Elham Mozafari ATheoreticalStudyofChargeTransportinMolecularCrystals

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Linköping Studies in Science and Technology Dissertation No. 1560

A Theoretical Study of Charge

Transport in Molecular Crystals

Elham Mozafari

LIU-TEK-LIC-2012:45

Department of Physics, Chemistry and Biology Linköping university, SE-581 83 Linköping, Sweden

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c

Elham Mozafari ISBN: 978-91-7519-731-9

ISSN 0280-7971

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A Theoretical Study of Charge Transport in Molecular Crystals

Abstract: The main objective of this thesis is to provide a deeper understand-ing of the charge transport phenomena occurunderstand-ing in molecular crystals. The focus is on the stability and the dynamics of the polaron as the charge carrier.

To achieve this goal, a series of numerical calculations are performed using the semi-emprical "Holstien-Peierls" model. The model considers both intra- (Holstein) and (Peierls) molecular interactions, in particular the electron-phonon inter-actions.

First, the stability of the polaron in an ordered two dimensional molecular lattice with an excess charge is studied using Resilient backPropagation, RPROP, algo-rithm. The stability is defined by the "polaron formation energy". This formation energy is obtained for a wide range of parameter sets including both intra- and inter-molecular electron-phonon coupling strengths and their vibrational frequen-cies, transfer intergral and electric field. We found that the polaron formation energies lying in the range of 50-100meV are more interesting for our studies. The second step to cover is the dynamical behaviour of the polaron. Using the stable polaron solutions acheived in the first step, an electric field is applied as an external force, pushing the charge to move. We observed that the polaron remains stable and moves with a constant velocity for only a limited range of parameter sets. Finally, the impact of disorder and temperature on the charge dynamics is consid-ered. Adding disorder to the system will result in a more restricted parameter set space for which the polaron is dynamically stable and mobile.

Temperature is included in the Newtonian equations of motion via a random force. We observed that the polaron remains localized and moves with a diffusive behaviour up to a certain temperature. If the temperature increases to values above this crit-ical temperature, the localized polaron becomes delocalized.

All this research work is coded in MATLAB software , allowing us to run the cal-culations, test and validate our results.

Keywords: Molecular Crystals, Charge transport, Polaron, Holstein model, Peierls coupling.

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Acknowledgement

Fisrtly and Foremost, I would like to thank my supervisor Prof. Sven Stafström for providing me with all his scientific support and the opportunity to study in this very interesting subject.

I would also like to thank Dr. Magnus Boman for his help in my first steps and for supporting me with his knowledge during this time.

My special thanks goes to Mathieu Linares for his kind attentions, for reading this thesis and providing me with his very useful advices which helped me to improve this work.

I would also like to give my gratitude to all administration team specially Lejla Kronbäck. Without you nothing was possible to go right and easy.

I am thankful to all my kind freinds and colleagues, every one in computational physics group and theoretical physics group. Thanks to Jonas Sjöqvist for bringing lots of sweatness to our days with his candy bar and his wonderful discussion topics during coffee times, and to my lovely friends who made all these years wonderfully enjoyable for me. Jennifer Ullbrand, Nina Shulumba, Parisa Sehati, Lida Khajav-izadeh, Fengi Tai, Peter Steneteg, Lars Johnson, Olle Hellman, Björn Alling, Johan Böhlin and Alexander Lindmaa thank you all.

Dear Hossein, my boyfriend, I would like to give my very special thanks to you for your very precious existance in all my moments in these years, for all your support and for enduring my bitterness specifically during the time of writing this thesis. I want you know how grateful I am for having you and I appreciate every moment of that.

I am definitely indebted to my family for all their emotional support and listening to my nagging phone calls.

Last but not the least, thanks to every one whom I may have forgotten to name but I want you to know that I am grateful for meeting you all.

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Chapter 1 Introduction

Contents

1.1 A brief introduction to charge transport in

molecular crystals . . . 1

1.1.1 Mobility. . . 1

1.1.2 Charge carrier mobility measurement . . . 3

1.1.3 Polaron Concept . . . 4

1.2 Thesis Outline. . . 5

M

olecular crystals have been of great interest during the past fewdecades and the research on these systems have been fueled both by academia and industry[Coropceanu 2007]. From the research point of view, molecular materials exhibit fascinating characteristics such as the interplay between the π-electronic structure and the geometrical structure, have given rise to a developing research field. Their energy gap (the energy difference between the highest occupied molecular orbital(HOMO) and the lowest unoccupied molecular orbital(LUMO)) is also comparable with that of the inorganic semicon-ductors. Due to their low cost, light wieght and also flexibility, these materials can be harnessed in technological applications such as OLEDs[Burroughes 1990], OFETs[Burroughes 1988] and photovoltaic cells[Sariciftci 1992], from the applied point of view.

1.1 A brief introduction to charge transport in

molecular crystals

Organic molecular crystals possess a rich vein of physical characteristics which are to some extent different from their inorganic counterparts. This may arise partly due to the generally weak Van der Waals inter-molecular interactions. An examples of a pentacene molecule and its molecular crystal is demonstrated in Fig.1.1. 1.1.1 _Mobility

Although the operating principles of organic devices were initially largely inspired by the inorganic counterparts, when it comes to electronic structure, their significant differences in the interaction between nuclear and the electronic degrees of freedom

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2 Chapter 1. Introduction

Figure 1.1: (a) Pentacene molecule structure (b) Sketch of the pentacene molecular crystal along the c axis of the herringbone arrangment of the crystal[Parisse 2007]. and also the type of the defect[Podzorov 2007], make this analogy to be of limited use. A typical way to macroscopically distinguish these two classes of materials, is to measure the charge mobility.

When an external electric field is applied to a system, it induces a drift in the charge carriers. The mobility,µ, is then defined as

µ = ν/E (1.1)

which is the ratio between the velocity of the charge,ν and the electric field strength, E. The mobility is usually expressed in cm2_{· V}−1_{· s}−1_.

In crystalline materials, the transport can be described as an adiabatic process in which the charge remains in the same eigenstate during the transport process. However, in systems with a high degree of disorder the electronic states become localized and the charge transport is always thermally activated. In this case, the charge is transported via a nonadiabatic process, i.e., the charge hops from one lo-calized state to another. This process can be described within the standard pertur-bation theory[Ashcroft 1976]. Room temperature charge mobilities of many organic molecules have been measured up to this date but none has exceeded a few tens ofcm2_{· V}−1_{· s}−1_{. For instance, the highest mobility reported for tetracene and}

pentacene are 2.4cm2_·V−1_·s−1_[_{Reese 2006}_{] and 35}_cm2_·V−1_·s−1_[_{Jurchescu 2004}_],

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1.1. A brief introduction to charge transport in molecular

crystals 3

for a FET based on rubrene crystal[Sundar 2004]. These data suggests that there may be an upper limit to the charge mobility achieveable in van der Waals bonded organic molecular crystals.In cases of very ordered systems, the mobility changes with temperature and decreases if the temperature increases. In this case, the electronic states are delocalized and the transport is band-like. At low tempera-tures, mobilities as large as a few hundreds of cm2· V−1_{· s}−1 _{were obeserved in}

the time-of-flight measurements[Karl 1991]. This indicate that the mobility can be described within the Drude theory[Ashcroft 1976]. However, the classical Drude theory is based on weak scattering and do not take the electron-phonon interac-tions into account. This issue is discussed in Chapter.2. High mobility values are associated with the degree of purity in the molecular crystals which is a drawback due to high cost and complexity of fabrication. However, using vaccum sublimation techniques, it is possible to synthesize molecular crystals with a very high degree of purity[Karl 1991,Warta 1985b].

