electronic and optical properties of conducting polymers from ...1145019/fulltext01.pdfelectronic...

Electronic and optical properties of conductingpolymers from quantum mechanical computations

AMINA MIRSAKIYEVA

Doctorate Thesis in PhysicsDepartment of Applied Physics, School of Engineering Sciences

KTH Royal Institute of TechnologyStockholm, Sweden 2017

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white text hereTRITA-FYS 2017:59ISSN 0280-316XISRN KTH/FYS/–17:59—SEISBN 978-91-7729-529-7

KTH School of Engineering ScienceSE-164 40 Kista, Sweden

Akademisk avhandling som med tillstand av Kungliga Tekniska hogskolanframlagges till offentlig granskning for avlaggande av teknologie doktoratexamen ifysik fredagen den 27 oktober 2017 klockan 10:00 i Sal C, Electrum, KunglTekniska Hogskolan, Isafjordsgatan 22, Kista.© Amina Mirsakiyeva, October 2017 All rights reservedTryck: Universitetsservice US-AB, Stockholm 2017

Dedicated to the memory of my grandfather Abai

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Abstract

Conductive polymers are also known as “organic metals” due to their semi-conducting properties. They are found in a wide range of applications in thefield of organic electronics. However, the growing number of experimentalworks is not widely supported with theoretical calculations. Hence, the fieldof conductive polymers is experiencing lack of understanding of mechanismsoccurring in the polymers. In this PhD thesis, the aim is to increase under-standing of conductive polymers by performing theoretical calculations.

The polymers poly(3,4-ethylenedioxythiophene) (PEDOT) together withits selenium (PEDOS) and tellurium (PEDOTe) derivatives, poly(p-phenylene)(PPP) and naphthobischalcogenadiazoles (NXz) were studied. Several compu-tational methods were applied for analysis of mentioned structures, includ-ing density functional theory (DFT), tight-binding modelling (TB), and Car-Parrinello molecular dynamics (CPMD) calculations.

The combination of CPMD and DFT calculations was applied to investi-gate the PEDOT, PEDOS and PEDOTe. The polymers were studied usingfour different functionals in order to investigate the full picture of structuralchanges, electronic and optical properties. Temperature effects were studied us-ing molecular dynamics simulations. Wide statistics for structural and molec-ular orbitals analysis were collected.

The TB method was employed for PPP. The formation and motion of theexcitations, polarons and bipolarons, along the polymer backbone was investi-gated in presence of electric and magnetic fields. The influence of non-magneticand magnetic impurities was determined.

The extended π-conjugated structures of NXz were computed using B3LYPand ωB97XD functionals in combination with the 6-31+G(d) basis set. Here,the structural changes caused by polaron formation were analyzed. The com-bined analysis of densities of states and absorption spectra was used for under-standing of the charge transition.

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Sammanfattning

Elektrisk ledande polymerer kallas även organiska metaller på grund av sinahalvledaregenskaper och har ett brett tillämpningsområde inom organisk elekt-ronik. Det växande antalet experimentella arbeten har inte erhållit tillräckligtmed stöd från teoretiska beräkningar. Därför upplever detta fält en viss bristpå förståelse av de olika mekanismerna som inträffar i polymererna. I den härdoktorsavhandlingen försöker vi därför erhålla en större förståelse av elektrisktledande polymerar genom att utföra storskaliga teoretiska beräkningar.

Vi beräknade de olika polymererna poly(p-fenyl) (PPP), naphthobischalco-genadiazoles (NXz) and poly(3,4-ethylenedioxythiophene) (PEDOT) tillsam-mans med sina selen- (PEDOS) and tellur- (PEDOTe) derivat. Flera beräk-ningsmetoder tilllämpades för nämnda stukturer, inkluderat täthetsfunktional-steori (DFT), tight-binding"(TB) modellering och Car-Parrinello molekylärdy-namik (CPMD).

En kombination av CPMD- och DFT-beräkningar tillämpades för att un-dersöka PEDOT, PEDOS och PEDOTe. Polymererna studerades med hjälp avfyra olika funktionaler för att erhålla en helhetsbild av strukturförändringar,elektroniska och optiska egenskaper. Molekylärdynamiken gjorde det möjligtatt simulera temperatureffekter, samt att samla ihop omfattande statistik fören strukturell och molekylärorbitalanalys.

TB-metoden användes för PPP. Bildning och rörelse av excitationer, po-laroner och bipolaroner längs med polymer-ryggraden undersöktes i fallet dåelektriska och magnetiska fält är närvarande. Påverkan av icke-magnetiska ochmagnetiska orenheter bestämdes. De utökade π-konjugerade strukturerna avNXz beräknades med hjälp av B3LYP- och ωB97XD-funktionaler i kombina-tion med 6-31+G(d)-basen. Här analyserades strukturförändringarna orsakadeav polaronbildning. Den kombinerade analysen av tillståndstäthet och absorp-tionsspektra användes för förståelse av laddningsövergångar.

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Аннотация

Свое второе название “органические металлы” электропроводящие по-лимеры получили благодаря своим полупроводниковым свойствам, нашед-шим применение в ряде областей органической электроники. И хотя попу-лярность данных полимеров растет так же, как и число эксперименталь-ных работ, количество теоретических исследований пока остается крайнемалым. Данная диссертация представляет собой теоретические рассчетыэлектропроводящих полимеров на основе ab initio методов.

Объектом исследований данной PhD-работы является ряд полимеров:поли(3,4-этилендиокситиофен) (PEDOT) и его селеновый (PEDOS) и тел-луриевый (PEDOTe) производные, поли(p-фенилен) (PPP), нафтобисгало-гендиазолы (NXz). Для анализа упомянутых структур применялась ком-бинация методов, основанных на теории функционала плотности (DFT),включая молекулярную динамику Кaра—Парринелло (CPMD) и расчетыс использованием гибридных функционалов, а также моделирование при-ближения сильно связанных электронов (TB).

Комплекс расчетов CPMD и DFT был применен для исследованияPEDOT, PEDOS и PEDOTe. Четыре разных функционала были исполь-зованы, чтобы продемонстрировать полную картину структурных измене-ний, электронных и оптических свойств. Молекулярная динамика позволи-ла смоделировать влияние температуры и предоставить обширную стати-стику для анализа структурных изменений, включая длины связей и углыкручения между мономерами, и молекулярных орбиталей.

Метод TB использовался для PPP, для изучения формирования и дви-жения экситонов, включая поляроны и биполяроны, вдоль основной цепиполимера под воздействием электрического и магнитного полей. В даннойже работе было определено влияние немагнитных и магнитных примесейна упомянутые экситоны.

Исследования нафтобисгалогендиазолов позволило рассмотреть рас-ширенные π-сопряженные структуры. Вычисления были выполнены с ис-пользованием функционалов B3LYP и ωB97XD в сочетании с базиснымнабором 6-31+G(d). В данной части тезиса были также проанализированыструктурные изменения, вызванные присутствием полярона. Анализируяплотности состояний и спектры поглощения, мы продемонстрировали воз-можность перехода заряда между основным и возбужденным состояниями.

viii LIST OF ABBREVIATIONS

List of Abbreviations

BO Born-Oppenheimer approximationBOMD Born-Oppenheimer molecular dynamicsCAM Coulomb-attenuating methodCPMD Car-Parrinello molecular dynamicsDFT density functional theoryDOF degrees of freedomDOS density of statesEDOS 3,4-ethylenedioxyselenopheneEDOT 3,4-ethylenedioxythiopheneEDOTe 3,4-ethylenedioxytelluropheneGGA generalized gradient approximationHF Hartree-Fock (theory)HK Hohenberg-Kohn (theorems)HOMO highest occupied molecular orbitalKS Kohn-Sham anzatsLC long-range correlationLCAO linear combinations of atomic orbitalsLDA local density approximationLED blue-light-emitting diodesLUMO lowest unoccupied molecular orbitalMO molecular orbitalNOz naphthobisoxadiazoleNTz naphthobisthiadiazoleNXz naphthobischalcogenadiazolesP3HTe poly(3-hexyltellurophene)PEDOS poly(3,4-ethylenedioxyselenophene)PEDOT poly(3,4-ethylenedioxythiophene)PEDOTe poly(3,4-ethylenedioxytellurophene)PPP poly(p-phenylene)PSS polystyrene sulfonatePSSH polystyrene sulfonic acidRG Runge-Gross (theorem)SSH Su-Schrieffer-Heeger (Hamiltonian)TB tight-binding (modelling or method)TD-DFT time-dependent density functional theoryTMA tetramethacrylateTOS tosylateUV ultravioletVEH valence effective Hamiltonian

List of Figures

1.1 Examples of conductive polymer applications taken from open sources:organic solar cells [1], organic transistors [2], OLED [3, 4], photograhicfilms [5], electroluminescence lamps [6]. . . . . . . . . . . . . . . . . . . 2

2.1 Conjugated polymers: a) polyaniline; b) polypyrrole; c) polyacetylene;d) polythiophene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Polyacetylene: a) trans-position; b) cis-position. . . . . . . . . . . . . . 42.3 Sulphur-containing conductive polymers: a) alkoxythiophene; b) Bayer

polythiophene [7]; c) PEDOT. . . . . . . . . . . . . . . . . . . . . . . . . 62.4 The band structures for undoped polymers, polarons and bipolarons. . . 72.5 Poly(p-phenylene): aromatic (a) and quinoid (b) structures. . . . . . . . 82.6 Timeline of conductivity values for PEDOT:PSS [8–17]. . . . . . . . . . 92.7 Aromatic (a) and quinoid (b) structure of the PEDOT. . . . . . . . . . 102.8 Effect of the chain length. Here, bond 1 links atom 2 and 3 according to

Figure 2.7, bond 2 connects atoms 3 and 4, bond 3 is between atoms 4and 5, and bond 4 characterizes the connection between monomer units. 11

2.9 Frontier orbitals of hexamer of PEDOT, calculated under B3LYP/6-31G(d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.10 PEDOT(a) and its derivatives: PEDOS(b) and PEDOTe(c). . . . . . . 142.11 Structures of benzochalcogenadiazoles and naphthobischalcogenadiza-

oles: a) 2,1,3-benzothiadiazole (BOz), b) 2,1,3-benzoxadiazole (BTz), c)2,1,3-benzoselenadiazole (BSz), d) naphtho[1,2-c:5,6-c]bis[1,2,5]oxadiazole(NOz), e) naphtho[1,2-c:5,6-c]bis[1,2,5]thiadiazole (NTz), f) naphtho[1,2-c:5,6-c]bis[1,2,5]selenadiazole (NSz). . . . . . . . . . . . . . . . . . . . . 17

3.1 Tight-binding models for (CH)x(a) and poly(p-phenylene) (b). . . . . . 27

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List of Publications

The results of present PhD thesis are reported in form of following manuscripts:

(i) QuantumMolecular Dynamical Calculations of PEDOT 12-Oligomerand its Selenium and Tellurium DerivativesAmina Mirsakiyeva, Håkan W. Hugosson, Xavier Crispin, Anna DelinPublished: Journal of Electronic Materials 46 (5), 3071-3075 (2017) [18]

(ii) Temperature dependence of band gaps and conformational disor-der in PEDOT and its selenium and tellurium derivatives: densityfunctional calculationsAmina Mirsakiyeva, Håkan W.Hugosson, Mathieu Linares, Anna DelinAccepted to Journal of Chemical Physics, Sep 2017

(iii) Polaron formation and optical absorption in PEDOT and its sele-nium and tellurium derivatives: density functional calculationsAmina Mirsakiyeva, Mathieu Linares, Anna DelinManuscript

(iv) Breakdown of Polarons in Conducting Polymers at Device FieldStrengthsM. Reza Mahani, Amina Mirsakiyeva, Anna DelinPublished: J. Phys. Chem. C, 2017, 121 (19), pp 10317–10324 [19]

(v) Charge transport via polarons in doped poly(p-phenylene) with im-purityM. Reza Mahani, Amina Mirsakiyeva, Anna DelinManuscript

(vi) Optical properties of Naphthobischalcogenadiazoles from densityfunctional perspectiveAmina Mirsakiyeva, Mathieu Linares, Anna DelinManuscript

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xii LIST OF PUBLICATIONS

My contribution to Papers I-III and VI includes a performance of calculations,analyses of obtained data, writing of the first draft.

