huzayyin ahmed a 201111 phd thesis

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Effect of Chemical Impurities on the Solid State Physics ofPolyethylene

by

Ahmed Ali Soliman Huzayyin

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

The Edwards S. Rogers Sr. Department of Electrical and Computer EngineeringUniversity of Toronto

© Copyright by Ahmed Huzayyin 2011



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Effect of Chemical Impurities on the Solid State Physics of

Polyethylene

Ahmed Ali Soliman HuzayyinDoctor of Philosophy

The Edwards S. Rogers Sr. Department of Electrical and Computer EngineeringUniversity 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 f ield 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.



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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.



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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



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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



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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



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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 (C 40H80O).

....................................................................................................................................................... 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



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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 In

a to

Ind................................................................................................................................................... 61

Figure 5.2 Binding energy per iodine atom ( E b[ I n]/n) for the most stable In configurations at each

n (n = 2 to 5). ................................................................................................................................ 61 Figure 5.3 Incremental binding energy ( E inc[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 ( E b 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



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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 (V KS

av) 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



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Chapter 1Introduction

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.



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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 andassessing 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



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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 chemicalimpurities?

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 FieldConduction 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 withthe 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



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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, E t , 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 ( E t ) 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 chargecarriers 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 ApproachMacroscopic 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 inexplaining 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.



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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, andthe 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]. Inaddition, 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 ofthe 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.



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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 implementDFT 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 2Brief 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 value2

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 tothe 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.



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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 muchheavier 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 N H Ψ(r , r ,............,r ) = E Ψ(r , r ,............,r ) (2.2)

where ˆ 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 N r , r ,....,r denote the location of each of the N electrons. The probability

of finding the N electrons in the locations1 2 N r , r ,....,r , is given by

2

Ψ . The Hamiltonian ˆ H is

ˆn-e e-e H = T + V + V . (2.3)

The term T represents the kinetic energy of electrons, the term V n-e is the potential acting on the

electrons from the nuclei and the term V e-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 2 N N M N N k

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 operator 2i , Z k 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

http://en.wikipedia.org/wiki/Identical_particles

http://en.wikipedia.org/wiki/Fermions

http://en.wikipedia.org/wiki/Quantum_state

http://en.wikipedia.org/wiki/Quantum_state

http://en.wikipedia.org/wiki/Fermions

http://en.wikipedia.org/wiki/Identical_particles



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

-1 H ψ (r) V (r) ψ (r) = E ψ (r)

2

(2.7)

where, H HF is the Hartree-Fock Hamiltonian and HF V (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 (V n-e) and interacts with other electrons in thesystem through the terms V Hartree and V x. The exchange term V x 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 V Hartree. The term V Hartree 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‖ (V c) which together with

the exchange potential V x 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 N 4) 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-FockMethods”

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 3 N 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 V ext , forexample that due to the nuclei of a solid, V n-e in (2.3). The first HK theorem states that the

external potential V ext 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, V ext [19]. Accordingly, the Hamiltonian (2.3) is determined to

within a constant, which in turn means that ground state Many Body wavefunction0Ψ , 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).

ext E[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 T 0[n] is the kinetic energy of a

system of non-interacting electrons of density n(r), and the fourth term E xc[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 T 0 is

different from the true kinetic energy T , the difference can be accounted for within the exchange

correlation term E xc[n]. This means that E xc[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

T 0 and T [16]. All terms of (2.11) can be evaluated exactly except for E xc[n], the approximationof 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 (V ext ) 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 exactlyin (2.12) which is not represented in (2.7).

2

KS i KS i i i

-1 H ψ (r) V (r) ψ (r) E ψ (r)

2

(2.12)



15

where, H KS is KS Hamiltonian, KS V (r) is the KS effective potential given in (2.13), ψ i(r) is the

wavefunction of the ith electron, E i 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, V xc is the exchange correlation potential given by (2.14).

xc xc

E [n]V (r)

n(r)

. (2.14)

The single particle wavefunctions ψ i(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 V xc[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 V xc[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 V xc[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 V KS (2.13) can be evaluated exactly, except for V xc, 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 n f (r) (2.15)

No

Self Consistent?

n f ≈ nin

Calculate Final Values; Energies, Wavefunctions and other ground state properties



17

E xc is available, and consequently none is available for V xc. Moreover, no clear guidelines are

available for a systematic approximation of E xc [16]. The approximation of E xc is central to how

well DFT performs. Two main classes of approximations of E xc 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 E xc[n] is defined as