The charge mobility is not affected only by disorder or temperature. There exist several other parameters that would change the mobility. One major factor is the molecular packing parameter[Sancho-García 2010]. The anisotropy which exists in the charge transport of the single crystals shows that the efficiency of the transport is crucially dependent on the positions of the molecules that are interacting with each other which in turn is related to crystal packing. Hence, the mobility can change depending on the direction in which it is measured[Sancho-García 2010]. For instance, the mobility anisotropy for Pentacene single crystal in contact with an electrod array is measured experimentally and the mobility (within the herringbone layer) is found to vary between 2.3 and 0.7 cm2_.V−1_.s−1 _{as a function of polar}

angle[Lee 2006]. Applying an external pressure can also influence the transport process that will reduce the adjacent intermolecular distances and therefore increase the mobility[Chandrasekhar 2001].

In addition to the discussed parameters above, it should also be noted that in the absence of chemical and physical defects, the transport depends on how the electronic and lattice vibrations (phonons) interact.

1.1.2 Charge carrier mobility measurement

The mobility is defined as a measure of the net charge speed per unit of applied electric field, Eq.1.1. This quantity in fact determines how fast a device or a circuit is responding and contributes to how much current they can carry for a given volt-age. Although there exist several methods to measure the mobility. The two most common methods are:

• Time of Flight(TOF)

This method is the most frequently used technique to measure the mobility. This technique is based on irradiating an organic film of a few micron thickness with a laser pulse. The film is sandwiched between two electrodes. Irradiation produces charges at the proximity of one electrode. When an electric field is applied the charges move. The current at the second electrode is then

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4 Chapter 1. Introduction

recorded as a function of time[Haber 1984]. The technique was first used by Kepler[Kepler 1960] and Leblanc[Leblanc 1960].

• Field Effect Transistor (FET) Configuration

In 1998, Horowitz[Horowitz 1998], showed that the current-voltage (I_{− V )} expressions derived for the inorganic transistors can also be applied to organinc transistors (OFET). In this method, the charge carrier mobility is measured in a FET configuration.

Some of these methods measure the mobility in the macroscopic distances of about ∼1 mm. These techniques are often dependent on the degree of the purity and order of the material while other techniques measure the mobility in a microscopic distances which usually is independent or less dependent on these characteristics. For instance, TOF measurements clearly show how the mobility changes due to the structural defects that are present in the material. In FET configuration measure-ments, the contact resistance at metal/organic interfaces is one of the parameters that plays an important role.

1.1.3 Polaron Concept

Injecting (removing) an electron to (from) an unsaturated organic system, will in-duce deformations in the lattice. The mutual interaction between the electron and the deformations in the lattice, result in quasiparticles of electrons surrounded by clouds of phonons. There are also other effects such as charge polarization surround-ing the charge, but since organic materials in general exhibit small polarizabilities, we focus on the lattice deformation here. Such quasiparticles are termed electron-(hole-) "Polaron"s, P− _(P+_{). Polarons can either be spatially extended, "large}

polaron" (Fröhlich polaron[Fröhlich 1954]), or localized in space, "small polaron" (Holstein polaron[Holstein 1959]).

The general concept was first introduced by Landau in 1933 followed by a de-tailed book of Pekar in 1951[Pekar 1963]. Landau and Pekar investigated the self-energy and the effective mass of the polaron subsequently which later on was showed by Fröhlich in 1954[Fröhlich 1954] to accord with adiabatic regime (large polaron). In 1959, Holstein[Holstein 1959] was the first to give a description of small polaron in molecular crystals. Thereafter, an enormous amount of research have been carried out on polaron properties in different systems in various situations.

The polaron concept is of interest not only because it gives a picture of physical properties of charge carriers in polarizable solids but also because it brings in an interesting area of theoretical modelling of systems in which a fermion (electron or hole) is interacting with a scalar bosonic field (phonons). One motive to study the polaron motion in molecular crystals is that it is beleived to play a crucial role in how charge is transported in materials and thus how the organic electronic devices function. In undoped materials, polarons are the predominant excitations responsible for the charge transport process.

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1.2. Thesis Outline 5

1.2 _{Thesis Outline}

In order to be able to harness the organic molecular materials in the technology, a challenging task is to develop adequate theories for modeling these systems and understanding the physical processes occuring during charge transport. Exploring the polaronic transport is of crucial importance since understanding the fundamental aspects of the transport process ultimately determines how a device will operate. In this thesis, one of the developing theories for the charge dynamics, mainly polaron dynamics, in organic molecular crystals is studied theoretically.

The calculations are done according to Holstein-Peierls model in one dimensional and two dimensional molecular crystal systems. In this model, the intra- and inter-molecular lattice vibrations are treated classically and the electronic part is treated quantum mechanically.

In Chapter 2, the basic theoretical knowledge of the physical concepts is sum-merized and the methodolgy is explained shortly but quite throughly. Chapter 3, deals with the modeling of the systems in 1D and 2D and also a description of the code used for nummerical calculations. It includes the methods used for geometry optimization and the dynamics (charge transport). The disorder and the temper-ature effects are also accounted for. Finally, Chapter 4 glances at the particular research topics covered in the supplemented papers.

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Chapter 2 Theory and Methodology

Contents

2.1 Charge Carrier Localization and Delocalization . 7

2.2 Charge Transport Models . . . 8

2.2.1 Delocalized Transport in Simple Electronic Band . 8

2.2.2 Electron Transport in the Polaron Model. . . 10 2.2.3 Weak Electron-Phonon Coupling . . . 10

2.2.4 Hopping Transport for Localized Carriers in Disor-dered Materials. . . 12

2.3 Transport in the Presence of Nonlocal Electron-Phonon Coupling . . . 14

2.3.1 Charge Carrier Dynamics in the Holstein-Peierls Model . . . 16

C

harge transport phenomena in molecular crystals are crucially depen-dent on how ordered the system is and at which temperature the system is operating. Inter- and intra-molecular interactions, in particular the trans-fer integral and the electron-phonon interactions are of great importance. In order to understand the transport phenomena, theoretical research has always been a necessitiy along with experiments. The models in which the charge transport is studied theoretically are either adiabatic band models or nonadiabatic hopping models.

2.1 _{Charge Carrier Localization and}

Delocaliza-tion

In organic molecular crystals (OMC) when an excess charge is added to a lattice, some pecularities would arise due to the nature of the charge carrier.

Generally the charge carrier in the lattice is either delocalized in the form of a Bloch wave or becomes localized as a result of the interaction with electronic or nuclear subsystems of the lattice. The strength of this interaction is characterized by a parameter named "transfer integral",Jmn, between two sitesm and n.

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8 Chapter 2. Theory and Methodology

in whichϕm andϕn are the molecular orbitals of two isolated molecules,m and

n, and Hel is the one electron Hamiltonian of the crystal. The transfer integral

value affects the width of the electronic energy bands in the solid. In broad band materials, the charge carriers are delocalized and move adiabatically as a Bloch wave. The delocalization of a charge carrier in a solid is always accompanied by an energy gain. Accordingly, the localization of the charge demands energy. This energy is called localization energy,Eloc> 0, which is a magnitude with a positive

sign.

