chemical impurities in charge transport through determining the energies and shape of
their impurity states which provide trapping sites or facilitate interchain charge transfer.
Understanding the role of chemical impurities in the process of charge injection at the electrodes
is also important to a rigorous understanding of the physical basis of the high field conduction in
PE and should be a topic of future research. Using density functional theory, molecular models
of the interface between polyethylene and metals which represent the electrode material can be
developed. Based on these models, which include the physical features of the PE/metal interface
and the deformation therein due to the dissimilarity of the materials, the barriers to injection of
holes and electrons can be determined from first principles. The widely used macroscopic
models based on Schottky and Fowler-Nordheim mechanisms include the effects of barriers to
charge injection for which accurate theoretical estimates can be provided by DFT. Common
chemical impurities can be included in the PE/metal interface models. Through such models, the
role of chemical impurities in facilitating charge injection by lowering barriers to injection can
be determined. A DFT study of PE/metal interface can provide valuable information about the
role which chemical impurities play in the injection process which cannot be determined using
experimental techniques. Such future research will be an important step towards a rigorous
physical explanation of high field conduction in insulating polymers.
88
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Appendix A
A Determination of the Vacuum Level
A.1 Vacuum Level in System Finite in Two Dimensions
The vacuum level of systems which are finite in two dimensions can be defined as the value at
which the planar average of the total local potential levels off [96-99]. Accordingly, the vacuum
level can be determined by plotting the planar average total local potential along one of the finite
dimensions (for instance X axis) where the average is taken over planes in the other two
dimensions (for instance ZY planes). Figure A.1 shows the vacuum level of the 40 C atoms chain
of which DOS plot is in Figure 3.4. Figure A.2 shows the vacuum level of the Core-Shell model
with a carbonyl impurity in the Core chain (Figure 3.11).
Figure A.1 The planar average total local potential of a 40 C atoms isolated chain of which the DOS plot is shown in figure 3.4. The chain is extended along the Z direction. The planar average is taken over ZY planes. The vacuum level is at -0.1 eV.
c) d)
X axis of the unit cell (Angstroms)
The
ZY
pla
nar
aver
age
tota
l lo
cal
pote
nti
al p
lanes
(eV
)
96
Figure A.2 The planar average total local potential of a Core-Shell model with a carbonyl impurity in the Core-Shell structure which is shown in Figure 3.11. The planar average is taken over ZY planes. The vacuum level is at -0.36 eV.
A.2 Vacuum Level in Bulk Periodic Systems
DFT can be used to determine the vacuum level and its energy relative to the VBM and CBM of
an insulator. Accordingly, the electron affinity of a specific surface of an insulator can be
computed. The procedure for doing so, which is known as the ―bulk plus band lineup‖ procedure,
involves two steps [14]. First, a bulk calculation for the insulator is carried out. The energy
difference between the VBM and CBM of the insulator and the planar average potential
(VKSbulkav
, where VKS is as in 2.12) along the direction of the surface for which the electron
affinity is to be is determined are specified. Second, a slab structure of the insulator is studied to
determine VKSslabav
along the direction of the surface of the slab. In the region away from the
surface of the slab and into the material, VKSslabav
retains its bulk profile. In such region, the
energy differences (CBM – VKSbulkav
) and (VBM – VKSbulkav
) determined from the first step, are
used to determine the energy levels of the CBM and VBM at the surface of the slab as shown in
Figure A.3. In the region away from the surface and into the vacuum, the VKSslabav
levels off
c) d)
The
ZY
pla
nar
aver
age
tota
l lo
cal
pote
nti
al p
lanes
(eV
)
X axis of the unit cell (Angstroms)
97
indication the vacuum level energy Evac. The energy difference between the CBM and the Evac is
the electron affinity of the slab surface.
Figure A.3 The plots of the planar average total local potential (VKSav
) showing the “bulk plus band lineup” procedure
applied to determine the electron affinity of PE (110). The unit cell of the PE slab structure is shown at the top of the figure, and the interface with vacuum is at the 110 surface. The gray dashed line is the VKS
slabav of the slab structure
which is determined by averaging the total local potential ( VKS in 2.12 ) on planes in the direction of the 110 surface. The black line is the VKS
bulkav of bulk PE determined from a calculation separate from the slab calculation. The
potentials from the bulk computations match with the potential of the slab structure further from the interfaces of PE and vacuum.
c) d)
Pla
nar
aver
age
tota
l lo
cal
pote
nti
al
(eV
)
Unit cell dimension along the direction of the surface of the slab (Angstroms)
98
Appendix B
B Pseudopotentials
B.1 Iodine Pseudopotential
The iodine pseudopotential was determined using the ATOM program which is provided with
the SIESTA release. The ATOM input file is shown in Figure B.1. The input values of input
parameters are based values in [100]. Basic relativistic effects were included through scalar
relativistic pseudopotential [101,102]. The resulting pseudopotential was able to reproduce the
bond length, binding energy and other features of the I2 molecule with reasonable accuracy as
previously discussed.
Figure B.1 Input file to generate iodine pseudopotential using ATOM program.
c) d)
99
Appendix C
C Main SIESTA Input Files
C.1 Core-Shell Systems
Below is the part of SIESTA input file which includes the main approximation parameters which
Such parameters were used in all Core-Shell systems.
MeshCutoff 150.0 Ry
PAO.BasisSize DZP
SolutionMethod diagon
MaxSCFIterations 50
DM.MixingWeight 0.2
DM.NumberPulay 3
DM.Tolerance 1.d-4
MD.TypeOfRun CG
MD.NumCGsteps 200
MD.MaxCGDispl 0.1 Ang
MD.MaxForceTol 0.05 eV/Ang
C.2 Bulkcell Crystalline System
Below is the part of SIESTA input file which includes the main approximation parameters. Such
parameters were used in the pure bulkcell crystalline system.
MeshCutoff 150.0 Ry
kgrid_cutoff 10.0 Bohr
PAO.BasisSize DZP
SolutionMethod diagon
MaxSCFIterations 50
DM.MixingWeight 0.2
DM.NumberPulay 3
DM.Tolerance 1.d-6
MD.TypeOfRun CG
MD.NumCGsteps 400
MD.MaxCGDispl 0.1 Ang
MD.MaxForceTol 0.005 eV/Ang
MD.VariableCell True
100
Appendix D
D Computers Main Specifications
Unit 1
Supermicro H8DAi-2 Main Board
2 X AMD Opteron 2352 CPU
4 X 2G DDR2 667 ECC/Registered RAM
Unit 2
Supermicro H8DAi-2 Main Board
2 X AMD Opteron 2352 CPU
8 X 2G DDR2 667 ECC/Registered RAM