LDA

xc xc E [n] n(r) ε ( n(r) ) dr (2.16)

where xc

ε ( n(r) ) is the exchange correlation energy of an electron at point r in a homogeneous

electron gas of density n(r). The term ε xc in (2.16) can be split into two parts, one which accountsfor exchange (ε x) and the other which accounts for correlation (εc) [20]. Explicit exact analytical

expressions for ε x 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 E xc[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 E xc 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 i

H 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 ψ i

[21]. In DFT codes, ψ i 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 ψ i through (2.18). Representing ψ i 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 ψ i, 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 (r c), beyond which the pseudopotential coincides with the true potential asshown 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 thetrue 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, whichincludes the structure to be studied, is in the center of each system and is outlined by a black line. The periodicreplicas created by the periodic boundary conditions are outlined by the dotted line. a) Isolated system where thestructure and its periodic replicas are far enough apart that they do not interact (supercell) b) Slab of a material withan infinite extension in 2D and a finite extension in the third dimension creating a surface c) Slabs of two differingmaterials creating an interface d) Bulk structure of orthorhombic crystalline PE with infinite extension in 3D.

2.3.6 Moving the Nuclei; Geometry Optimization and MolecularDynamics

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].

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 widelyused [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 ofelectrons 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 DFTcombined with geometry optimization or molecular dynamics is applied to study a system of electrons and nucleiforming 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, V KS , 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 E i and ψ i 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



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 3Quantum Mechanical Modeling of Chemical Impurities in

Polyethylene

3 Quantum Mechanical Modeling of Chemical Impuritiesin Polyethylene

3.1 Review of DFT Studies of Polyethylene and ChemicalImpurities 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 affinityof 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 ref erred 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; FromSingle 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 C nH2n+2 rather thanCnH2n 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 cellare 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 theprimitive 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 toprevent interaction of the chain and its periodic replicas in X and Y directions. b) The unit cell of a finite PE chainconsisting of 10 carbon atoms and terminated by methyl groups (C10H12). The unit cell dimensions in X, Y, and Zdirections 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

( E f ) is indicated in DOS plots; however, the E f which is determined by SIESTA for a bulk

dielectric has no physical significance except to indicate the occupancy of states. The ―true‖ E f 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 thenumber of carbon atoms in the chain (n) (CnH2n+2). The band gap of infinite PE chain is indicated by the gray dashedline.

Figure 3.4 DOS of a pure 40 C atom finite chain (C40H82). The contribution of the orbitals of terminal methyl groups isshown 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 aninfinite chain = 8.5 eV

Number of carbon atoms in a finite chain CnH2n+2

E n e r g y ( e V )

D O S ( a r b . U n i t 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 inXY plane with lattice constant La and Lb in X and Y directions, respectively. Z direction is in the paper with a latticeconstant 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 forme d 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 the40 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 thecarbonyl 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)

D O S ( a r b . U n i t 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 N 3, 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 distanceseparating 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

E n e r g y s e p a r a t i n g

i m p u r i t y s t a t e 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 presentwork 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 theblack frame. Each chain in the supercell has 8 ethylene groups in the Z direction (into the paper). The neighboringreplicas, 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) consistsof 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 inZY 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 thecrystalline bulkcell structure is shown to the left. Bending of the Shell chains in the vicinity of the carbonyl group isevident 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 acrystalline 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 4Effect of Chemical Impurities on Solid State Physics of

Polyethylene

4 Effect of Chemical Impurities on Solid State Physicsof 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 impuritieson 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 C b). The atoms C, O, Ca, and C b 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, C b, H1, and H2 are in one plane. Adouble 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 C b). 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

CarbonylC=O1.23

OCCa, OCC b 121.27, 120.8

Vinyl C1=C2, C2-H1, C2-H2 1.35, 1.11, 1.11

C2C1Ca, C2C1C b,121.77, 121.71

Double BondC1=C2, C1-H1, C2-H2

1.35, 1.12, 1.12

CaC1H1, C bC2H2,

118.51, 118.31

Conj. Double

Bond

Ca-C1, C1=C2, C2-C3, C3=C4, C4-C b,

1.49, 1.36, 1.44, 1.36, 1.49

C1C2C3, C2C3C4, C3C4C b,

124.41, 124.3, 124.72

The formation energy E f 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 theimpurity. 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 takenas reference. The occupied and unoccupied impurity states are formed mainly by the carbonyl group orbitals. Theshallow impurity state is created indirectly by the carbonyl through the physical disorder it causes to adjacentethylene 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]. DFTcomputations 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

Carbonyloccupied state

Carbonylunoccupied state

Shallow impurity state

D O S ( a r b . u n i t 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.