In the dynamic process of delocalization, the interaction between the charge carrier and the lattice has to be taken into account. This interaction causes local polarization,Epol< 0, which is competing with the delocalization.

In addition, during the localization process, the charge carrier may form local bonds with a particular molecule or a group of molecules. This results in the forma-tion of a molecular ion or a small-radius molecular polaron (a polaron formed due to the interaction of the charge and the intra-molecular vibrations). This process will also naturally result in an energy gain, the charge bonding energy,Eb< 0.

The fate of the charge carrier in the lattice is then determined by these three factors[Silinsh 1994].

δEloc+ δEpol+ δEb= δE (2.2)

The delocalization occurs whenδEpolandδEb are small andδE > 0. This can

happen if the transfer integral value is large and the bands are broad. The larger the polarization gets, the narrower the bandwidths get, hence eventually results in localization. If the contribution of the polarization comes from the interaction with the lattice, the delocalized state can then be described by the quasiparticle picture, the "polaron" models[Holstein 1959,Eagles 1966,Eagles 1969].

2.2 _{Charge Transport Models}

2.2.1 Delocalized Transport in Simple Electronic Band When the charge carrier interacts weakly with the nuclei of the system or in other words when the local electron-phonon coupling is neglected, the simple band model based is a proper choice. In Bloch theorem description, the state of the system is described by

|Ψk(r)= fN

X

n

exp(ik.rn)|ϕk(r− rn) (2.3)

in which k is the wave vector of the carrier and_{|ϕ(r − r}n) ≡ |ϕn are the basis

functions which for instance can be molecular orbitals centered at site n. The prefactorfN is just a normalization constant.

To achieve the band structure one should be able to introduce a proper Hamil-tonian. Considering tight-binding model, the electronic Hamiltonian is written as

Hel= X n εnc†ncn+ X n,m6=n Jnmc†ncm (2.4)

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2.2. Charge Transport Models 9

c†n andcn are the annihilation and creation operators, respectively. The diagonal

elements of the Hamiltonian,εn =ϕn| Hel| ϕn, are the on-site energies which

are all identical for a periodic structure,ε. The off-diagonal elements, Jnm, are the

transfer integrals, eq.2.1. Note that the Coloumb interactions between the excess charges are neglected in the tight-binding model.The spatial overlap between the electronic states of the adjacent molecules,Sm,n=m±1=< ϕm|ϕn>, has to be also

taken into account.

If the material is well-ordered and periodic, then all the on-site energies and transfer integral values and also the spatial overlap integrals will be identical. The band structure[Ashcroft 1976] can be obtained by solving the Schrödinger equation

Hel| Ψk= Ek| Ψk (2.5)

and multiplying it withΨk| from the left side. Then

Ek=

< Ψk|Hel|Ψk>

< Ψk|Ψk>

= ε + 2J0cos(ka)− 2J00sin(ka)

1 + 2S0_cos(ka)_{− 2S}00_sin(ka) (2.6)

in which the transfer integral value,J = J0+iJ00and the spatial overlapS = S0+iS00 are complex values to maintain generality of the relation 2.6. It has to be noted that 2.6is derived for a 1D system. In the case of 2D or higher dimensions, the transfer integral and spatial overlaps will have more components to be included in the numerator and denominator in the above relation.

The carriers under this circumstance are completely delocalized and charge trans-port can be described by the Boltzman equation[Ashcroft 1976]. In band theory, carriers are scattered during the interplay with impurities and phonons, leading to transitions between Bloch states changing the wave vector from k to k0_{. The}

mobility can then be predicted from Drude theory[Ashcroft 1976].

In summary, there are several parameters needed to be taken into account for determining the nature of the band model. The parameterJnmmust be sufficiently

large in addition to translational symmetry (periodicity) which would provide us with the possibility to harness Bloch wave functions for describing the motion of a delocalized carrier[Silinish 1980].

It should be noted that the band model and the Drude theory hold if the charge carrier does not undergo strong interactions with phonons or scattering due to the impurities in the lattice. This is true since Drude theory is based on the assumption of weak scattering. This implies that the band width of the material should be larger than the change in the energy due to the scattering. The band model, discussed above, in combination with this condition, implies that the mobility has to have a lower limit ofµ > ea2

2~ witha as the lattice constant. Considering the fact that

molecular crystals have a typical intermolecular distance of a = 0.3-0.4 nm, in comibination with the band width condition, yields to the conclusion that the band model, in the case of narrow band materials, can be applied only when the mobility is higher than 1cm2_.V−1_.s−1_[_{Grozema 2008}_].

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2.2.2 Electron Transport in the Polaron Model

The next step forward, is to consider a more general model than the simple band model that includes the effect of the local electron-phonon coupling. This intra-molecular coupling can be devided into two general cases. (i) When the electron-phonon coupling is weak: in this case the spatial extension of the polaron is larger than the lattice spacing (large polaron). This case was first studied by Fröhlich[Fröhlich 1954] and (ii) when the coupling is strong: The subject was inves-tigated in details in the pioneering papers of Holstein[Holstein 1959] in which the self-induce localization caused by an excess charge is of the same order of the lattice constant. In the following section the way to treat the system in these two cases will be discussed. To summarize, one can refer to Fig. 2.1depicting the temperature dependence of the mobility as predictied by the Holstein polaron model for limit-ing cases of weak electron-phonon coupllimit-ing (g2 1) and strong electron-phonon coupling (g2 _{1) where g}2 _{is the coupling strength. In the Holstein paper, the}

electron-phonon coupling constant is denoted byA. In fact A is a parameter which accounts for the energy gain due to the polaron formation. The parametersg and A are linked by the relation[Zoli 2000]

g = dA

2_~

2M ω0

(2.7) d being the system dimension and M the reduced molecular mass and ω0 denotes

the intra-molecular vibrational frequency.

As can be seen, in the case of weak local coupling the mobility decreases with µ_{∼ T}−n, n > 0 indicating a band-like transport.

In the other extreme limit of strong local coupling, the temperature dependence is devided into three regions: (i) at low temperatures,T T1, tunneling transport;

(ii) at intermediate temperatures,T1< T < T2, dominance of the hopping

compo-nent as indicated by the tepmperature-activated behaviour; (iii) as the temperature increases to high values,T > T2, the thermal energy overcomes the polaron energy

resulting in polaron dissociation, hence the residual electron is scattered by thermal phonons and as a result mobility decreases with temperature increase.

2.2.3 Weak Electron-Phonon Coupling

If the electron-polaron coupling is weak, it can be treated as a small perturba-tion, thus the wave function undergoes a slight modification due to the interaction with the phonons. The problem can then be treated using the known perturbation theories such as Rayleigh-Schrödinger perturbation theory, Brillouin-Wigner per-turbation theory[Lowdin 1964] or other various advanced methods[Mahan 2000]. A simple but yet sufficient result achieved using Rayleigh-Schrödinger theory is that the electron-phonon coupling impacts the effective mass of the carrier[Mahan 2000].

¯

m∗= m

∗

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T1 T2

Figure 2.1: Temperature dependence of the mobility for the limiting cases of weak and strong electron-phonon couplings (g2_{) as predicted by Holstein polaron}

model[Coropceanu 2007].