My contribution to Papers IV-V is mainly including calculations using the tight-binding model of my colleague M. Reza Mahani and participation in analysis.

Contents

List of Abbreviations viii

List of Figures ix

List of Publications xi

Contents xiii

1 Introduction 1

2 Conductive polymers 32.1 A brief history of conductive polymers . . . . . . . . . . . . . . . . . 32.2 Mechanism of conductivity . . . . . . . . . . . . . . . . . . . . . . . 52.3 Poly(p-phenylene) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Poly(3,4-ethylenedioxythiophene) (PEDOT) . . . . . . . . . . . . . . 82.5 Derivatives of PEDOT . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Naphthobischalcogenadizaoles . . . . . . . . . . . . . . . . . . . . . . 16

3 Theoretical background 193.1 Density Functional theory . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Hartree-Fock method . . . . . . . . . . . . . . . . . . . . . . . 193.1.2 Born-Oppenheimer approximation . . . . . . . . . . . . . . . 203.1.3 Hohenberg-Kohn theorems . . . . . . . . . . . . . . . . . . . 203.1.4 The Kohn-Sham ansatz . . . . . . . . . . . . . . . . . . . . . 213.1.5 Methods of approximations . . . . . . . . . . . . . . . . . . . 21

3.2 Hybrid functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.1 B3LYP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.2 CAM-B3LYP . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.3 LC-ωBE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.4 ω97XD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Time-dependent DFT . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5 Tight-binding and the SSH Hamiltonian . . . . . . . . . . . . . . . . 263.6 Car-Parrinello molecular dynamics . . . . . . . . . . . . . . . . . . . 28

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xiv CONTENTS

4 Computational details 314.1 CPMD calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Gaussian calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Tight-binding modelling . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Results and discussion 37

6 Conclusion 41

7 Acknowledgement 43

Bibliography 45

Chapter 1

Introduction

Begin at the beginning, theKing said gravely, and go ontill you come to the end: thenstop.

—Lewis Carroll

“Organic metals” is a relatively new term, used for polymers showing a com-paratively high electrical conductivity. Such materials initially appeared in the late1970s, when they first conquered the scientific world and then the industry. It ishard to overestimate the influence of those materials, since they are the core ofthe organic electronics. They found application in biosensors [20], electrochemicaltransistors [21], photovoltaics [22], touch screens [23], solar cells [24], supercapaci-tors [25], etc.

Basically, organic metals are conductive polymers. It may feel confusing toput together “conductivity” and “polymers” in the same phrase, since most of theplastics are found to be good insulators. However, in 1977 a series of experimentsperformed by Alan J. Heeger, Alan G. MacDiarmid and Hideki Shirakawa [26–28]demonstrated a significant conductivity of polyacetylene thin films. The oxidationof polyacetylene with chlorine, bromine and iodine resulted in conductivity value105 S/cm, which is 109 times greater than usual polymer materials. This value canbe compared to the conductivity of silver and copper, which is equal to 108 S/cm [29].The first investigated materials unfortunately showed disadvantageous propertiessuch as stability against air humidity and water solubility, which prevented anindustrial implementation. That has not changed until the end of 1980s, when Bayercompany patented poly(3,4-ethylenedioxythiophene), or PEDOT, [30–32] which wasa significant breakthrough in organic electronics.

The fascinating number of experimental work and engineering solutions for PE-DOT were not supported with a sufficient amount of theoretical studies. Only inthe beginning of 2000s the structure of PEDOT was determined from the ab initiopoint of view [33]. Currently, a number of the computational studies is focusing onelectrical structure, optical properties, morphology, and charge transport effects of

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2 CHAPTER 1. INTRODUCTION

Figure 1.1: Examples of conductive polymer applications taken from open sources:organic solar cells [1], organic transistors [2], OLED [3, 4], photograhic films [5],electroluminescence lamps [6].

different conductive polymers. Some examples of conductive polymer applicationsare illustrated on Figure 1.1.

This PhD work is focusing on the theoretical studies of conductive polymers,or so-called organic metals. In this study, poly(3,4-ethylenedioxythiophene) and itsderivatives, poly(p-phenylene) and naphthobischalcogenadizaoles were investigatedusing the combination of density functional theory-based calculation together withCar-Parrinello molecular dynamics and tight-binding modelling.

Chapter 2

Conductive polymers

The more original a discovery,the more obvious it seemsafterwards.

—Arthur Koestler

2.1 A brief history of conductive polymersNormally, plastic materials or polymers, are known for their insulator properties.However, the main research objects of this thesis are the conjugated polymers (Fig-ure 2.1), which are characterized by a relatively high conductivity. Those plasticsin neutral state belong to the semiconductors, but after oxidation or reduction theirelectrical conductivity significantly increases. Therefore, such materials are com-monly called synthetic or organic metals [26–28].

The first of the conductive polymers were discovered back in the XIX century,when the chemistry professor at the College of the London Hospital H. Lethebyelectropolymerized aniline sulfate to a blue-black solid layer on a platinum elec-trode [34]. It took around a 100 years since the first experiment with polyaniline(Figure 2.1a) for the conductive properties of this polymer to be discovered. TheFrench group of R. Buvet with a major impact of M. Jozefowicz [35] demonstratedthe value of electrical conductivity from 10−4S/cm to 10 in 1967. Six years later

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Figure 2.1: Conjugated polymers: a) polyaniline; b) polypyrrole; c) polyacetylene;d) polythiophene.

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4 CHAPTER 2. CONDUCTIVE POLYMERS

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Figure 2.2: Polyacetylene: a) trans-position; b) cis-position.

the electroconductivity of the polyaniline was found in the range between 5 and30 S/cm [36]. Despite a wide number of studies, polyaniline could not reach ahigher success in the industry due to its intensive color.

The second significant mark in the history of conductive polymers is the dis-covery of the first conductive poly(heterocycles) - the polypyrroles (Figure 2.1b).Since 1960s this group was and still remains one of the most attractive polymersfor practical implementation. In 1963 the research group of R. Weiss investigatedneutral polypyrrole and polypyrrole-iodine complexes by charge-transfer electronspin resonance absorption and electrical resistivity [37]. It was shown that polymercan behave as either intrinsic or extrinsic semiconductor with n- or p-type charac-teristics. Later in 1969 an Italian research group studied the oxidation of pyrroleitself to pyrrole black and reached an electrical conductivity of 7.54 S/cm at theambient temperature [38]. Ten years later the electroconductivity increased up to10-100 S/cm [39]. Nevertheless, the polypyrroles were too toxic, with strong inten-sive color and poor transparency of the thin films [40] - all together it preventedpolypyrroles from their use in “metallic” applications.

The next type of the polymers with clear conductive properties was polyacety-lene (Figure 2.1c), for which oligomers were obtained in 1874. The successfulsynthesis of the polymer chains was performed by G. Natta in 1958 [41, 42], whodiscovered black powder insoluble in organic solvents. Later in 1961 a Japanesegroup of Hatano studied its semiconductivity [43]. Using Ziegler-Natta catalyststhey demonstrated that more crystal structures had higher values of conductivity.However, the greatest result achieved was 10−5 S/cm. At that time polyacetyleneswere materials of interest, but had wide range of technical challenges. For example,the long polymer chains were not soluble. The breakthrough in the field of con-ductive polymers and particularly polyacetylenes was done in 1960s and 1970s byAlan J. Heeger, Alan G.MacDiarmid and Hideki Shirakawa [44, 45], who later wasawarded with a Nobel Prize in Chemistry [26–28]. Their discovery was initiated bya laboratory failure. After Natta’s experiment in 1958 [41, 42], the Ikeda’s group inTokyo Institute of Technology started investigation of the polyacetylene polymer-ization mechanism, when Shirakawa joined the group. The resulting black powderof the polyacetylene in trans-position (Figure 2.2a) was not suitable for furtherstudies due to its insolubility and infusibility. However, the usual concentration ofZiegler-Natta catalyst was increased 1000 times due to misunderstanding between

2.2. MECHANISM OF CONDUCTIVITY 5

Shirakawa and his co-worker [28] (instead of mM the usual M were used). As aresult, the reaction rate significantly increased and the acetylene gas polymerizedon the surface of the catalyst as a thin film with a metallic lustre. The furtherinvestigations showed a strong dependence of double bond formation from reac-tion temperature [46]. At 150◦C the monomers are organized into trans-position(Figure 2.2a), which leads to a higher conductivity 10−5-10−4 S/cm. Meanwhile,at -78◦C the 98% of monomers are found in cis-position (Figure 2.2b), and itsconductivity is equal to 10−9-10−8 S/cm.

By that time MacDiarmid and Heeger have investigated (SN)x materials. Theyfound that pure thin films have less conductivity than the ones doped with Br2. Itwas decided to add bromine to polyacetylene. Therefore, the conductivity increasedof approximately four orders of magnitude from 10−5 to 0.5 S/cm. The polyacety-lene synthesis reproduced with iodine showed a rise in conductivity up to 30 S/cm.The following experiments showed that polyacetylene reaches up to 560 S/cm con-ductivity. In 1987 Naarmann and Theophilou set an experiment for iodine-basedpolyacetylene with a final conductivity around 105 S/cm [47].

Despite the high conductivity, the polyacetylene could not guarantee a suc-cessful industrial implementation for conductive polymers, because of its high air-sensitivity. The sensitive π-electrons could be stabilized by heteroatoms, such as Nand S atoms, for which the electron-donating nature will stabilize the conjugatingsystems. The nitrogen-containing compounds - polypyrroles - were described above.However, the sulphur-containing polymers - polythiophenes (Figure 2.1d)- will bediscussed below as the next important point in conductive polymers history.

Polythiophenes were also known before their conductivity was observed in 1967.A. G. Davies and co-workers discovered that furan, pyrrole and thiophene havecyclic structures and show some electrical conductivity [48]. The nature of con-ductivity was found to be ionic, unlike in polyaniline and polypyrrole, where theconductivity occurs due to electrons/holes. In 1982 Tourillon and Garnier discov-ered true electronic conductivity [49] in polythiophene. This polymer demonstratedconductivity values between 10-100 S/cm. Nevertheless, the stability against airand humidity was low, and the bipolaron state (explained in more details in thesection “Mechanism of conductivity”), important for technical applications,was not achieved [42]. Thus, the polythiophenes were stabilized further with oxy-gen atoms at the positions 3 and 4 (Figure 2.3a, where O atoms are located at R1and R2 positions), which allowed to stabilize the bipolaronic state of the conju-gated polymer [40]. Bayer company registered several patents for substituted poly-thiophenes (for example, Figure 2.3b), which was the last step towards poly(3,4-ethylenedioxythiophene), or PEDOT (Figure 2.3c).

2.2 Mechanism of conductivity

Before a further overview of this thesis main research objects will be given, themechanism of conductivity in conjugated polymers will be introduced.


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Figure 2.3: Sulphur-containing conductive polymers: a) alkoxythiophene; b) Bayerpolythiophene [7]; c) PEDOT.