ImpurityOccupied state, Energyabove VBM (eV)

Unoccupied state, Energybelow 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 conjugateddouble bond

π2p , 1.53 π*2p, 1.85

Figure 4.3 Square of the wavefunctions (probability density) of the unoccupied impurity states. Carbonyl impuritystate (top left), double bond impurity state (top right), side vinyl impurity state (bottom left), and conjugated doublebond impurity state (lower right) in the Core-Shell structure plotted in the XY plane. The plots are at planes whichinclude the highest probability value of each impurity state.

E l e c t r o n p r o b a b i l i t y d e n s i t y



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 bondimpurity state (lower right) in the Core-Shell structure plotted in the XY plane. The plots are at planes which includethe highest probability value of each impurity state. The probability scale in the plot is the same as that in Figure 4.3and 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

E l e

c t r o n p r o b a b i l i t y d e n s i t y



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 shallowimpurity state indirectly introduced by carbonyl through the physical distortion of its neighboring ethylene groups isonly 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


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 whichimpurity 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 modelof the figure. Hydroxyl impurities have been identified experimentally in PE [68]; however, they are not as common asthe other impurities in the figure.

4.3 Discussion of Chemical Impurity States in the Context ofHigh 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.

D O

S ( a r b . u n i t 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 doublebond unoccupied state and a CBM state is shown. To the right, overlap surface between carbonyl occupied impuritystate and a VBM state is shown. The difference in the spatial features of the overlap in both cases is due to thedifference in the spatial features of CBM and VBM states in addition to difference of the spatial features of thecarbonyl and vinyl impurity states.

Electron traps created by chemical impurities can be defined as the unoccupied orbitals of theimpurity 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 bycarbonyl, 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 PresentThesis and Estimates in Literature

4.3.2.1 Comparison with Estimates Based on Macroscopic Modeling andMeasurements

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

E n e r g y ( e V )



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-Fmeasurements.

Single trap level 1 eV - 1.15 eV - 0.85eV - 1.08 eV.

71 Fitting model to I-Fmeasurements.

Continues parabolic distribution. 0.01 to 1 eV

69 Fitting computations tomobility measurements

Two distributions Shallow in the rangeof 0.3-0.6

Distribution of deep

traps around 1.5 eV

80 Fitting model to I-Fand Space charge (SC)measurements.

Single trap level for holes and single trap levelfor electrons

Holes 1eV Electrons 1 eV

79 Fitting model to SCmeasurements.

Exponential distribution with deepest levelindicated for electrons and holes

Electrons 0.85 eV Holes 0.79 eV


Exponential distribution with deepest levelindicated for electrons and holes

Electrons 0.6 eV to1eV for various datasets

Holes 0.53 eV to 0.83eV for various data

sets


Single trap level for holes and single trap levelfor 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 electronaffinity 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 ofconduction in PE which is about ~ 1 eV, and is in general agreement with the ―effective tr ap

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) carbonyloccupied 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. Thecarbonyl and vinyl impurity states are extended towards neighboring chains compared double bond and conjugateddouble bond states which are more localized around the impurity atoms. The double bond occupied impurity state isshown in b) for comparison.

a) b)

c) d)



59

Chapter 5Effect 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 I n 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 n E I / n = E I / n - E I (5.1)

where, E[I n ] 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 I 2

binding energy per atom ( E b[I 2 ]/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 E b[n] [86]. Based on present DFT computations, linear configurationsand 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 ( E b[I n ]/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, E inc[I n ], 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 n E I = E I + E I - E I (5.2)

Figure 5.2 Binding energy per iodine atom (E b[I n]/n) for the most stable In configurations at each n (n = 2 to 5).

Most stable Unstable

relaxes to I3a

Most stable

Unstable relaxes to I4a Most stable Stable

StableStable

Stable

I3a I3

b I3c

I4

a

I4

b

I4c I4

d I5a I5

b

B i n d i n g e

n e r g y p e r a t o m ( e V )

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 E b

PE-In / n is defined as

b PE-In n n E / n = E PE+I - E PE - E[I ] / n (5.3)

where, E[PE+I n ] 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 E b 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 ( E b[I n ]/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.