α = ∆E/~ω is the polaron stabilization energy due to lattice deformation. Weak electron-phonon coupling impliesα < 6. Delocalized charges can then be described by a semiclassical model and a renormalized effective mass,m¯∗_{. The semiclassical}

theories were first introduced in 1950s, discussing the idea that the carrier moving in an ionic crystal, carries a polarization cloud (Fröhlich polaron) with itself giving rise to a slight increase in the effective mass. Since the polarity in organic crystals is small, the Frölich polaron theory has not been applied for them, however the theory has recently been revived for describing the transport phenomena in the interface between organic thin films and inorganic polar insulators[Hulea 2006].

2.2.3.1 Strong Electron-Phonon Coupling

Another approximation scheme considers a strong electron-phonon interaction. Un-der this assumption,Velis then considered as a small perturbation, i.e., the molecular

crystal is pictured by a collection of isolated molecules.Velis the electronic coupling

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coupling interaction, ˜Vel = eSVele−S, will then carry the electron-phonon

interac-tion. eS _{is a unitary operator defined as}

S =_−gX n b†_n_{− b}n c†_ncn (2.9)

in whichbn andcnare the bosonic and fermionic components respectively.

If the phonon occupation number does not change during the transport process (inelastic processes are not important), one can then substitute ˜Velwith its thermal

average, DV˜el

E

T, hence the bosonic component can be neglected[Troisi 2010]. It

should be kept in mind that this method can be applied if only the carrier is delo-calized. The following relation can then be derived for the transformed electronic coupling interaction ˜ Vel= Je −2g2 Nω+12 X n c†_ncn+1 (2.10)

withNω= [exp(~ω/kBT )− 1]−1. Comparing2.10withVel=−JPnc†ncn+1, shows

that the effect of the phonons on delocalized carriers is to reduce the effective hop-ping integral by a factor ofe−2g

2 Nω+12

. This temperature dependent reduction factor causes a reduction in band width and also the inverse effective mass. This implies that the effect of phonons is similar to that of a large polaron theory. It is thus reasonable to use a semiclassical approach to describe the charge dynamics, however one must be aware to use the polaronic band model in place of the sim-ple band model[Holstein 1959]. The inverse effective mass decreases with increasing temperature resulting in a noticeable decrease of carrier mobility in polaronic bands.

2.2.4 Hopping Transport for Localized Carriers in Disor-dered Materials

If the material exhibits a static structural disorder, i.e, the arrangments of the molecular units vary from one site to the next implies that the polarization energy and consequently the on-site energy in equation 2.4 to have static disorder. In addition, the orientation of adjacent molecules causes a static disorder in the transfer integral, eq.2.1. For disorder large enough to cause localization the charge carrier then hops between the neighbouring sites while the phonon occupation number of the two sites will change. Generally, two different modes of hopping are distinguished: (i) phonon-assisted hopping without polaronic effects and (ii) polaronic hopping accompanied by a lattice deformation (small polaron)[Grozema 2008]. Although this kind of transport is outside of the scope of the work we have done which in general deals with ordered systems (molecular crystals), it is worthwhile to be explained.

In the case where the polaronic effects are absent, hopping transport process is described in terms of Miller-Abrahams model[Miller 1960]. The Miller-Abrahams

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hopping rate is ususally expressed as

κif = κ0exp(−2αRif)  exp −εf−εi kBT εf> εi 1 εf≤ εi  

κ0 is the attemp hopping frequency, proportional to the square of the magnitude

of the transfer integral. Rif denotes the spacial seperation between the initial and

the final sites;α is called the decay factor taking into account how much the charge transfer integral decays with distance and the whole exponential term accounts for the decrease of the electronic coupling with distance. εi and εj are the site

energies and the second exponential term is just the Boltzman factor for an upward jump in energy which will be equal to 1 for a downward jump. Equation 2.2.4 demonstartes that the hopping rate is determined by the competition in between these two exponential factors as Mott discusses[Mott 1979].

This model was originally developed to describe the charge transport mechanism in doped inorganic materials[Miller 1960] but has recently been applied to organic materials as well.

In the presence of the electron-lattice interaction, the charge induces a defor-mation in the lattice, hence the hopping rate should be calculated from a model described by semic-classical Marcus theory of electron transfer rates[Marcus 1993]. In this case, the charge carrier is assumed to couple to harmonic nuclear vibrations in the lattice. In other words, the precise form of the hopping rate expression in this case is dependent on nuclear vibrational frequencies coupled to the charge carrier. There is a general expression for hopping rate that formulated for different temper-ature regimes[Jortner 1976]. In the limit of high temperatures, kBT ~ωm (ωm

is the intermolecular vibration frequency), Marcus theory defines the hopping rate expression as κif = 2π_|J2 if| ~ s 1 4πλreorgkBT exp " −(εf− εi+ λreorg) 2 4λreorgkBT # (2.11)

The first term in the above equation2.11, 2π|J

2 if|

~ , denotes the electronic tunneling

of the charge carrier between the initial and the final site. Reorganization energy, λreorg, is the energy cost due to geometry modifications to go from a neutral state

to a charged state and vice versa. Note that equation 2.11 first increases with the magnitude of ∆G◦ _{(normal region) for a negatice driving force, and gains its}

maximum when∆G◦_{= ε}

f− εi=−λreorg. In the case where∆G◦<−λreorg, the

hopping rate decreases with∆G◦ decreasing. This is the so-called Marcus inverted region which is totally absent in Miller-Abrahams formalism.

It is worthwhile recalling that both Marcus and Miller-Abrahams theories are the two limiting cases of a more general expression obtained by the time-dependent per-turbation theory with the assumption of a weak electronic coupling. The later one is applied for weak electron-phonon coupling at low temperatures while in contrast the former one is valid for large electron-phonon coupling values at high temperatures.

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14 Chapter 2. Theory and Methodology Delocalized states

+

Electron-phonon coupling, and Reorganization energy Band Transport, Drude theory Marcus theory Hopping transport, Miller-Abrahams theory

+

Localized states Electron-phonon coupling, and Reorganization energy

Figure 2.2: Schematic representation of transport models.

In disordered materials, however, due to the variation of hopping rates as a result of the variation in site energies from one site to the next and also charge transfer integral values, the above general discussion cannot be applied. Therefore, the theroretical study of the mobility in disordered materials is a highly demand-ing task. To overcome this difficulty, the on-site energies are usually considered to exhibit a Gaussian distribution with adjustable width in order to achieve a better agreement between theory and experiment on charge transport. Different distributions can be used, they can either be spatially correlated[Gartstein 1995,

Dunlap 1996] or uncorrelated[Pasveer 2005,Coehoorn 2005]. Different approaches have also been used to study the charge transport in disordered materials based on Miller-Abrahams hopping rates, namely the analytical effective medium approach (EMA)[Fishchuk 2001], the master equation approach[Pasveer 2005], or by Monte Carlo simulations[Hilt 1998,Martin 2003,Kreouzis 2006,Olivier 2006]. There has also been several theoretical studies on polaronic hopping transport in recent years for variety of materials such as polymers[Kreouzis 2006,Athanasopoulos 2007,

Jakobsson 2012] andπ-stack molecular materials[Kirkpatrick 2007].

A schematic represenattion of all transport models discussed in this section is summerized in Fig. 2.2.