The unique properties of the conjugated polymers can be explained by the al-ternation of single and double bonds along the polymer backbone. For instance,the polyacetylene is a standard example of the conjugated polymers (Figure 2.2).Each carbon atom is forming four bonds. Three of them are σ-bonds (two with aneighboring carbon atoms and one with a hydrogen atom) and the remaining oneengages in a π-system formation [26]. If all the carbon-carbon bonds were equallylong, then the π-electron would be found in a one-and-a-half filled continuous band.However, the Peierls transition states that oscillations of atomic positions in 1D-crystal leads to instability or distortion of the chain [42]. As a result, the symmetryof the system is reduced and the levels of the orbitals are rearranged in such waythat the orbitals with a lower energy are filled first. Hence, in the polyacetylenechain the shorter bond (double) is followed by the longer bond (single). This struc-ture is typical for all conjugated polymers. Therefore, all conjugated polymers aretypically semiconductors in their pristine state with a lower conductivity.

To improve the conductivity of the conjugated polymers one can use severalmethods to introduce a charge into the system and significantly increase the elec-tronic properties. The first method is so-called chemical doping. For example, whenpolyacetylene is treated with iodine, the polymer chain is oxidized, the polyacety-lene obtains a positive charge, while iodide is forming a counterion. In general, bothn- and p-type dopants could be used for conjugated polymer treatment. The othermethods is electrochemical doping, such as photo-doping, in which an electron-ionpair is formed due to light absorption.

As it was mentioned in “A brief history of conductive polymers”, the pro-cess of doping and its influence on the conductivity of those materials was discoveredin the 1970s during investigations of polyacetylene [26–28]. For conductive poly-mers the term “doping” means the implementation of the charge transfer throughthe process of oxidation or reduction, the associated insertion of a counter ion, andthe simultaneous control of the Fermi level or chemical potential [50, 51]. Intro-duced charges are changing the electronic structure of the polymers and stored onnewly formed states which result both in charge and lattice distortion - polarons,bipolarons, and solitons. On Figure 2.4 the valence band, which associates with thehigh occupied molecular orbitals (HOMO), and conducting band, which associates

2.3. POLY(p-PHENYLENE) 7

CB

VB

CB

VB

Fermi

Level

E1△ ε

(a) (b)

Figure 2.4: The band structures for undoped polymers, polarons and bipolarons.

with the lowest unoccupied molecular orbitals (LUMO), are shown. Here, one canfirst see the undoped structure (Figure 2.4a). Then the electron is added (n-type)or is removed (p-type) and newly formed polaron is obtained in valence band (Fig-ure 2.4b). The newly appeared half-filled level has a spin 1/2 for both p- and n-typematerials. A combination of the two polarons with the same sign is forming bipo-laron. For n-type bipolaron both newly formed energy levels are occupied and forp-type - both are empty. The spin of the bipolaron is equal to zero. The third typeof the excitons - soliton, appears only in polymers with degenerate structure, suchas polyacetylene [52], and are not investigated in this study.

The investigation of the polarons and bipolarons is one of the fundamental ques-tion for computational studies. Those excitons could be tracked through the changesin the geometrical structures [50], such as bond lengths, the point-charges distri-bution together with the spin densities distribution. For instance, the combinedHartree-Fock (HF) and density functional theory (DFT) calculations using BLYPfunctional by Moro and co-workers demonstrated the delocalization of the struc-tural distortion in the center of the polythiophene chain and localization of thenet charge values and atomic spin densities distortion [53]. Another example ofDFT studies, which used BHandHLYP functional, indicated a self-localization forcharges, spin densities, and geometrical distortions [54]. Therefore, it is importantto use an appropriate method to investigate polarons and bipolarons in conductivepolymers.

2.3 Poly(p-phenylene)

Poly(p-phenylene) (PPP), or poly(1,4-phenylene), is an insoluble polymer with a lowmolecular weight [55]. It was first synthesized in 1962 by direct coupling of benzenecations and Lewis acid [56]. Its conductivity attracted attention starting 1979, whenit was found that undoped PPP has conductivity value around 10−14 S/cm, whichcan be further increased up to 106 S/cm during the doping process [57]. Today that


(b)(a)

nn

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

Figure 2.5: Poly(p-phenylene): aromatic (a) and quinoid (b) structures.

value reached 500 S/cm [52]. As all conductive polymers, PPP found applicationin different areas, such as organic rechargeable batteries [58, 59] and blue-light-emitting diodes (LED) [60].

In 1985 Brédas determined the aromatic-like structure of the undoped PPP [61]with the carbon-carbon bond to be equal to 1.39Å inside the ring and 1.57Å forconnecting monomers (Figure 2.5).

Both optical and electronic properties of PPP polymer are resulting from largeconjugation along the chain, when π-electrons are highly delocalized. Therefore, thedegree of conjugation depends on planarity of the chain. In 1984 HF calculationsshowed the torsion to be 22.7◦ [62]. Later in 1995, the local density approximation(LDA) calculations performed by Ambrosch-Draxl and co-workers showed that tor-sion angle between PPP monomer units is equal to 27◦ for single chains and 17◦for crystals [63], and MP2/6-31G(d,p) based investigation in 2005 determined thelowest energy minimum at 45◦ [64].

Using the valence effective Hamiltonian (VEH) pseudopotential technique Bré-das calculated the band gap for the undoped PPP polymer to be 3.37 eV [61], whichis in good agreement with experimental value 3.43 eV [65]. The band gap decreasedwith increasing of the degree of the quinoid contribution to the structure due toshifting of the highest occupied molecular orbitals up and lowest unoccupied onesdown. Hence, the aromatic structure is characterized with a higher ionization poten-tial and a smaller electron affinity compare to the quinoid-like one. Therefore, thesystem will show structural distortions (polarons and bipolarons) around charges indoped polymers.

2.4 Poly(3,4-ethylenedioxythiophene) (PEDOT)

Poly(3,4-ethylenedioxythiophene) (PEDOT) was invented in April 1988 by F. Jonas,G. Heywang, and W. Schmidtberg [30–32], who could already show its implementa-tion in capacitor applications [66]. The novel material was an insoluble polymer inits pure state with unpleasant odor [40] with a high conductivity (around 300 S/cmat that time) [8]. However, PEDOT was found to be stable against air humidity -an important property for water-solubility and further industrial implementation.

Short after the first application, the new opportunity for PEDOT appeared:the Agfa-Gevaert AG, a Bayer subsidiary, was looking for a new antistatic materialfor photographic films. The sodium salt of poly(styrenesulfonic acid), or PSS-Na,

2.4. POLY(3,4-ETHYLENEDIOXYTHIOPHENE) (PEDOT) 9

Table 2.1: Selected Physical Properties of EDOT [40].

Viscosity (20◦C) 11mPa·sDensity (20◦C) 1.34 g/cm3

Melting point 10.5◦CBoiling point (1013mbar) 225◦CVapor pressure (20◦C) 0.05mbarVapor pressure (90◦C) 10mbarSolubility in water (20◦C) 2.1 g/lFlash point 104◦CIgnition point 360◦C

2000 2005 2010 2015Year

0

1000

2000

3000

4000

Conduct

ivit

y, S

cm

-1

Figure 2.6: Timeline of conductivity values for PEDOT:PSS [8–17].

which was currently used at that time, has a high sensitivity to air humidity. Studiesgave rise a water-soluble PEDOT:PSS complex, where PSS became a counter ionfor positively charged PEDOT [40, 67]. Today other well-known doping agents forPEDOT are tosylate (TOS) [10, 68] and tetramethacrylate (TMA).

To sum up the brief history of PEDOT, let us have a look on Table 2.1, wheresome selected physical properties are shown [40], and Figure 2.6, where the dra-matic growth of the electroconductivity value for PEDOT is shown: the maximumvalue around 3000-4000 S/cm was obtained before 2015 [13–15, 69, 70], the most ex-perimental works report about 1000-2000 S/cm [8–12, 16, 17]. Together with otherunique features of PEDOT, such as stability at the room temperature and at thepresence of air, transparency, thermal stability and others, that guaranteed PEDOThigh popularity in the industry. However, despite of the large number of experi-mental research, the lack of understanding of the processes in conductive polymersin general and in PEDOT particularly brings computational studies to be highlyessential and valuable.

The computational investigations of PEDOT can be divided into two main seg-ments: geometrical structure and electronic properties. One of the early structuralstudies of PEDOT, which determined its molecular structure, are the investigations


⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

(b)(a)

n

5

43

2

n

S

OO

S

OO

Figure 2.7: Aromatic (a) and quinoid (b) structure of the PEDOT.

done by Dkhissi and co-workers in 2002 [71]. Here, the neutral oligomers of PEDOTwere found to have two stable structures: aromatic (Figure 2.7a) and quinoid (Fig-ure 2.7b). Later, the following research [72–74] indicated that the neutral PEDOTis most probably in the aromatic-like state, which is changing to the the quinoid-likestructure during the doping process.

Structural studies were concentrated on bonds, angles and dihedral angles. Stud-ies done by Kim and Brédas [74] investigated PEDOT oligomers with a chain lengthN=2-4 using Car-Parrinello molecular dynamics (BLYP/GGA). Those computa-tions confirmed the polymer backbone structure as it is mentioned above (neutralaromatic-like structure and charged quinoid-like one) and demonstrated the effectof the chain length on the bond length: the longer the polymer chain, the greaterits conjugation (Figure 2.8). In addition, it has also described the influence ofthe oxygen and sulphur atoms in the general planarization of the polymer chain.Later, Burkhardt and co-workers [75] discovered that the π-bonding interactionshave a stronger influence on the planarity rather than non-bonding interactions be-tween sulphur and oxygen atoms. Moreover, Poater and co-workers investigateda number of polythiophenes and its derivatives [76] and concluded that the pro-posed intramolecular interactions between the sulphur and oxygen atoms in factdestabilize the structure of PEDOT, but that is compensated by interaction of theoxygen atoms with the carbon atoms in the positions 3 and 4 (according to Figure2.7) [75]. The research conducted by Burkhardt and co-workers under B3LYP/6-31+G(d,p) level of physics showed that C-O bond length and C-O-C angles aremore typical for sp2-hybradized oxygen atoms, while it was expected to find oxygenin sp3-hybridization. Hence, the π-electrons at the p-orbital of the oxygen atomsare participating in the overall conjugation of the system, which stabilizes it andincreases the delocalization of the charge.

Next attempting structural studies are related to angles and dihedral angles.Kim and Brédas [74] stated the influence of the monomer rotation on the chargecarriers: the rotation for 9◦ around the polymer backbone for doped PEDOT:TOScrystal leads to the intrachain effective mass increase for holes and decrease forelectrons. Bendikov group showed by using B3LYP/6-31G(d) for infinite polymer,that twisting of the PEDOT oligomers requires only 0.4 kcal/mol to twist monomeron 15◦ and 6.2 kcal/mol - on 90◦ [77]. Franco-Gonzalez and Zozoulenko [78] in-

2.4. POLY(3,4-ETHYLENEDIOXYTHIOPHENE) (PEDOT) 11

1 2 3 4Bond number, count

1.4

1.45

Bon

d le

ngth

, Å2EDOT4EDOT10EDOT20EDOT

Figure 2.8: Effect of the chain length. Here, bond 1 links atom 2 and 3 accordingto Figure 2.7, bond 2 connects atoms 3 and 4, bond 3 is between atoms 4 and 5,and bond 4 characterizes the connection between monomer units.

vestigated bending of the PEDOT chain θ in general and the twisting betweentwo monomer units φ using ωB97XD/6-31+G(d) for optimization of the PEDOToligomers (N=2-18) with further molecular dynamics applied. Calculations weresimulating the experimental procedure [79], so it was performed in the presence ofthe water molecules with a following gradually decreasing of the water concentra-tion. The φ found to be the same for the oligomers with a different length, whileθ is growing for the longer chains. The overall bending of the polymer chains isalso dependent on water content due to less distance between the chains and moreinfluence on each other when the water is absent.