I n c r e m e n t a l b i n d i n g e n e

r g y ( e V )




63

Figure 5.4. Core-Shell model of PE with I2 molecule (PE-I2) shown from two perspectives. The I2 molecule residesbetween 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


P E - I n

b i n d i n g e n e r g y p e r I a t o m ( e V )



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 tothe right, a plane far from the I2. Charge density contour lines extend between PE chains only in the vicinity of the I 2 molecule. The plots are in the XY plane where the Z direction is along the chains backbones and into the paper. Thescale units are in electron/Å3

The ―difference electron charge density‖, which is the PE-I2 electron charge density aftersubtracting 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)



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 statesrelative to the VBM and CBM, which result from the introduction of I2, are indicated. Unoccupied states are in greenand 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 byI2 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 squareof 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 (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 throughthe 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 gapimpurity state, b) the occupied band gap impurity state 0.9 eV above the VBM which is very similar to that at 0.7 eVabove the VBM, c) and d) the mixed impurity states in two XY planes at locations along the Z axis with relatively largevalues 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)

E l e c t r o n p r o b a b i l i t y d e n s i t y



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 (V KS 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 impuritystate 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 extendedacross chains compared to the localized band gap impurity in b) and to the VBM in c) which is extended along chainsunlike 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 indirectlythrough 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 onconduction in PE as compared to iodine [12]. A Br 2 molecule was studied in a Core-Shell model

similar to that in Figure 5.4. The DFT computations for the isolated Br 2 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 belimited. 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 I n 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 Br 2 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 theexisting 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 ofcharge 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



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 mixedimpurity state at the VBM edge, b) the iodine occupied band gap impurity state 0.7 eV (hole trap), c) the carbonyloccupied band gap impurity state 0.95 eV (hole trap), and d) the vinyl occupied band gap impurity state 1.06 eV (holetrap). The isosurfaces in the various figures have the same value of probability. The carbonyl and vinyl impuritystates 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 combinedeffect 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 statewhich 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, asthose 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 mixedimpurity state of a carbonyl impurity 0.2 eV below the VBM edge and b) the mixed impurity state of iodine impurity atthe 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 andhave 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. Theelectron spatial probability density (square of the electron state wavefunction) of impurity states created by a vinyland a double bond impurity in adjacent chains and 3.5 Å apart is shown a) an unoccupied impurity state 1.22 eVdeep, b) an unoccupied impurity state 0.86 eV deep, and d) CBM state. The isosurfaces in the various figures havethe same value of probability and is similar to that used in all probability surface plots in the whole thesis.

b) a) c)



79

Chapter 6Summary, 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 VBMstates [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 impuritiessimultaneously 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 ofPolyethylene

The effect of several common chemical impurities on the solid state physics of PE was studied incrystalline 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 VBMinto 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 proposedmechanism 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 impur ity 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 byiodine 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 featuresof 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 iodinecharged 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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95

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 infigure 3.4. The chain is extended along the Z direction. The planar average is taken over ZY planes. The vacuumlevel is at -0.1 eV.

c) d)

X axis of the unit cell (Angstroms)

T h e Z Y p l a n a r a v e r a g e t o t a l l o c a l p o t e n t i a l p l a n e s ( e V )



96

Figure A.2 The planar average total local potential of a Core-Shell model with a carbonyl impurity in the Core-Shellstructure which is shown in Figure 3.11. The planar average is taken over ZY planes. The vacuum level is at -0.36eV.

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

(V KS

bulkav

, where V KS is as in 2.12) along the direction of the surface for which the electronaffinity is to be is determined are specified. Second, a slab structure of the insulator is studied to

determine V KS slabav along the direction of the surface of the slab. In the region away from the

surface of the slab and into the material, V KS slabav retains its bulk profile. In such region, the

energy differences (CBM – V KS bulkav) and (VBM – V KS

bulkav) 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 V KS slabav levels off

c) d)

T h e Z Y p l a

n a r a v e r a g e t o t a l l o c a l p o t e n t i a l

p l a n e s ( e V )

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 (V KS av

) showing the “bulk plus band lineup” procedureapplied to determine the electron affinity of PE (110). The unit cell of the PE slab structure is shown at the top of thefigure, and the interface with vacuum is at the 110 surface. The gray dashed line is theV KS

slabav of the slab structurewhich is determined by averaging the total local potential (V KS in 2.12 ) on planes in the direction of the 110 surface.The black line is the V KS

bulkav of bulk PE determined from a calculation separate from the slab calculation. Thepotentials from the bulk computations match with the potential of the slab structure further from the interfaces of PEand vacuum.

c) d)

P l a n a r a v e r a g e t o t a l l o c a l p o t e n t i a l

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)



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

huzayyin ahmed a 201111 phd thesis

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