2.3 _Transport

_in

_the

_Presence

_of

_Nonlocal

Electron-Phonon Coupling

The transport models discussed up to now lack an important component when ap-plied to organic materials: the nonlocal electron-phonon coupling (Peierls coupling ). This coupling corresponds to the modulation of the hopping integral triggered by phonons.

The interplay between the band theory and the hopping model was first observed in studies of Naphthalene crystals[Schein 1978,Warta 1985a] and the experimental data in low temperature region have been ascribed to Holstein model which causes the bands to narrow[Holstein 1959]. Holstein model considers only local

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electron-2.3. Transport in the Presence of Nonlocal Electron-Phonon

Coupling 15

phonon coupling acting purely on-site of the electronic excitation. Considering non-local couplings, one reaches to models such as Su-Schrieffer-Heeger model[Su 1979] in which the inter-site vibrations are considered and have been investigated by many authors, amongst all most notably by Munn et al.[Munn 1985] and also Zhao et

al.[Zhao 1994]. Hannewald et al.[Hannewald 2004] in their study generalized the

Holstein model by adding nonlocal couplings so that both local (intra-molecular) and nonlocal (inter-molecular) electron-phonon couplings were treated in a close-run.

In their work, Dalla Valle and Girlando[Della Valle 2004], extensively explored the possibility of seperating intra- and inter-molecular vibrations. They performed several Raman spectroscopies on pentacene polymorphs and analyzed their results with computations in order to check the inter-molecular vibrational modes effect on the intra-molecular modes. They showed modes above 200cm−1have a100% intra-molecular characteristics. In this case the inter-intra-molecular coupling can be neglected due to the very high frequency of modes whereas most modes in the intermediate range between 60cm−1_{to 200}_cm−1_{possess a significant mixing of intra- and}

inter-molecular characteristics in an unrecognizable trend.

This term cannot be treated with the techniques and approximations mentioned before. On the one hand, the small polaron picture and the band model are useful as long as the charge is delocalized which is not the case in the presence of thermal disorder that produces localization. On the other hand, if the average intermolecular coupling is stronger than both local and nonlocal couplings, the simple hopping theories are also of no use. However, it is still possible to study the charge carrier dynamics using a simplified model system. The total Hamiltonian which considers both intra- (local) and inter- (nonlocal) molecular interactions for a one dimensional system with N molecules (note that each molecule represents a single "site" in Holstein model) can be expressed as

H = Hel,intra+ Hel,inter+ Hlatt,intra+ Hlatt,inter (2.12)

withHel,intrabeing the diagonal elements ofH (Holstein model plus disorder

con-siderations) defined as Hel,intra= N X n=1 (εn+ Aun)ˆc†nˆcn (2.13)

and the off-diagonal terms (SSH model) Hel,inter=− N X n=1 J0+ α(vn+1− vn) ˆ c†_n+1ˆcn+ ˆc†nˆcn+1 (2.14) un andvn are the intra- and inter-molecular displacements respectively. εn is

on-site energy which is subjected to disorder (for a well-ordered system,εn= 0) and A

denotes the coupling strength between a single internal phonon and the electronic system. J0is the transfer integral value (assumed to be the same for all sites) and

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To consider the role of the lattice (Hlatt) in semi-classical treatment adopted

dur-ing this thesis, the phonon system is devided into two seperate harmonic oscillators, one for intra- and the other for inter-molecular vibrations.

Hlatt,intra= K1 2 N X n=1 u2_n+m 2 N X n=1 ˙ un2 (2.15) Hlatt,inter= K2 2 N X n=1 v2_n+M 2 N X n=1 ˙ vn2 (2.16)

The force constantsK1 andK2 and also the masses m and M refer to the

intra-and inter-molecular oscillators, respectively.

The driving force for the charge carrier to move is supplied via an extrenal electric field which can be introduced in the system by a vector potential defined asΛ(t) =_−cEt[Kuwabara 1991,Ono 1990]. The effect of the field is denoted by a phase factor,exp(iγΛ(t)), included in the inter-molecular transfer integral.

Jn+1,n= (J0+ α(vn+1− vn))e(iγΛ(t)) (2.17)

withγ≡ ea/~c (c is the speed of light).

2.3.1 Charge Carrier Dynamics in the Holstein-Peierls Model

The dynamics of a charge carrier moving in an electric field (but not a magnetic field) in a non-relativistic quantum mechanical regime, is governed by time dependent Schrödinger equation (TDSE).

i~∂Ψ(t)

∂t = ˆHelΨ(t) (2.18)

The focus of interest is to study the dynamical behaviour of the total system which requires solving both the TDSE and the eqautions of motion for the lattice. By classical definition, the force acting on a particle is equal to the negative derivative of the total energy with respect to its position.

M ¨rn=−∇rnEtot (2.19)

The total energy of the system can be expressed as

Etot=hΨ| ˆH|Ψi (2.20)

|Ψi is the total wavefunction consisting of all molecular orbital wavefunctions, ψk.

For a 1D system described via the Hamiltonian2.12, the Newton’s equations of motion for intra-molecular and inter-molecular vibrations are then written as

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2.3. Transport in the Presence of Nonlocal Electron-Phonon

Coupling 17

M ¨vn=−K2(2vn− vn+1− vn−1)− 2αe(iγΛ(t))(ρn,n−1(t)− ρn+1,n(t)) (2.22)

respectively.ρ is the density matrix and its elements in the mean-field approximation are defined as

ρn,m(t) =

X

k

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Chapter 3 Computational Details

Contents

3.1 Model Systems. . . 19

3.1.1 One Dimensional Molecular Chain. . . 19 3.1.2 Two Dimensional Molecular Lattice . . . 20

3.2 Geometry Optimization and Polaron Stability . . . 22 3.3 Polaron Dynamics . . . 24

3.3.1 Dynamics in the Presence of Disorder . . . 27

3.3.2 Temperature Impact on Dynamics . . . 27

T

o this date, the best model to describe polaron motion in molecularcrystals is most likely the Holstein model[Holstein 1959], although it only considers local electron-phonon coupling. In order to develop this model, nonlocal electron-phonon coupling should also be added, Peierls coupling[Munn 1985,Zhao 1994], stating that considering lattice contribution will enhance the hopping behaviour. The model is descibed in detail in the previ-ous chapter. This chapter deals with the calculations that have been done using Holstein-Peierls model Hamiltonian in a one dimensional and a two dimensional system to describe polaron dynamics in the presence of some important factors such as disorder or temperature that affect the transport.

3.1 _{Model Systems}

Holstein model was originally presented for a one dimensional molecular system. Some years later, D. Emin and T. Holstein[Emin 1975] did a study on the role of the dimensionality on the polaron characteristics. This was later also studied by Kalosakas et al [Kalosakas 1998].

3.1.1 One Dimensional Molecular Chain

The considered system for a set of calculations in a one dimensional chain of molecules is represented in Fig.3.1in which the intra-molecular displacements,uis

and the inter-molecular ones, vis are shown. To be more didactic, the molecules

are schematically demonstrated by big circles in which their constituents, atoms, are shown by small circles. The gray colored circles show the system before the

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20 Chapter 3. Computational Details

i-1

i

i+1

Equilibrium Displaced

v

i

u

i0

u

i

Figure 3.1: Schematic representation of a one dimensional Holstein-Peierls system.

arrival of the charge carrier. When the charge arrives at a certain site, it induces a lattice deformation resulting in vibrations of sites. They then adjust themselves to a new position, shown by dashed circles in Fig.3.1. After the carrier is passed, the molecules go back to equilibrium positions of a neutral system (The dashed circles are shifted a bit downward just for the clearity).