The same research demonstrated the formation of the crystallite aggregates - the3-6 π-π-stacked chains. Those chains behave as percolative paths for the system.It is shown that the size of the crystallite aggregates is depending on water con-centration, while the π-π-stacked distances are constant regardless of chain length,water and charge carriers. This bring us to the last challenging part of the struc-tural studies of the PEDOT. It turned out that after several decades of research,the scientific society is still unaware of the polymerization process, crystallizationand morphology of the PEDOT films. The topic essential for understanding of thetransport mechanism, is lightly covered in a short list of publications, includingaforementioned work of Franco-Gonzalez and Zozoulenko [78].

The computations of PEDOT crystal made by Kim and Brédas [74] were basedon the orthorhombic unit cell, where a, b, and c were taken as 7.935, 10.52, and7.6Å. The a was optimized, while two other parameters were based on X-ray mea-surements. Each cell contained two chains, each with two EDOT units. Using the


HOM

OLUM

O

Figure 2.9: Frontier orbitals of hexamer of PEDOT, calculated under B3LYP/6-31G(d).

crystal of pristine PEDOT and its isolated chain, Kim and Brédas characterizedthe effect of the interchain interaction on the electronic structure, such as expectedreduction of band gap. For doped crystal, it was shown that the presence of TOSmolecules cancels the chain rotation mentioned above.

One more investigation of PEDOT crystal was performed by Lenz and co-workers [73]. In their study, the crystals of the undoped PEDOT, PEDOT+:PSS−,PEDOT+0.5:PSS−/PSSH were computed using GGA/Pω91. The demonstrateddensities of states show the shift of Fermi level towards valence bands with in-creasing of the doping degree. This was supported with the visualization of thehighest occupied (HOMO) and lowest unoccupied molecular orbitals (LUMO). Au-thors demonstrated that the HOMO is moving from the PEDOT molecules to PSSduring the doping. More than a decade earlier it was illustrated that in poly-thiophene’s derivatives the ethylenedioxy substitutes were destabilizing the HOMOmore than LUMO [80]. This was an effect of the donor-acceptor behavior in theatoms 3 and 4 (according to the Figure 2.7). Figure 2.9 demonstrates strong local-ization of the HOMO in PEDOT polymer chains accumulates over donor subunits,while strong delocalization of the LUMO concentrates over the bridges betweenmonomer units [81].

In principle, the value of HOMO-LUMO gap is comparable to the optical gapobtained in experiments. In Table 2.2 we collected those values calculated withdifferent computational methods. The experimental values of the undoped PEDOTare found around 1.5 eV with a decrease to 1 eV during the doping process [82].The leading choice of B3LYP as a functional together with 6-31G(d) basis set was

2.5. DERIVATIVES OF PEDOT 13

Published work Method Oligomer/polymer band gap,eV

Dkhissi and co-workers, 2003 [33]

6-31G/ B3LYP Undoped oligomerslinearly extrapolatedto infinite polymer

1.65 eV

Aléman and co-workers, 2005 [83]

6-31G(d)/B3LYP

Oligomer of 8 units 2.2 eV

Kim and Brédas,2008 [74]

GGA/BLYP Isolated chain ofN=4 was scaled for200%

1.8 eV

Patra and co-workers, 2008 [84]

6-31G(d)/B3LYP PBC 1.84 eV

Shalabi and co-workers, 2012 [81]

6-31G(d)/B3LYP

Extrapolated poly-mer

1.72 eV

Lenz and co-workers, 2014 [73]

GGA/Pω91 Undoped crystal 0.7 eV

Oligomer of 8 units 1.3 eV

Table 2.2: Calculated band gaps for PEDOT.

obvious at that time. However, calculations with other types of functionals couldbring new insights to that topic.

Finally, the computational studies are focusing on charge carriers in conductivepolymers. Kim and Brédas [74] demonstrated that the influence of the interchaininteractions on the charge carrier effective mass is taking over the influence of theethylenedioxysubstitutes. As a result, the electrons are lighter than holes in undopedcrystal of PEDOT. Munoz and co-workers [85] investigated the densities of statefor π-π-stacked crystalittes of PEDOT and concluded the shrinking of the distancebetween the valence band and bipolaron band with increasing of the charge carriersconcentration.

To sum up, PEDOT is one of the most studied conductive polymers nowadays.Its role in organic electronics cannot be overestimated. Some of the calculationsmentioned above have described the structural and electronic properties of PE-DOT. However, there is a number of remaining questions about the physics of thatpolymer. For instance, most of the calculations are present at a ground tempera-ture behavior of the polymer chains, while the real experiment is occurring at anambient temperatures. Furthermore, the effect of the calculation methods such asa choice of functional is not fully understood and could be questioned.

2.5 Derivatives of PEDOTAs PEDOT is the most successful commercial conductive polymer, the interest to itsderivatives - poly(3,4-ethylenedioxyselenophene) (PEDOS) and poly(3,4- ethylene-


⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

nn

(c)(b)(a)

n

Se

OO

S

OO

Te

OO

Figure 2.10: PEDOT(a) and its derivatives: PEDOS(b) and PEDOTe(c).

dioxytellurophene) (PEDOTe) - is growing (Figure 2.10), although the PEDOTewas not yet synthesized (as author knows to date). Due to differences between se-lenium and sulphur atoms, it was expected that selenium-containing polymers willhave several advantages over sulphur-containing compounds, such as lower oxida-tion and reduction potentials, higher tendency to polarization, greater interchaintransfer [84]. Regarding to that, a significant work was done by M. Bendikov’sgroup [77, 84, 86].

The initial interest to selenium derivatives of PEDOT originated from polythio-phene and polyselenophene studies, when it was demonstrated that the band gap ofthe selenium-containing polymers was lower [87]. Using B3LYP functional togetherwith 6-31G(d) basis set, the number of the oligothiophenes and oligoselenopheneswere studied and the band gaps for their polymers extrapolated by second-orderpolynomial were determined. The value for oligoselenophenes was 0.1-0.2 eV lowerthan the value for oligothiophenes. The same difference was found for PEDOS andPEDOT, too: 1.66 eV for PEDOS and 1.83 eV for PEDOT [88]. The experimentallyobtained optical band gaps are 0.2-0.4 eV less than those calculated [89].

Following studies added that for oligomers up to N=12 the internal reorgani-zation energy λ was larger for PEDOS in comparison to PEDOT [90, 91]. Forlonger polymer chains this value was almost equal to 0 (studied up to N=50) dueto the greater positive charge delocalization in longer oligomers. The value of λ ischaracterizing a charge transfer since this parameter is controlling a charge hopping.

The comparison of the twisting energy of the PEDOT and PEDOS chains deter-mined that the twisting of the selenium-containing polymer systems required moreenergy compared to the sulphur-containing polymers both in pentocycles and in thePEDOT/PEDOS cases [77]. The higher twisting energy of the polyselenopheneswas found to be the result of the low aromaticity of the selenophene ring whichprefers quinoid structure of the polymer over aromatic [86, 92]. Hence, it makedthe conductive polymer more planar, as it was expected [93].

Concerning the fact that polyselenophenes could also be doped with a greaternumber of the dopants than polythiophenes [91], one can conclude that PEDOSshould be more conjugated than PEDOT, and as a result, the conductivity of PE-DOS should exceed the value for PEDOT.

The polytellurophenes are less studied compared to the polythiophenes andpolyselenophenes. The dramatically smaller number of reported investigations on

2.5. DERIVATIVES OF PEDOT 15

tellurium-containing polymers indicates great challenges in the synthesis of stablemolecules. However, we can predict some properties of such material based on theS-Se-Te comparison. First of all, the electronegativity of the tellurium atom is only2.10, while for sulphur it is 2.58 and for selenium it is 2.55. Hence, tellurium is ametal with a higher degree of polarizability. The band gap for PEDOTe should beeven lower than for PEDOS and PEDOT.

The most studied tellurium-based polymer is poly(3-hexyltellurophene) (P3HTe).The experimental result of the band gap for P3HTe equals to 1.37 eV [94], forselenium analogue the band gap is equal to 1.6 eV [95]. The Bendikov’s group,whose contribution into the polyselenophenes investigation is highly valuable, alsomade an attempt in obtaining polytellurophenes: Patra and co-workers have synthe-sized 3,4-dimethoxytellurophene [96]. The resulting compound was unstable, butthey recorded the UV-spectrum during the electropolymerization, which allowedthem to determine the optical band gap - 1.51 eV, and to compare it to calcu-lated value - 1.64 eV. Computations were performed using B3LYP functional andLanL2DZ basis set for both tellurium and selenium analogues. As expected, 3,4-dimethoxytellurophene had a smaller band gap than 3,4-dimethoxyselenophene forabout 0.2 eV.

Later, Jahnke and co-workers [97, 98] have published several experimental andtheoretical papers on the synthesis, electronic and optical properties of the polytel-lurophenes. In 2010, they have studied the influence of the halogenes on the frontierorbitals of the polymers by using B3LYP functional together with LanL2DZ basisset. The found reduction of the HOMO-LUMO band gap from 3.18 eV to 2.19 eV andchanging of the molecular orbital position states a formation of the charge-transfercomplexes. Two years after that the absorption spectra for polytellurophenes werecalculated using the same DFT method [99]. In this study, a deeper understand-ing of the increasing stability during the polymerization process was shown, whenBr2 molecules were added. Authors concluded that two-electron oxidative additionto polytellurophenes is promising for catalyzing energy storage reactions. Morederivatives of polytellurophenes were studied with the same method and publishedin 2013 [94].

A comparison between electronic structures of the polythiophene and its sele-nium and tellurium compounds was performed using B3LYP/LanL2DZ. The cal-culated HOMO-LUMO band gaps were 2.5 eV for polythiophene, 2.1 eV for poly-selenophene, and 1.9 eV for polytellurophene [100]. The density of states (DOS)analysis demonstrated that the heteroatoms had minimal influence on the forma-tion of the bands. However, the dipolar interaction dominated exciton splitting hadsignificant influence on the HOMO-LUMO gaps of these oligomers.

Recently, Kaya and Kayi published theoretical investigations on selenium- andtellurium-containing polymer [101]. The polymers named 4,7-di(selenophen-2-yl)-benzo[c][1,2,5]selenadiazole and 4,7-di(tellurophen-2-yl)benzo[c][1,2,5]telluradiazolewere investigated with B3LYP and LC-BLYP functionals and LanL2DZ basis set.Their band gaps were found to be lower than all other newly synthesized and studiedconductive polymers: with B3LYP it is 1.08 eV for selenium polymer and 0.69 eV


for tellurium, and with LC-BLYP - 0.97 eV and 0.93 eV respectively.To sum up, polyselenophenes and polytellurophenes have a more quinoid char-

acter of the molecular structure compared to polythiophenes. They also have lowerband gaps and a higher polarizability. Previous studies demonstrated that in poly-selenophenes intermolecular Se-Se interaction led to a wide bandwidth and strongerinter-chain charge transfer. Consequently, it is reasonable to assume that the largersize of the Se and Te will provide higher charge accumulation probability. Thehuge industrial success of the PEDOT makes PEDOS and PEDOTe interestingcandidates for further studies, both experimental and theoretical.