3.1.2 Two Dimensional Molecular Lattice

In order to develope the models to a more realistic system, one should perhaps consider a system in higher dimensions than one. Generally, molecular crystals are highly anisotropic. They exhibit a strong in-plane electronic overlap whereas the overlap in the perpendicular direction to these planes is weaker, see Fig.1.1. It sounds then logical to restrict the study to a two dimensional system, Fig. 3.2. In this model, each molecular site is represented by two indices(i, j), i (x-direction) andj (y-direction). The intra- and inter-molecular displacements, ui,js and vi,js

(consisting of two componentsvx

i,jandvyi,j), are also distinguished by two indices,

in this case. The same thing as Fig. 3.1occurs when a charge carrier arrives at a site. Both intra- and inter-molecular distances vibrate back and forth till the charge passes. However, in the dynamics section, Sec.3.3, it will be shown that the lattice deformation and the charge density are coupled, hence one follows the other.

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3.1. Model Systems 21

i

i+1

i-1

j

j+1

j-1

x

y

Figure 3.2: Schematic representation of a two dimensional Holstein-Peierls system.

y components of transfer integral and displacements also into account.

Hel =

X

i,j

(εi,j+ Aui,j)ˆc†i,jˆci,j

+ X

i,j

(J_i+1,j;i,jx cˆ†_i+1,jˆci,j+ H.C.)

+ X

i,j

(J_i,j+1;i,jy cˆ†_i,j+1ˆci,j+ H.C.) (3.1)

with

J_i+1,j;i,jx = J₀x− α(vx

i+1,j− vxi,j)eiγΛx(t) (3.2)

and

J_i,j+1;i,jy = J₀y− α(vi,j+1y − vyi,j)eiγΛy(t) (3.3)

The parameters have the same definition as introduced in Sec.2.3. The lattice dis-tribution in the extended total Hamiltonian, eq.2.12, to include all components in

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both directions will be

Hlatt = K1 2 X i,j (ui,j)2+m 2 X i,j ( ˙ui,j)2 +K2 2 X i,j (vx i+1,j− vi,jx)2 +K2 2 X i,j (vy_i,j+1_{− v}_i,jy )2 +M 2 X i,j [( ˙vx i,j)2+ ( ˙vi,jy )2] (3.4)

With these in mind, it is then possible to solve Newtonian equations of motion m¨ui,j(t) = −K1ui,j(t)− Aρi,j;i.j(t) (3.5)

and

M ¨vx

i,j(t) = −K2(2vi,jx(t)− vi+1,jx (t)− vxi−1,j(t))

− αe−iγΛx(t)_(ρ

i,j;i−1,j(t)− ρi+1,j;i,j)(t))

− αeiγΛx(t)_(ρ

i−1,j;i,j(t)− ρi,j;i+1,j(t)) (3.6)

M ¨v_i,jy (t) = _−K2(2vi,jy (t)− vi,j+1y (t)− vyi,j−1(t))

− α(ρi,j;i,j−1(t)− ρi,j+1;i,j(t)

+ ρ_i,j−1;i,j(t)_{− ρ}i,j;i,j+1(t)) (3.7)

along with TDSE, eq. 2.18, simultaneously to achieve the dynamical behaviour presented in Paper II and discussed shortly in Sec.3.3.

3.2 Geometry Optimization and Polaron Stability

In our calculations, which are mostly concentrating on a two dimensional molec-ular lattice, the initial geometry is always optimized using Resilient backPRO-Pogation algorithm (RPROP) created by Martin Riedmiller and Heinrich Braun in 1992[Riedmiller 1993]. RPROP is an efficient algorithm in which a weight step is directly adapted based on local gradient information and hence the adaptation process is not blindfolded by gradient behaviour. This means in this algorithm in-spite of other techniques, only the sign of the partial derivative is taken into account in order to perform minimization and get the optimum value. In other words, the size of the step used to update the values is defined by the signs rather than the magnitude of the derivatives.

The formation of Polarons, quasi particles formed due to the self-trapping of a quantum particle such as an electron or hole or an exciton via the interaction with molecules or atoms in the lattice is described in the first chapter[Pekar 1946,

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3.2. Geometry Optimization and Polaron Stability 23 0 2 4 6 8 10 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 n x n y

Molecular Charge Density

Figure 3.3: A typical ground state molecular charge density of a two dimensional (10_{× 10) polaron in Holstein-Peierls model achieved by RPROP (n}xandny are the

number of sites inx and y direction, respectively).

Marcus 1956,Emin 1975,Holstein 1959]. The polaron formation energy,Ep, is

ex-pressed as the difference between the energy of the neutral ground state of the system with molecules in their equilibrium geometries at their equilibrim positions in the lattice and the energy of the system in its new relaxed configuration with molecules at their new equilibrium geometries and positions when an excess charge is introduced into the system. The ground state energy of the neutral structure in the model described above is equal to∆J (J is the transfer integral value and ∆ is the dimensionality)[Stafström 2010]. In our calculation the total energy of the charged system is obtained by RPROP,E_p±for an added hole(+) or electron (−). For a 2D lattice

Ep= 2(J0x+ J0y)− E±p (3.8)

For the polaron to be stable,Ephas to be negative (Ep< 0) if J0x,y> 0. It is shown

that in the frame of Holstein model the polaron is always defined as the ground state of the lattice with the additional charge. The solutions in this case can cover a continuous transition from a small polaron (Ep J where the polaron is localized

mostly on a single site) to a large polaron (Ep∼ J where the polaron is extended over

several sites)[Emin 1975,Holstein 1959]. Taking the lattice role (Peierls electron-phonon interaction) also into account will result in a slightly less localized polaron but increases the stability[Mozafari 2012]. One can then conclude that both intra-and inter-molecular electron-phonon couplings, A and α, play important roles in polaron stability in a way that increasingA and α, eq.3.1will enhances the polaron

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formation energy. The choice of parameters will affect the polaron stability as well as its shape which is discussed in detail in Paper I. A typical ground state molecular charge density of a polaron is depicted in Fig.3.3. As can be seen, the charge is centered on a single site. Polaron stabilities which lie in the range of50_{−100meV are} of more ineterst. They are in agreement with the stabilities reported for Pentacene and Rubrene,55meV [Kera 2009] and78meV [Duhm 2012] respectively.

3.3 _{Polaron Dynamics}

In order to do the dynamical simulations and solve the differential equations , eq. 3.5, 3.6, 3.7along with eq. 2.18in MATLAB, an ODE solver is harnessed which is implemented in the software. There exist several classes of ordinary differential equation, ode (ode15, ode23, ode45) solvers that can be used to solve differential equations of different orders numerically. Apart from ode15, the others use the very well-known Runge-Kutta algorithm with varying time step. ode23 uses the second and the third order formulas whereas ode45 takes the forth and the fifth formulas into account. From the accuracy part of view, both ode23 and ode45 are quite equivalently accurate, albeit ode23 needs more time steps though each time step is calculated in a faster speed. We have used ode45 in our calculations.

In Fig.3.4, the results for the dynamics calculation of a one dimensional system consisting of20 sites is demonstrated. The calculations are carried out considering an adiabatic approximation in which the wavefunction of the charge is assumed to be a single eigenstate (the lowest LUMO) of the total Hamiltonian in eq.2.12and the evolution of this eigenstate is studied and depicted in our results. In other words, the charge associated with the polaron remains in the same state during transport process and only this state changes its position with time.

After the electric field is applied, it takes a while (about250f s) for the charge to start moving but then it moves with a constant velocity of about 25 Å/ps. This velocity can be calculated from the molecular charge denisty (panel (a)) by counting the number of the sites that has been travelled during the simulation time. Beside the charge density, the intra-molecular displacements, uis (panel (b)) and

inter-molecular bond lengths,(vi+1− vi)s (panel (c)) are also demonstrated.

The calculation is done with the assumption of periodic boundary conditions which makes the polaron to have a circular motion in a way that it would appear on the first site (i = 1) when reaches to the end of the system (i = 20) which is not shown in Fig.3.4. Note that both intra-molecular displacements and inter-molecular distances are following the localized moving charge which proves that the moving polaron exhibits a Holstein-Peierls polaron nature.

The potential energy according to eq.2.12 is devided into two parts,Hlatt,intra

andHlatt,inter. The former deals with the local vibrations,uis, Fig.3.4(b). This

oscillatory behaviour have a frequency expressed as q

K1 m.

Inspite of intra-molecular vibrations, inter-molecular distances, Fig.3.4(c), form traveling waves in the system which move with approximately the same velocity

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3.3. Polaron Dynamics 25

T1 T2

T3

Figure 3.4: Polaron dynamics in a one dimensional system: (a) molecular charge distribution, (b) intra-molecular displacement,ui, (c) inter-molecular bond length

in thex direction, vx

i+1−vixforA=1.5 eV/Å, K1=10.0 eV/Å2,α=0.5 eV/Å, K2=1.5

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in the opposite direction of the polaron motion. In summary, when the polaron is situated on sitei, the bond length between sites i_{− 1 and i (in this case where the} polaron moves in the_{−x direction) will get contracted (blue color in panel (b) and} (c)) which in turn will result in an expansion in the bond length between sitesi andi + 1 (red color in panel (b) and (c)). It should be noted that the oscillatory behaviour follows the classical spring-mass-spring oscillator with a frequencyqK2

M.

It is important to get a view over the polaron transport phenomena in our system. In Fig.3.4 (a), it can be seen that the charge is localized on two site in the beginning. It starts to move due to the applied force of electric field. One can describe the motion as an adiabatic process in which at each instant of time the charge goes from being centered on a single molecule(T1) to being shared equally

between two neighbouring molecules(T2) and then drifts and gets centered on the

next molecule(T3). More details on this issue and how the transport is affected by

changing different parameters can be found in Paper II in which the more general case of a two dimensional molecular lattice is studied.

The dynamic calculations are not limited to only one set of parameters. We have performed several calculations using different parameters most importantly the intra- and inter-molecular elecron-phonon couplings,A and α, respectively. Fixing the valuesK1andK2and the masses,m and M , the effects of the other parameters

including the electric field strength is studied individually. To summerize, we found that the polaron remains localized and moves with a constant velocity for values of A between 1.2-1.7 eV /Å. These values correspond to the polaron formation energies of 25meV to 44 meV . For every value of intra-molecular coupling below 1.2 eV /Å, the polaron is unstable and delocalizes into a band state due to electric field force and for values above 1.7eV /Å, the polaron is immobile. The same trend is observed for varrying the inter-molecular coupling strength, α. In this case, the polaron formation energies lie in a slightly higher range between 35meV to 64 meV . The range ofα values for which we have a moving polaron is limited to 0.5 eV /Å, up to (and including) 0.6eV /Å. The values of the polaron formation energies show that increasing α will result in a more extended polaron whereas increasing A makes the polaron more localized. finally, by varrying the transfer integral value, J_i,jx,y, we observed that the polaron is stable for the values in the range of 40-80meV corresponding to the formation energies of 78meV at J_i,jx,y= 40 meV and 44 meV atJ_i,jx,y= 80 meV . For values of transfer integral below J_i,jx,y= 40 meV , the polaron is immobile while for values larger thanJ_i,jx,y= 80 meV , it becomes a band state.

When the strength of the external electric field becomes larger, the driving force on the polaron gets larger. This excess energy in the system causes the polaron to destabilize into a band state above a certain critical field strength. Fixing other parameters to our standard parameter set, see paper II, we found that the highest limit of the field strength for which the polaron is dynamically stable is 3.8mV /Å. For higher field strengths the polaron becomes unstable and dessociates into a band state.

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3.3.1 Dynamics in the Presence of Disorder

When the crystal is deviated from its perfect structure, it is called to be disordered. For instance, thermal vibrations can cause disorder as well as introducing impurities into the system either willingly (doping) or unwillingly (defects). The very first model to consider disorder was introduced by P. W. Anderson[Anderson 1958] in which the on-site energies are randomly distributed in a box with the width W with equal probabilities, εi,j∈ [−W/2, W/2]. In his study, Anderson showed once

the disorder exceeds than a critical value of(W/B)crit(B is the bandwidth), the

solutions of the Schrödinger equation are not the Bloch extended states anymore, but become spatially localized so that the charge can transport from one site to the next by just exchanging energy with lattice phonons. However, it should be noted that the transition between extended and localized states has only been observed in three dimensional lattices. In lower dimensions, any non-zero value of disorder will result in a localization. More detailed description of what happens in a two dimensional molecular lattice is provided in Paper II.

3.3.2 _{Temperature Impact on Dynamics}

Depending on the temperature, the transport process can be devided into dif-ferent types such as band transport, tunneling, temperature activated adiabatic transport and also nonadiabatic transport. At low temperatures, mobilities as high as a few hundred cm2_V−1_{s can be obtained via time-of-flight experimental}

measurements[Karl 1991]. A value up to 300cm2_V−1_{s is achieved for hole}

mobil-ity in Naphthalene at T = 10K.[Warta 1985a]. In general, a room temperature mobility in the range of1− 50cm2_V−1_{s is obtainable for -acene family molecular}

crystals[Karl 1991,Jurchescu 2004], ruberen[Podzorov 2003] or perylene[Karl 1999]. This values are indicating a band-like transport for which the mobility decreases with increasing temperature. However, in the systems with localized electronic states in which the charge carriers should overcome a potential in order to pass the energy barriers, the mobility will increase with temperature enhancements. The maximum mobility can be observed in highly ordered molecular materials such as pentacene[Nelson 1998] to lie between temperatures from200K up to room tem-perature. However, when the material is cooled down to a critical temperature, the mobility may drop siginificantly[Podzorov 2004,Zeis 2006,Dunlap 1996] which can be interpreted as a sign of the presence of traps. In other words, for the tem-peratures lower than the range which gives the maximum mobility, the transport process is temperature activated. Above this critical temperature, the role of the traps become less important whereas the lattice phonons become dominant in the transport process which is not the case in our studies. Our system is an ordered molecular crystal.