2.6 Naphthobischalcogenadizaoles

First naphthobischalcogenadizaoles, or NXz, were reported in 1991 [102], whenMatako and co-workers published their work on reaction of explosive N4S4 [103]with variety of naphtols. The interesting structures were not investigated enoughthen due to complicated and dangerous synthesis route. However, recently Osakaand Takamiya [104] reported an alternative scheme, where NXz molecules could beobtained through naphthalene-containing compounds by reaction of reduction inthe presence of HCl.

Although those molecules are quite new, the similar polymers were synthesizedthroughout last two decades. The main advantage of those naphthalene-containingpolymers is the extended π-conjugated system. Thus, the gap between valence andconducting bands is expected to be lower than in polymers like PEDOT [104, 105].They also could provide the extended absorption range with a broader peak around700-800 nm [106], which is a promising advantage for solar cells. Besides that, thestronger donor-accepting characteristics could be reached through the enhancedconjugation of stalked rings [105].

The DFT methods are popular for investigation of extended π-conjugated sys-tems. The combination of B3LYP functional and 6-31G(d) basis set was used to cal-culate several newly synthesized chemicals, such as single monomer of naphtho[1,2-c:5,6-c]bis[1,2,5]oxadiazole (NOz), naphtho[1,2-c:5,6-c]bis[1,2,5]selenadiazole (NSz),naphtho[1,2-c:5,6-c]bis[1,2,5]thiadiazole (NTz) [107], naphtho[2,3-b:6,7-b]dithiophe-nediimide (NDTI) [108]. The molecular orbital analysis was provided for a singlemonomer of mentioned polymers in order to investigate the band gaps. The re-sults were compared to benzene analogs of those polymers (2,1,3-benzothiadiazole(BOz), 2,1,3-benzoxadiazole (BTz), 2,1,3-benzoselenadiazole (BSz)) (Figure 2.11).It was shown that for naphthalene-containing polymers the both molecular orbitals,HOMO and LUMO, shift to lower values. The gap between HOMO and LUMOdecreases for around 0.3 eV.

To sum up, the field is now represented with a wide range of experimentalprojects. Therefore, more consistent theoretical calculations should be performedon longer oligomer chains with more accurate methods of computations in orderto understand the true mechanisms of conductivity occurring in the extended π-

2.6. NAPHTHOBISCHALCOGENADIZAOLES 17

(f)(e)(d)

(c)(b)(a)

NN

NN

O

NN

S

NN

Se

NN

S

S

O

NN

N

O

N

Se

NN

N

Se

N

Figure 2.11: Structures of benzochalcogenadiazoles and naphthobischalcoge-nadizaoles: a) 2,1,3-benzothiadiazole (BOz), b) 2,1,3-benzoxadiazole (BTz),c) 2,1,3-benzoselenadiazole (BSz), d) naphtho[1,2-c:5,6-c]bis[1,2,5]oxadiazole(NOz), e) naphtho[1,2-c:5,6-c]bis[1,2,5]thiadiazole (NTz), f) naphtho[1,2-c:5,6-c]bis[1,2,5]selenadiazole (NSz).

conjugated systems.

Chapter 3

Theoretical background

All things are difficult beforethey are easy.

—Dr. Thomas Fuller

To follow this PhD work, it is important to understand the density functionaltheory (DFT), tight binding modelling (TB) and Car-Parrinello molecular dynamics(CPMD). In this chapter, the core of the mentioned methods will be covered briefly.

3.1 Density Functional theoryThe quantum mechanics describes the physics of electrons and nuclei. The fun-damental equation defining the behavior of particles and their interactions is themany-body Schrödinger equation [109], where the Hamiltonian H determines allpossible states |Ψ(r)

⟩corresponding to the given energy E:

H|Ψ(r)⟩

= E|Ψ(r)⟩. (3.1)

Here, the energy E of the state |Ψ(r)⟩consists of kinetic and potential v(r)

energies. Since the many-body Schrödinger equation has complicated solution, anumber of approximations were suggested.

3.1.1 Hartree-Fock method

In 1928, Douglas Hartree assumed that every electron moves in an effective potentialwhich is created by the external potential of the nuclei and average electron-electroninteractions [110]. In this approach, the total Hamiltonian H is represented as asum of the effective one-particle operators h:

h =(− ~2

2me52 +v(r)

), (3.2)

19

20 CHAPTER 3. THEORETICAL BACKGROUND

where ~ is reduced Planck constant and me is a mass of an electron. Hence, ac-cording to Hartree, the many-body wavefunction could be written as a productof the single-particle orbitals, which are solutions to single-particle Schrödingerequations. This approximation ignores the antisymmetric nature of the total wave-function which is demanded by Pauli exclusion principle. However, two years laterVladimir A. Fock proposed to use a Slater determinant as a trial function [111].The so-called Hartree-Fock theory (HF) characterizes exact exchange effects. Anyeffect beyond HF method is called correlation effects.

The many-body Hamiltonian H can be presented as following:

H =− ~2

2

nucl∑j

52Rj

Mj− ~2

2me

elec∑i

52ri−

elec∑i

nucl∑j

e2Zj|ri −Rj |

+

+ 12

elec∑i 6=j

e2

|ri −Rj |+ 1

2

nucl∑i 6=j

e2ZiZj|ri −Rj |

,

(3.3)

where ~ is the reduced Planck constant, Rj is the nuclear coordinate for jth nucleus,ri and Rj are the electronic coordinates for the ith and jth electrons, correspond-ingly, Mj is the mass of jth nucleus, me is the mass of the electron, and Zj isthe nuclear charge. The first two terms characterize kinetic energies for the nu-clei and electrons, respectively. The remaining terms describe the electron-nucleus,electron-electron and nucleus-nucleus interactions.

3.1.2 Born-Oppenheimer approximationAs nuclei are significantly heavier than the electrons, the nuclei can be consideredas fixed and their kinetic energy can be omitted while electrons can be consideredto be moving in the external potential Vext generated by static nuclei. This iscalled Born-Oppenheimer approximation (BO). The Hamiltonian H for it can besimplified to the form of the equation 3.4. Here, the Hartree atomic units are used(~=me=4πε0=e=1):

H = −12

elec∑i

52ri −

elec∑i

nucl∑j

Zj|ri −Rj |

+ 12

elec∑i⊂j

1|ri −Rj |

, (3.4)

or

H = T + Vext + Vee, (3.5)where T describes the kinectic energy of the electrons, Vext refers to the externalpotential, Vee characterizes the electron-electron interactions.

3.1.3 Hohenberg-Kohn theoremsThe simplified H (Equation 3.5) still allows for a significant degree of freedom. Thus,solving the Schrödinger equation remains challenging. Therefore, the Hohenberg-Kohn theorems (HK) were introduced in 1964 [112]:

3.1. DENSITY FUNCTIONAL THEORY 21

Theorem I. For any system of interacting particles in an external potentialVext(r) , the potential Vext(r) is determined uniquely, except for a constant, by theground particle density n0(r).

Theorem II. A universal functional for the energy E[n] in terms of the densityn(r) can be defined, valid for any external potential Vext(r). For any particularVext(r), the exact ground state energy of the system is the global minimum valueof this functional, and the density n(r) that minimizes the functional is the exactground state density n0(r).

These two theorems form the basis of the density functional theory. They statethat the density n(r) consisting of three variables can be used instead of ψ(r). Theenergy functional can be written as

EHK[n] = FHK[n] +∫Vext(r)n(r)dr, (3.6)

where FHK[n] = T [n] + Eint[n] is a universal functional of the electron density,consisting of the kinetic energy T [n] and interacting energy of the particles Eint[n].The second term in equation 3.6 is the interaction energy with the external potential.

Although the HK theorems resulted in solvable equations, the systems to whichthis method could be applied were close to the mathematical model of the non-interacting gas.

3.1.4 The Kohn-Sham ansatzIn search for practical computational method, Kohn and Sham [113] stated thatinteracting many-body system can be replaced with auxiliary non-interacting sys-tems. In essence, Kohn-Sham (KS) ansatz states that the ground state densitiesof such two systems are equal. This results in a number of independent-particleequations for the non-interacting system. Such equations can be solved exactly,with built-in exchange-correlation functional densities.

Applying the KS ansatz, the HK energy functional to the ground state can bereformulated to

EHK[n] = TS[n] +∫Vext(r)n(r)dr + EH[n] + EXC[n]. (3.7)

Here, TS[n] is the independent-particle kinetic energy, Vext(r) is the externalpotential affected by the nuclei, EH[n] is the classical Coulomb interaction energyof the self-interacting electron density n(r). The remaining EXC[n] term representsall many-body effects.

3.1.5 Methods of approximationsSince many-body effects EXC[n] can only be approximated, this sub-chapter intro-duces two well-known methods of approximation.

In local density approximation (LDA), the unknown EXC[n] is substituted with

ELDAXC =∫εXC[n(r)]n(r)dr, (3.8)


where εXC is the exchange-correlation energy for homogeneous electron gas.In generalized gradient approximation (GGA), both density and its gradient are

taken into account:

EGGAXC =∫f [n(r),5n(r)]dr. (3.9)

3.2 Hybrid functionalsHybrid functionals are approximations of the exchange-correlation component ofthe total energy in DFT. The coupling of the Hartree-Fock theory together withlocal density-functional approximations for dynamical correlation was introducedfor the first time by Axel D. Becke back in 1993 [114]:

EXC =∫ 1

0UλXCdλ. (3.10)

Here, λ is an interelectronic coupling-strength parameter. It “switches on” the1/r12 Coulombic repulsion between electrons. The UλXC is the potential energyof the exchange-correlation corresponding to certain λ. When λ=0, the systemis considered as non-interacting, and when λ=1, the system is found to be fully-interacting. The intermediate values of λ mix the non-interacting and interactingsystems in adiabatic way.

3.2.1 B3LYPOne of the most used hybrid functionals is B3LYP [115]:

EB3LY PXC = (1− α0)ELSDAX + α0E

HFX + αX∆EB88

X + αCELY PC + (1− αC)EVWN

C .

(3.11)

Here, the term ELSDAX represents local exchange functional [116, 117] (here,LSDA is referred to the local spin density approximation), ∆EB88

X is Becke’s gradientcorrection to the exchange functional [118], and EVWN

C is the local correlationfunctional of Vosko, Wilk, and Nusair (VWN) [119]. The EVWN

C term supports theELY PC correlation functional [120], since LYP does not have an easily separated localcomponent. Concerning parameters, Becke suggested coefficients α0=0.2, αX=0.72,and αC=0.81.

3.2.2 CAM-B3LYPThe second hybrid functional used in this thesis, CAM-B3LYP [121], combinedB3LYP with the long-range correlation (LC) [122]. Here, CAM refers to “Coulomb-attenuating method”. This functional is based on mixing of short-range and long-range-corrected exchange functional schemes:

1r12

= 1− [α+ β · erf(ωr12)]r12

+ α+ β · erf(ωr12)r12

. (3.12)

3.2. HYBRID FUNCTIONALS 23

In equation 3.12, the first term represents the short-range exchange interactionand the second term describes the long-range orbital-orbital exchange interactionusing HF exchange integral. For CAM-B3LYP, parameters α and β should satisfythe following conditions:

0 =< α+ β =< 1,0 =< α =< 1,0 =< β =< 1.

(3.13)

The parameter α regulates the HF exchange contribution over the entire rangeby a factor of α, while the parameter β describes the contribution of DFT exchangeover the full range by a factor of 1 − (α + β). Exchange contribution of B3LYPfunctional can be obtained as a special case of the exchange component of CAM-B3LYP, when α = 0.2 and β = 0.0 :

EB3X = (1− α)ESlaterX + αEHFX + cB88∆EB88

X , (3.14)

where ESlaterX is Slater exchange, ∆EB88X is Becke’s gradient correction for ex-

change [118], cB88=0.72 is a semi-empirical parameter [114].

3.2.3 LC-ωBE

In this functional’s name [123–125], “LC” stands for to long-range-corrected, ω isan adjustable parameter and PBE is Perdew-Burke-Ernzerhof GGA-method [126].The LC can be obtained from equation 3.12 by setting α = 0.0, β = 1.0 [122]:

1r12

= 1− erf(ωr12)r12

+ erf(ωr12)r12

. (3.15)

When ω=0, then long-range contribution vanishes and the short-range term isrepresented through full Coulomb operator, and vice versa for ω=1 [127]. For thisfunctional ω is chosen to be 0.4 bohr−1 [123]. Hence, the LC-ωPBE can be describedas follows:

ELC−ωPBEX = ESR−DFTX (ω) + ELR−HFX + EDFTC . (3.16)

3.2.4 ω97XD

The last functional used in this thesis, ωB97XD (here “D” refers to dispersion),describes long-range-correlated exchange and atom-atom dispersion correction [128].It builds upon ωB97X functional [129], which is constructed similar to LC-ωPBE.Hence, only the dispersion term ED in ωB97XD is detailed here.

The total energy in the ωB97XD functional reads

EωB97XD = EωB97X + ED, (3.17)


where EωB97XD is ωB97X functional [129], and ED is the empirical atomic-pairwisedispersion correction:

ED = −Nat−1∑i=1

Nat∑j=i+1

Cij6R6ij

fD(Rij). (3.18)

Here, Nat is the number of atoms, Cij6 dispersion coefficient for atom pair ij, R6ij

is an interatomic distance, and fD is a damping function. The last term is necessaryto correct zero dispersion on short distances and asymptotic pairwise van der Waalspotential.

fD(Rij) = 11 + α(Rij/Rr)−12 , (3.19)

where α is a non-linear parameter which controls the strength of dispersion and Rris the sum of van der Waals radii of the pair of atoms at the site i and j.

3.3 Basis setsIn general, a basis set refers to a collection of orthonormal vectors, which definesa solution space for the problem. From quantum-mechanical point of view, basissets are sets of one-particle function used to build molecular orbitals. In otherwords, basis sets describe orbitals mathematically. Larger basis sets impose fewerrestrictions on the electron locations, and are, therefore, usually more accurate.

The basis sets implemented in Gaussian code use linear combinations of atomicorbitals (LCAO) to form the molecular orbitals. The atomic orbitals are representedby atom-centered Gaussian functions:

gijk = Nxiyjzk exp[−αr2], (3.20)

where i, j, k are non-negative integers, N is a normalization constant and α is apositive orbital exponent.

The basis sets used throughout this thesis are:

• 6-31G(d) is a polarized basis set, characterized by a mix of several angularmomenta (here, the d-polarization is added to the split-valence double-zetabasis set 6-31G, for which the valence orbitals are described by two basisfunctions) [130, 131];

• 6-31+G(d) is a basis set with additional diffuse function, which is essentialfor structures with electrons being far from the nuclei [130, 131];

• CEP-31G is a pseudopotential-based basis set, which uses the basic functionsthe pseudospinors with associated pseudopotentials [132–134];

• LanL2DZ is also a pseudopotential-based basis set. Like CEP-31G basis set,it halts the core electron calculations and computes the free atomic states,but it is based on another type of d-functions exponent [135–137].

3.4. TIME-DEPENDENT DFT 25

The Car-Parrinello molecular dynamics uses plane waves, which are more pop-ular and useful for periodic materials, such as polymers. The plane wave methodenrols basis of functions appropriate for the entire system, unlike LCAO-based meth-ods, where the molecular orbitals are described through a set of atomic orbitals. Inthis case, a basis set of an infinite number of plane waves with corresponding coef-ficients cn,kK reads

ψk(r) =∑Kcn,kK exp [(k + K) · r]. (3.21)

To simplify the solution, the maximum value of K in the reciprocal lattice islimited to Kmax. The free-electron energy corresponding to Kmax is called cut-offand can be expressed as

Ecut−off = ~2K2max

2me. (3.22)

In Car-Parrinello molecular dynamics, one more approximation is used togetherwith plane waves basis sets - pseudopotentials. These are effective potentials con-structed to replace the complicated effects of core-electrons motion in close prox-imity to a nucleus. Since all chemical bonding enrol the valence-electrons far awayfrom nucleus, applying pseudopotentials allows to decrease the number of electronsand sizes of basis sets. The valence-electrons are described by pseudowavefunctionwith the same accuracy, but with lower computational costs.

3.4 Time-dependent DFTTime-dependent density functional theory (TD-DFT) studies the many-body sys-tem in the presence of time-dependent potentials, e.g., electric fields. It is basedon an analog of HK theory - the Ronge-Gross (RG) theorem, which was introducedin 1984 [138]. This theorem introduces an additive split of the time-dependentHamiltonian H into a time-independent H0 and a time-dependent V (t) terms:

H = H0 + V (t). (3.23)

Here, the time-dependent term V (t) is a small perturbation of the time-independentHamiltonian H0. Meanwhile, the time-dependent wavefunction ψ(r1, r2, ..., rN ; t)for a given initial state ψ0 is assumed to be equivalent to the time-dependent elec-tronic density n(r, t) [138]. Therefore, the time-dependent Schrödinger equationcan be expressed and solved from ground state:

H0|n(r, t)⟩

= En|n(r, t)⟩, (3.24)

and state of the system |ψ⟩at time t then can be written as

|ψ⟩

=∑n

dn(t) exp[−iEnt/~]|n(r, t)⟩. (3.25)


Here, coefficient dn(t) isolates time-dependent contribution from the perturbationV (t). The system is at its ground state when dn(−∞)=δn0. The perturbation V (t)by electric field F (t) is described by the following equation

V (t) = −µαFα(t), (3.26)

where µα is the dipole moment operator. The excitation energy En0 from groundstate to excited state n is connected to the dipole moment operator through theoscillator function:

fn0 = 2me2~e2En0

∑α=x,y,z

|⟨0|µα|n

⟩|2, (3.27)

where me is electron mass and e is electron charge.In addition, existence of a unique mapping between the time-dependent external

potential vext(r, t) and the time-dependent density n(r, t) is assumed in TD-DFT.The main difference between HK and RG theorem is the absence of the generalminimization principle for the time-dependent case.

TD-DFT is highly useful for computations of small linear perturbations of theinitial ground state of the system, which were caused by external fields. As a result,wide number of physical properties can be investigated, including polarizability,dielectric functions, excitation energies to calculate absorption spectra, etc.

3.5 Tight-binding and the SSH Hamiltonian

The tight-binding (TB) method is a computational scheme for electronic structurecalculations. A common approximation within this scheme is that only nearest-neighbor interaction is taken into account. Usually, the required parameters aredetermined experimentally, resulting in semi-empirical models. An important ad-vantage of TB is low computational cost combined with a relatively good accuracy,especially for carbon-containing systems, such as poly(p-phenylene), studied in thisthesis. An early success of the method was the modelling of band structures ofsemiconductors, using parameters from experiment.

The main idea of the TB method is to express the Hamiltonian using an orthog-onalized linear combination of atomic orbitals ψn(r) centered on the atomic sites ofthe lattice, instead of the total wavefunction Ψ(r):

Ψ(r) =∑n

bnψn(r), (3.28)

where bn are expansion coefficients. When TB is applied to periodic structures, theelectronic state can be expressed in the form of a Bloch wave [139]:

Φ(r) =∑R

exp (ik · r)Ψ(r−R), (3.29)

3.5. TIGHT-BINDING AND THE SSH HAMILTONIAN 27

1 2 3 NN-1 1

2 3

4

N N-1

(a) (b)

Figure 3.1: Tight-binding models for (CH)x(a) and poly(p-phenylene) (b).

where k is the crystal momentum and R is the lattice vector. It is convenient towrite the Hamiltonian H in the general form

H = Hat + ∆V (r), (3.30)

where Hat is the atomic Hamiltonian and ∆V (r) is the potential resulting from theother ions of the lattice. For each atomic orbital ψn(r) the following Schrödingerequation holds:

Hatψn(r) = Enψn(r). (3.31)

Using equations 3.29 and 3.30, the overall Schrödinger equation can be formulated:

HΦ(r) = HatΦ(r) + ∆V (r)Φ(r) = E(k)Φ(r). (3.32)

By multiplying with ψ∗ and integrating over r, the matrix form of the Hamil-tonian with on-site matrix elements and overlap matrix elements is obtained. Theon-site energy is described with the diagonal matrix elements, whereas the overlapmatrix elements characterize the interactions with the nearest neighbors.

This thesis includes tight-binding modelling for poly(p-phenylene) (Figure 3.1b)using the Su-Schrieffer-Heeger (SSH) Hamiltonian [140]. In this model, the focusis on the coupling between the π-electrons that make up the valence band and, inaddition, the ionic motion along the polymeric chain. The SSH model was first usedfor studying solitons in trans-polyacetylene (Figure 2.2a). The model is very usefulfor analyzing charge transport and charge self-localization in π-conjugated carbon-based structures, studied in this project. Figure 3.1a illustrates the (CH)x-model,where each blue circle represents a CH-group. Each carbon atom forms three σ-bonds and one π-bond. In the SSH model, it is assumed that each σ-electron movesadiabatically with its nucleus.

The nearest-neighbor approximation is used, i.e., it is assumed that the hoppingbetween two atomic sites is possible only for neighboring atoms (for example, theelectron can hop from atom 2 in Figure 3.1a to atom 1 and atom 3 only). To beable to address polaron formation and polaron transport in an electric field, theSSH Hamiltonian needs to be generalized, as described below. The generalized SSHHamiltonian has two parts: the electronic term Hel and the lattice term Hlatt.


HSSH = Hel + Hlatt,

Hel = −∑n

(t0 − α(un+1 − un))(eiγAc†ncn+1 + e−iγAc†n+1cn),

Hlatt = K

2∑n

(un+1 − un)2 + 12∑n

Mu2n,

(3.33)

where t0 is the hopping integral between two adjacent atomic sites, α is the electron-phonon coupling constant for the π-electrons, un is the displacement of the n-thCH-group from its equilibrium position, c†n is the creation and cn is the annihilationoperators for a π-electron on the n-th site, K is the elastic constant associated withthe σ-bonds and M is the mass of the CH group. Thus, in equation 3.33, Helcontains both hopping and electron-phonon coupling terms.

With the help of this Hamiltonian, the motion of excitons can be studied, as isshown in Paper IV-V of this thesis.

3.6 Car-Parrinello molecular dynamicsThe ab initio molecular dynamics is based on computations of interatomic forces ata given time. The Born-Oppenheimer molecular dynamics (BOMD) is fully basedon BO approximation, when the nuclei are considered to be fixed. For each time-step in BOMD scheme, the electronic structure with stationary ions is solved, thenthe ionic movement with stationary electronic structure is computed, and, finally,the electronic structure with new ionic positions is recalculated.

Conversely, in the Car-Parrinello molecular dynamics (CPMD) method, the elec-tronic degrees of freedom (DOF) are represented as fictitious dynamical variables.This method is also based on DFT. It was first introduced by Car and Parrinelloback in 1985 [141], and caused a significant growth of ab initio molecular dynamicsdue to its strong advantages over BOMD. The CPMD does not require accurateforce calculations or self-consistently solution for Kohn-Sham Hamiltonian at eachtime step. As a result, the required computational time is reduced.