In our studies, the temperature effect is simulated by adding a thermal ran-dom forces,Rn(t)[Berendsen 1984,Wen 2009,Ribeiro 2011] with zero mean value,

Rn(t)= 0, and the varianceRintrai,j (t)Rintrai0,j0 (t0)

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Rinter

i,j (t)Rinteri0,j0 (t0)

= 2kBT mλδi,j;i0,j0δ(t− t0). These forces are added to the

New-tonian equations of motion, Eq.2.21 and 2.22. In order to keep the temperature constant at its initial value, it is necessary to introduce a damping factor,λ. The lattice dynamics will then be goverened by the following expressions (in a 1D case). m ¨ui=−K1ui− Aρi,i(t)− mλ ˙ui(t) + Riintra(t) (3.9)

and

M ¨vi = −K2(2vi− vi+1− vi−1)− 2αe(iγΛ(t))(ρi,i−1(t)

− ρi+1,i(t))− Mλ ˙vi(t) + Rinteri (t) (3.10)

These equations are no longer ordinary differential equations, they are stochastical differential equations (SDE). It is then important to find a proper integrator for solving SDEs. One way is to use the Langevin dynamics. In our calculations, an integrator called BBK[Brunger 1984,Izaguirre 2001] is used. The method is also called half a kick and the algorithm is explained in the following.

To avoid any complication, from now on the general letter X is used for the intra-molecular,uior the inter-molecular, vi, displacements. Accordingly, ˙X and

¨

X will represent the velocities and accelerations, respectively. To simplify more, the site indices(i) are also removed. M can demonstrate either of the intra- or inter-molecular oscillators’ masses,m or M . The algorithm will then be

half a kick ˙ X+12 = (1−1 2λ∆t) ˙X n₊1 2M −1_∆t(Fn_{+ R}n₎ _(3.11) drift Xn+1= Xn+ ∆t ˙Xn+12 _(3.12) half a kick ˙ Xn+1= ˙ Xn+12 +1 2M−1∆t(Fn+1+ Rn+1) (1 +1₂λ∆t) (3.13) where Rn ₌ q2λkBT

∆t M1/2Zn with Zn being a vector of independent Gaussian

random numbers of zero mean and variance one. n is the counter of the time step ∆t meaning that for every time step t_{→ t + ∆t, n goes to n + 1. Thus n +}1₂ means the time has moved forward for half a time step, t _{→ t +}∆t

2. Fn is the

intra-molecular, eq.3.9or inter-molecular, eq.3.10, equation of motion.

Apart from the lattice, the time evolution of the wave function can be obtained using time dependent Shrödinger equation, eq.2.18. The solution of TDSE at each instant of time can be expressed as[Ono 1990]

ψ(n, t + ∆t) =X l X m φ∗_l(m)ψ(m, t) e −iεl∆t/~ φl(n) (3.14)

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where φl(m) and εl are the instantaneous eigenfunctions and eigenvalues of the

electronic part of the Hamiltonian,Hel, at timet.

Solving equations, eq.3.9,3.10and3.14, simultaneously, we were able to study the dynamics of the polaron in a 1D system. In the following the results for the 1D case is shown and discussed.

(a)

(b)

(c)

i+1−vixforA=1.5 eV/Å, K1=10.0 eV/Å2,α=0.5 eV/Å, K2=1.5

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(a)

(b)

(c)

i+1−vxi forA=1.5 eV/Å, K1=10.0 eV/Å2,α=0.5 eV/Å, K2=1.5

eV/Å2_,_J

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Starting with a zero-temperature polaronic ground state, obtained from RPROP, we performed a series of calculations on a 1D system consisting of 20 sites. The results are obtained for the temperature range from 50K to 300 K for the same parameter set used in Fig.3.4and the electric field is set to zero.

The calculations to trace the behaviour of the polaron are performed for 5ps. Fig. 3.5(panel (a)), demonstrates the molecular charge density at 200 K. It can be seen that the polaron is localized and has a diffusive motion due to the fluc-tuations caused by the temperature. Panels (b) and (c) show the corresponding intra-molecular displacements, ui, and inter-molecular bond lengths, (vi+1− vi),

respectively. It is apparent that the fluctuations are following the charge density. These quantities are quite large but the deformation corresponding to the polaron is still clearly visible.

The total vibrational energy of the lattice is achieved by P_i (m ˙ui2/2) +

(M ˙vi2/2). At thermal equilibrium, this energy is equal to the inner energy of

the lattice,PN kBT /2 (N≡ degrees of freedom). It is obvious that if the

temper-ature increases, the random forces will also increase. The larger the forces become, the higher the amplitude of the lattice vibrations become. Therefore, the total vi-brational energy increases. Fig.3.6, shows the behaviour of the system at 300K. As can be seen from the molecular charge density, panel (a), the initial localized polaron becomes delocalized at this temperature. One can conclude then that there exist a critical temperature for which the localized polaron destabilizes. In our case it should lie between 200K and 300 K. Running more calculations for this range of temperatures, we found that the critical temperature for this system is around 250 K. We also observed that in each calculation above the critical temperature, it will take a while for the localized polaron to evolve into a delocalized state depending on the magnitude of the temperature. As the temperature increases, this time becomes shorter.

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Chapter 4 Comments on Papers

Contents 4.1 Paper One. . . 33 4.1.1 Overview . . . 33 4.1.2 My Contribution . . . 33 4.2 Paper Two . . . 33 4.2.1 Overview . . . 33 4.2.2 My Contribution . . . 34

T

he intorduction given in this thesis, is just a brief theoretical expla-nation of what is presented and published in the following papers. In summary, the subject deals with the charge transport phenomena in the so-called Molecular Crystals. Considering a model Hamiltonian, Holstein-Peierls, the effect of several parameters such as disorder and tempera-ture is studied on the charge dynamics.

4.1 _{Paper One}

4.1.1 _Overview

The stability of polarons in a two dimensional molecular crystal is studied apply-ing the semiclassical Holstein-Peierls model. Calculations are performed usapply-ing this model for a wide range of intra- and inter-molecular parameters in order to obtain a detailed description of polaron formation energies and stabilities in a system with an excess charge but no external force.

4.1.2 My Contribution

I wrote the code getting help from Magnus Boman, performed all the calculations and obtained the results. I also wrote some parts of the paper.

4.2 _{Paper Two}

4.2.1 _Overview

Harnessing the semiclassical Holstein-Peierls hamiltonian, the charge transport is studied in a two dimensional molecular lattice with and without disorder. Both

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34 Chapter 4. Comments on Papers

intra- and inter-molecular electron-phonon couplings are cosidered in the model and the paper describes the dynamics of the charge carrier. In this study only the dynamically stable polaron solutions are considered for the dynamics studies. We found that the parameter space in which the polaron can move adiabatically is quite confined. Increasing the on-site electron-phonon coupling,A, will result in a more localized polaron whereas enhancing the inter-molecular one, α, will reduce this effect and increases the width of the polaron. We observed that for a large value of electron-phonon coupling and a weak inter-molecular electron interaction the polaron is very much localized and immobile whereas for small electron-phonon coupling and a strong inter-molecular electron interaction, is dynamically unstable and dissociates into a band state decoupled form the lattice. Adding disorder to the system will further restrict the parameter space in which the polaron is mobile. 4.2.2 _{My Contribution}

The code is written mostly by me. I also did all the calculations. I wrote some parts of the introduction, the methodology section and took part in writing the discussion section.

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