In the space of the coefficients ci,G, the fictitious dynamics is studied throughfictitious velocities dci,G/dt:

µd2

dt2ci,G = − ∂EKS

∂c∗i,G−∑j

λijci,G, (3.34)

where µ is a fictitious electronic mass, EKS is Kohn-Sham functional and λ is anextended Lagrangian which enforces the orthonormality of the wave function. For afixed nuclear configuration, the quantity EKS is considered as an effective potentialenergy (including both kinetic and potential energies of the quantum-mechanicalelectron gas). It is also supposed that the system is trapped in a local minimum.However, the system can theoretically overcome the surrounding barriers due to the

3.6. CAR-PARRINELLO MOLECULAR DYNAMICS 29

fictitious kinetic energy. In principle, it is possible to reach a global minimum bygradually reducing the fictitious temperature down to zero.

The motion of the ions can be described as

MId2

dt2RI = − ∂E

∂RI, (3.35)

where RI are nuclear coordinates. Forming the system of coupled equations forions (Equation 3.35) and electrons (Equation 3.34) leads to the calculation of theequilibrium nuclei configuration and achieves self-consistency simultaneously. Themolecular dynamics which results from these equations is different from the BOMD.Thus, the ions are affected by the forces different from the BO generated ones. How-ever, if the fictitious electronic kinetic energy stays small, then the ionic trajectoriescan be considered as good approximations to the BO trajectories. To achieve asmall fictitious kinetic energy, the following steps are required:

• the given electronic configuration needs to be in its ground state at the ini-tial time (for that the Kohn-Sham Hamiltonian is required to be solved self-consistently in the first molecular dynamics step);

• the initial fictitious velocity must be small;

• during the simulation, the exchange of energy between the nuclei and thefictitious electronic DOF must be small.

In case all three requirements are fulfilled, then the fictitious kinetic energywill remain small along the trajectory. The fictitious electronic and true ionic mo-tions can be decoupled if the system has an energy gap or the fictitious mass µis significantly small. In the case of metals which do not have an energy gap, theCar-Parrinello method needs to be combined with two thermostats [142] which arecoupled to the ionic and the fictitious electronic DOF, and they are set at differentT . The use of separate thermostats for the electrons allows the electronic systemto stay at desired temperature due to exchange of energies with the ionic system.Thus, the fictitious kinetic energy stays very small. Therefore, two subsystems canbe held at different temperatures while the whole system is kept in metastable state.

Chapter 4

Computational details

MICRO CREDO - NEVERTRUST A COMPUTERBIGGER THAN YOU CANLIFT.

—Gaussian 09 code database

The initial guess of the PEDOT structure was taken from Kim and Brédas ar-ticle [74]. It was modified to longer oligomers using AVOGADRO software [143],which was also used to build the rest of the structures, including oligomers of PE-DOS, PEDOTe, and NXz.

The computational details can be divided into three main sections: (i) calcula-tions using CPMD software package [141, 144], (ii) calculations using Gaussian 09software package [145], and (iii) tight-binding modelling using Python [146].

4.1 CPMD calculations

The oligomers of N=6-12 were calculated using CPMD method. In this methodol-ogy the calculations were performed in the orthorombic box with PBC applied inthe x, y- and z-directions. The distance between the studied oligomers and the wallof the box was scaled to 4Å in all three dimensions [147]. The introduced fictitiouselectrons have an electron mass of 800 a.u.. The equation of motion was integratedwithin ∆t = 0.1 fs timestep. The Troullier-Martins norm-conserving pseudopoten-tials [148] were used in this study together with the BLYP [118, 120] functional.The cut-off of 90Ry was used for the plane-wave basis set.

The ground-state conformation was calculated from the initial planar structurethrough quenching and annealing optimization for 2.5 ps. The second step was toperform an equilibration of the system during 5 ps. At that point, the temperaturewas controlled by rescaling the velocities in order to keep the system within a 40Ktolerance window around the studied temperature (T=200K, 300K and 400K).

31

32 CHAPTER 4. COMPUTATIONAL DETAILS

Oligomers 6-31G(d)/6-31+G(d) CEP-31G LanL2DZPEDOT + + -PEDOS + + -PEDOTe - + +

Table 4.1: The basis sets used for different type of oligomers.

Those values of T were chosen in order to investigate the effect of temperature onthe oligomer properties and compare to experimental results.

The final step was MD run of 50 ps for each target temperature. The statisticsfrom that step were collected and used for the further analysis. Here, temperaturecontrol was implemented for both the atomic and electronic degrees of freedom usingNosé-Hoover thermostats [149, 150] with a characteristic frequency of 10000 cm−1

for the electrons and 480 cm−1 for the atoms.

4.2 Gaussian calculationsThe oligomers of PEDOT and its selenium and tellurium derivatives PEDOS andPEDOTe were computed using Gaussian 09 software package [145]. The lengthsof the studied oligomers were taken from 2 to 20 monomer units. For chargedoligomers, the minimum length was set to 6 monomers in order to allow enoughspace for the polaron. Both flat and twisted conformations were investigated.

First, all oligomers were optimized using four hybrid functionals: B3LYP [114,120], CAM-B3LYP [121], LC-ωPBE [123–125], and ωB97XD [128]. The choice ofbasis sets for each of the oligomer type depended on the heteroatom (S, Se or Te).Table 4.1 is showing which of oligomers were computed under which basis sets.

Those calculations allowed to chose the most suitable functionals+basis set pairfor further investigations. In addition, the effects of twisting and oligomer lengthon (i) total energy and (ii) polaron formation were studied. It was decided toperform the further analysis of optical properties for 15-monomer units oligomerusing ωB97XD functional and CEP-31G basis set. The density of states were plottedfor all calculations in Paper III using GaussSum software package [151].

The calculations of absorption spectra were done using TD-DFT method. Forthe first 10 excited states the energy and oscillator strength were found and demon-strated using gaussian broadening of 0.2 eV.

The HOMO-LUMO gaps were also studied and presented as a function of twist-ing angle for oligomers of EDOT, EDOS and EDOTe using B3LYP functional to-gether with the 6-31+G(d) and the LanL2DZ basis sets. Linear regression givesincorrect result. Therefore, the results were extrapolated up to infinite polymersusing finite size scaling on computed data for the dimers, tetramers and hexam-ers [152]. Computed data was fitted to the following equation:

E = E0

√1 + 2 k

′

k0cos π

N + 1 . (4.1)

4.3. TIGHT-BINDING MODELLING 33

This finite size scaling approach is based on a simple model of the polymer as a chainof identical oscillators, and was first used by W. Kuhn [153] to address polyene.Here, E is the gap and N is the number of double bonds along the shortest pathconnecting the terminal carbon atoms of the molecular backbone. E0 and k′/k0 aretreated as free parameters in the fit. In the model, E0 corresponds to the vibrationenergy of a single oscillator, k0 is the force constant of the isolated oscillator, andk′ is the force constant coupling two adjacent double bonds. Finally, the effectof conformational disorder on the gap size was modeled by performing a weightedaverage of the computed gap with respect to the dihedral angle distribution.

4.3 Tight-binding modellingThe PPP polymer was calculated using tight-binding modelling. The effect of elec-tric field on polaron motion was investigated together with the influence of impu-rities on the polaron break down. Twenty-monomers-long polymer was describedthrough SSH Hamiltonian with periodic boundary conditions applied:

HSSH = Hel +Hlatt +Hnm/mag,

Hel = −∑n,σ

(t0 − α(un+1 − un))(eiγAc†n,σcn+1,σ + e−iγAc†n+1,σcn,σ),

Hlatt = 12∑n

K(un+1 − un)2 + 12∑n

Mu2n,

Hmag =∑i

V magi (c†i↑ci↑ − c

†i↓ci↓), Hnm =

∑i

V nmi (c†iσciσ),

(4.2)

Similar to the general SSH Hamiltonian, the equation above contains the electronicpart Hel and the lattice part Hlatt. The remaining term Hnm/mag describes theimplemented impurities. The model is constructed in such way that both magneticor non-magnetic impurities can be incorporated into the certain position of thestructure. Both the hopping terms and the electron-phonon coupling terms areincluded in Hel. In our model we assumed that the hopping integral t0 is essentialonly for nearest-neighbor carbon sites. The electron-phonon coupling constant forthe π-electrons is expressed through α. The model investigates the displacement ofthe n-th CH-group from its equilibrium position by tracking un term, the creationc†n and annihilation cn operators for a π-electron at the n-th site. The σ-bondsare described by the elastic constant K, while M is assumed as the mass of theCH-group. The strength of impurities is expressed through V mag/nm term.

Since the poly(para-phenylene) structure is very similar to polyacetelyne, weintroduced the set of parameters in the Hamiltonian equal to those for polyacetelynedescription [154]: t0 = 2.5 eV, K = 21 eVÅ−2, α = 4.1 eVÅ−1, a = 1.22Å andM = 1349.14 eVfs2Å−2.

The electric field is implemented to the model using Peierls substitution [154,155] applied to a time-dependent vector potential A. The parameter γ is defined as

34 CHAPTER 4. COMPUTATIONAL DETAILS

γ = ea/~c, where e, a, c are the absolute value of the electronic charge, the latticeconstant, and the speed of light, respectively. Hence, the time derivative of thevector potential determines the electric field:

E = −1c

∂A

∂t. (4.3)

In present research, the electric fields of several values were applied through thegradual increasing of the electric field strength during first 50 fs of simulations.

The initial geometry of the charged polymer was determined in its static statewith a zero electric field. This allows polaron or bipolaron formation. The dis-placement un of the n-th CH-group was defined by equation 4.4, which was self-consistently computed through the ground state total energy minimization withinthe adiabatic approximation under zero electric field.

un = 12(un+1 + un−1) + 2α

K

′∑k

(ψ∗k(n)ψk(n+ 1)− ψ∗k(n− 1)ψk(n)), (4.4)

where ψk(n) is an eigenfunction at site n. Here, the sum is going over all occupiedstates.

The system was studied by solving the time-dependent Schrödinger equation:

i~∂ψ(n, t)∂t

= (−t0 + α(un+1 − un))eiγAψ(n+ 1, t)+

(−t0 + α(un − un−1))e−iγAψ(n− 1, t), (4.5)

and the equation of motion for the lattice displacement:

Mun(t) = Fn(t) = −K(2un − un+1 − un−1)+

α′∑k

eiγA(ψ∗k(n)ψk(n+ 1)− ψ∗k(n− 1)ψk(n)) +H.c. (4.6)

simultaneously. In the equation of motion 4.6 the forces Fn(t) were derived fromthe total potential, which includes the electronic potential and the lattice harmonicpotential. The coupled differential equations were solved numerically using the pro-tocol of Ono and coworkers [154]. The solutions of the time-dependent Schrödingerequation are

ψk(t) = Te−i/~∫ t

0 h(t′)dt′ψk(0), (4.7)

where h(t) is the single particle Hamiltonian and T the time ordering operator. Forthe present work the time step ∆t is chosen to be 0.025 fs, which is relatively smallon the scale of the bare phonon frequency ωQ =

√4K/M of the system. Hence, the

Hamiltonian can be rewritten as following:

ψk(tj+1) = e−ih(tj)∆t/~ψk(tj). (4.8)

4.3. TIGHT-BINDING MODELLING 35

By expanding the electronic wavefunction in terms of the eigenfunction (φl) andeigenvalues (εl) of the single-particle Hamiltonian h(t) at each time step, the wave-function becomes

ψk(n, tj+1) =∑l

[∑p

φ∗l (p)ψk(p, tj)]e−iεl∆t/~φl(n). (4.9)

This set of coupled equations can be numerically integrated using the followingalgorithm:

un(tj+1) = un(tj) + un(tj)∆t, un(tj+1) = un(tj) + Fn(tj)M

∆t , (4.10)

resulting in the pertinent time-dependent electronic wavefunctions.To analyze the motion of the excitations, the polaron and bipolaron positions

and velocities need to be computed. The position of the excitation is defined byconsidering the center of mass xc for the excess charge density ρn, taking the periodicboundary conditions into account. [154] Thus,

xc =

Nθ/2π, 〈cos θn〉 ≥ 0 and 〈sin θn〉 ≥ 0,N(θ + π)/2π, 〈cos θn〉 ≤ 0,N(θ + 2π)/2π, otherwise,

(4.11)

where

〈cos θn〉 =∑n

ρn cos (2πn/N), 〈sin θn〉 =∑n

ρn sin (2πn/N),

θ = arctan( 〈sin θn〉〈cos θn〉

),

and the excess charge density ρn is given by

ρn(t) =′∑k

| ψk(n, t) |2 −1. (4.12)

From the computed xc, the velocity is calculated as an average velocity over 400time steps according to

v(tj) = xc(tj)− xc(tj − 400∆t)400∆t . (4.13)

As mentioned above, the present model does not include the effects of electron-electron correlations.

Chapter 5

Results and discussion

Good enough is great!

The aim of this PhD work was to investigate conductive polymers in generaland understand the electronic, optical and charge transport properties of thosematerials. The results presented in Papers I-VI can be divided into two maingroups:

• structural, electronic and optical investigations of conductive polymers at fi-nite temperature using a DFT-based approach consisting of a combination ofcomputations using the Gaussian 09 code [145] and the CPMD code [144];

• tight-binding studies of polaron formation and breakdown, including the in-fluence of electric field and impurities.

In order to understand the electronic and optical properties of conductive poly-mers, particularly PEDOT and its derivatives, their structures were first investi-gated in quite some detail. For all three cases (PEDOT, PEDOS and PEDOTe) theswitching from aromatic to quinoid conformations (according to Figure 2.7) afterapplying the external charge was observed. This as accomplished by performinga bond-length analysis of the carbon-carbon bonds along the polymer backbone.In particular, a detailed bond length analysis is presented in Paper III, Figures1-3. Here, the calculations were done using four different approximations to theexchange-correlation functional: B3LYP, CAM-B3LYP, LC-ωPBE and ωB97XD.For all three types of studied oligomers, B3LYP was found to only weakly self-localize the polaron. The calculations using CAM-B3LYP functional demonstratedslightly better localization. Not surprisingly however, the LC-ωPBE and ωB97XDfunctionals were found to give a superior description of the polaron formation, witha sharp polaron localization over the center of the oligomer chain obtained. Thereason for this is the way the long-range exchange is included in those functionals.In addition, the aromatic structure was shown for neutral oligomers at ambienttemperature using CPMD calculations (Paper I, Figure 2).

37

38 CHAPTER 5. RESULTS AND DISCUSSION

The polaron localization was further quantified with help of spin density analysis,where the Mulliken spin density per monomer unit was computed and graphed. InPaper III, Figure 4 the polaron localization is clearly demonstrated for the CAM-B3LYP, LC-ωPBE and ωB97XD functionals. In contrast, the B3LYP functionalagain demonstrates only weak localization. Based on these two types of analysis,it is concluded that B3LYP is less preferable for polaron studies in conjugatedpolymers.

The influence of heteroatom (S, Se, Te) and temperature on an angle 2-S-5(according to Figure 2.7 of this thesis) is demonstrated in Paper I. As expected, thevalue of the angle decreases while heteroatoms atomic number increases.

An important part of the structural studies of PEDOT and its derivatives is toinclude the effect of twisting along the axis of two connected monomer units. Firstof all, our CPMD calculations demonstrate how the dihedral angle distributionchanges with temperature (Paper II, Figure 2). In agreement with the calculatedground state structure, finding local minimum of the energy, the twisting betweentwo monomers with a tolerance window 40-50◦ was confirmed for both cis- andtrans-structures (according to Figure 1 in Paper II). Furthermore, the influence ofoligomer length on twisting is demonstrated (Paper III, Supporting information).For neutral oligomers, the energy difference per monomer unit ∆E caused by twist-ing increases with the length of the oligomer chain. However, the difference isminimal for oligomers with a length of 15 monomer units and higher. For radicalcations the ∆E is instead decreasing with increasing oligomer chain length.

The effect of twisting is essential, since the value of the HOMO-LUMO gapdepends on the oligomer conformation and the difference can be up to 2.5 eV. Thisclaim is supported by Figure 5 in Paper II, where in addition the weighted averagesfor the HOMO-LUMO gaps is determined using statistics from a CPMD run over50 ps. This work also shows the influence of temperature on the HOMO-LUMOgap in all three studied cases: for PEDOT the gap did not change appreciably whittemperature, while PEDOS and PEDOTe demonstrated a growth of the gap ofaround 0.2-0.4 eV after increasing T from 200K to 400K. Here, the choice of basisset was also seen to have some impact on the results for the HOMO-LUMO gapcalculations.

The molecular orbital analysis was supported by density of states and absorptionspectra figures in Paper III. The spin contamination analysis presented in this Papershows that among the tested long-range corrected functionals, ωB97XD is found tobe the most suitable one for further investigations of the optical absorption. Thetransition of the charge carrier from HOMO to LUMO and formation of polaronstate is discussed in Paper III in detail. This is valuable for understanding theelectronic properties of future organic electronics devices. Especially, for PEDOTderivatives, since PEDOS is still not widely used in industry and PEDOTe is notsynthesized, as far as the author of this PhD thesis is aware.

For extended π-conjugated structures of NXz polymers our results are shown inPaper VI. Here, the length of the oligomers is taken to be three monomer units, andall results were first compared to the single-monomer unit calculations presented by

39

Osaka [104]. This length of the oligomer is chosen to enable a reasonable inves-tigation of the polaron formation, since for polyacetylene the length of polaron isfound to be around 20 bonds [40]. The polaron localization was investigated usingthe ωB97XD functional, which was identified as the most suitable one in our earlierwork. Figure 2 in Paper VI shows the bond length analysis, where the localizationof distortions is concentrated over the center of the oligomer. The Mulliken spindensities per monomer unit are shown on Figure 4. A small decline of localizationis observed in the middle of the chain. We speculate that this may be caused bythe influence of the extended π-conjugated structure, when the polaron is spreadingaround the naphthene-ring. An interesting observation can be done for our re-sults for the HOMO-LUMO gap. Here, both B3LYP and ωB97XD functionals wereused. The B3LYP is showing values which are closer to the experimental ones, whileωB97XD overestimates the gaps. However, the B3LYP calculations show almost nochange in energy gaps, while for ωB97XD the HOMO-LUMO gap is decreasing for0.3 eV after introducing the charged state.

The visualization of molecular orbitals is shown for all studied polymers in thecorresponding Papers.

The polaron nature and behavior was investigated for poly(p-phenylene) usingtight-binding modelling. Table 1 in Papers IV shows the sensitivity of the polaronand bipolaron transport to the strength of the applied electric field. The main mes-sage of Paper IV is that polarons appear to break down under standard electric fieldstrengths used in organic electronics devices. Paper V deals with a similar analy-sis when magnetic and non-magnetic impurities are incorporated into the polymericstructure. Overall, we find that the mechanism of charge transport based on polaronhopping along the polymer chains can cause significant changes.

Chapter 6

Conclusion

In conclusion, I would like to summarize the work which was performed within lastfive years and demonstrated in a shape of attached manuscripts.

The main aim of this project was to perform a series of theoretical calculationson conductive polymers to contribute to the scientific area of organic electronics.The combination of tight-binding modelling, density functional calculations togetherwith Car-Parinnello molecular dynamics was applied in several types of conduc-tive polymers, including poly(p-phenylene) (PPP), naphthobischalcogenadiazoles(NXz), poly(3,4-ethylenedioxythiophene) (PEDOT) and its selenium (PEDOS) andtellurium (PEDOTe) derivatives.

The choice of those polymers is explained by their unique structural properties.The PPP is presented as a chain of phenylene rings, which could be modelledas polyacetelyne, the basic conductive polymer. This allowed investigation of thepolaron motion in electric field via tight-binding modelling.

PEDOT was chosen due to its high conductivity, air stability, water solubilityand high transparency. Its selenium and tellurium derivatives are attractive sincePEDOT found a wide application in modern industry. The success in PEDOSsynthesis demanded deeper understanding of that material features. PEDOTe wasstudied since its synthesis is considered to be difficult. This thesis demonstratedwhich method would be more preferable for PEDOT-like electronic and opticalinvestigations. The results of band gaps analysis valuable for organic photovoltaicsand organic transistors are calculated for 0K, 200K, 300K and 400K.

NXz materials are characterized with extended π-conjugation. Thus, the lowerband gaps are expected to possess better charge transport properties. This thesisprovides analysis of polaron formation, molecular orbital analysis, densities of statesinvestigation together with absorption spectra.

In addition, the choice of functionals and of basis sets for conductive polymersstudies was investigated in the present thesis

41

Chapter 7

Acknowledgement

During the last five years I have gone long way from Masters in Chemical Engineer-ing to PhD in Physics, and it would be impossible without brilliant people I wassurrounded by. It is hard to express how valuable was their support, but I hopeeach of them knows what I am trying to say in this Chapter.

First of all, I would like to express my sincere gratitude to my supervisorProf. Anna Delin for the continuous support of my PhD study and in all relatedresearch. I am grateful for her patience, motivation, and knowledge she has sharedwith me. For me, Anna became a great example of a scientist. I could not haveimagined having a better supervisor.

My sincere thanks goes to Dr. Håkan W. Hugosson for his co-supervising duringmy first years of studies. Without his guiding it would not be possible to accumulateall that necessary knowledge for computational scientist.

I am highly grateful to Dr. Mathieu Linares for every Skype and face-to-facemeeting, for every scientific discussion we had and critical reviews he expressed. Allthis has made me a better researcher.

I also would like to express my gratitude to Prof. Lars Bergqvist, who was alwaysthere to reply to each of my questions and created such a friendly atmosphere inour group.

To my dear PhD-fellows, Dr. Fan Pan and Dr-to-be Karim Elgammal, I wouldlike to say Xièxiè and Shukraan! Having you both next to me every day duringlast five years was greatest luck and biggest joy! I hope your after-PhD-life will bebright and successful independent from career path you will choose.

I would like to thank all former and present group members: Simone Borlenghi,Mikael Råsander, Long Bui, Johan Hellsvik, Pavel Bessarab, M. Reza Mahani,Michele Visciarelli. It was a pleasure to work with you and learn from each of you.Thank you for all discussions, presentations, lunches and fika we had together.

I am also grateful to my colleagues from ICT and SCI schools: Sergei Popov,Elena Vasiliyeva, Markus Soldemo, Karin Törne, Alexander Forsman, Federico Pe-vere, Babak Taghavi, Anna Fucikova and many other wonderful people in KTHKista Campus.

43

44 CHAPTER 7. ACKNOWLEDGEMENT

I am thankful to Prof. Muhammet Toprak, who was internal reviewer fromKTH. Special gratitude goes to everyone who participated in reviewing of this thesis:Dr. Hubertus Braun, Dr. Danny Thonig, Dr. Andrii Grytsan, Dr. Alexandra Firsova,Mikhail Selivanov, and Sonja Friman.

Finally, I am grateful to my family, my mother Gulnara, my father Khal-makhamed and my brother Iskander, for supporting me at the distance over 5449km. Their constant belief in myself gave me a strength to continue PhD, when Iwas ready to give up.

Amina MirsakiyevaStockholm, 26 Sep 2017

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