a theoretical study of charge transport in molecular crystals

Linköping Studies in Science and TechnologyDissertation No. 1560

A Theoretical Study of ChargeTransport in Molecular Crystals

Elham Mozafari

LIU-TEK-LIC-2012:45

Department of Physics, Chemistry and Biology

Linköping university, SE-581 83 Linköping, Sweden

Linköping 2012

c©Elham MozafariISBN: 978-91-7519-731-9

ISSN 0280-7971Printed by LiU-Tryck, Linköping, Sweden, 2012

A Theoretical Study of Charge Transport in MolecularCrystals

Abstract: The main objective of this thesis is to provide a deeper understand-ing of the charge transport phenomena occuring in molecular crystals. The focus ison the stability and the dynamics of the polaron as the charge carrier.To achieve this goal, a series of numerical calculations are performed using thesemi-emprical "Holstien-Peierls" model. The model considers both intra- (Holstein)and inter- (Peierls) molecular interactions, in particular the electron-phonon inter-actions.First, the stability of the polaron in an ordered two dimensional molecular latticewith an excess charge is studied using Resilient backPropagation, RPROP, algo-rithm. The stability is defined by the "polaron formation energy". This formationenergy is obtained for a wide range of parameter sets including both intra- andinter-molecular electron-phonon coupling strengths and their vibrational frequen-cies, transfer intergral and electric field. We found that the polaron formationenergies lying in the range of 50-100 meV are more interesting for our studies.The second step to cover is the dynamical behaviour of the polaron. Using the stablepolaron solutions acheived in the first step, an electric field is applied as an externalforce, pushing the charge to move. We observed that the polaron remains stableand 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 setspace 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 behaviourup 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.

iii

Acknowledgement

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

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

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

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

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

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

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

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

Contents

1 Introduction 11.1 A brief introduction to charge transport in molecular

crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Charge carrier mobility measurement . . . . . . . . . 31.1.3 Polaron Concept . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theory and Methodology 72.1 Charge Carrier Localization and Delocalization . . . . . . 72.2 Charge Transport Models . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Delocalized Transport in Simple Electronic Band . 82.2.2 Electron Transport in the Polaron Model . . . . . . 102.2.3 Weak Electron-Phonon Coupling . . . . . . . . . . . . 102.2.4 Hopping Transport for Localized Carriers in Dis-

ordered Materials . . . . . . . . . . . . . . . . . . . . . . 122.3 Transport in the Presence of Nonlocal Electron-Phonon

Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 Charge Carrier Dynamics in the Holstein-Peierls

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Computational Details 193.1 Model Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 One Dimensional Molecular Chain . . . . . . . . . . . 193.1.2 Two Dimensional Molecular Lattice . . . . . . . . . . 20

3.2 Geometry Optimization and Polaron Stability . . . . . . . . 223.3 Polaron Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 Dynamics in the Presence of Disorder . . . . . . . . . 273.3.2 Temperature Impact on Dynamics . . . . . . . . . . . . 27

4 Comments on Papers 334.1 Paper One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1.2 My Contribution . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Paper Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.2 My Contribution . . . . . . . . . . . . . . . . . . . . . . . 34

Bibliography 35

5 Papers 43

Chapter 1

Introduction

Contents1.1 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

Molecular crystals have been of great interest during the past fewdecades and the research on these systems have been fueled both byacademia and industry[Coropceanu 2007]. From the research point ofview, molecular materials exhibit fascinating characteristics such as the

interplay between the π-electronic structure and the geometrical structure, havegiven rise to a developing research field. Their energy gap (the energy differencebetween the highest occupied molecular orbital(HOMO) and the lowest unoccupiedmolecular orbital(LUMO)) is also comparable with that of the inorganic semicon-ductors. Due to their low cost, light wieght and also flexibility, these materialscan be harnessed in technological applications such as OLEDs[Burroughes 1990],OFETs[Burroughes 1988] and photovoltaic cells[Sariciftci 1992], from the appliedpoint of view.

1.1 A brief introduction to charge transport inmolecular crystals

Organic molecular crystals possess a rich vein of physical characteristics which areto some extent different from their inorganic counterparts. This may arise partlydue to the generally weak Van der Waals inter-molecular interactions. An examplesof 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 inspiredby the inorganic counterparts, when it comes to electronic structure, their significantdifferences in the interaction between nuclear and the electronic degrees of freedom

2 Chapter 1. Introduction

Figure 1.1: (a) Pentacene molecule structure (b) Sketch of the pentacene molecularcrystal 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 limiteduse. A typical way to macroscopically distinguish these two classes of materials, isto measure the charge mobility.

When an external electric field is applied to a system, it induces a drift in thecharge 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 processin which the charge remains in the same eigenstate during the transport process.However, in systems with a high degree of disorder the electronic states becomelocalized and the charge transport is always thermally activated. In this case, thecharge 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 organicmolecules have been measured up to this date but none has exceeded a few tensof cm2 · V −1 · s−1. For instance, the highest mobility reported for tetracene andpentacene are 2.4 cm2 ·V −1 ·s−1[Reese 2006] and 35 cm2 ·V −1 ·s−1[Jurchescu 2004],respectively. Sunder et al. reported a room temperature mobility of 15 cm2·V −1·s−1

1.1. A brief introduction to charge transport in molecularcrystals 3

for a FET based on rubrene crystal[Sundar 2004]. These data suggests that theremay be an upper limit to the charge mobility achieveable in van der Waals bondedorganic molecular crystals.In cases of very ordered systems, the mobility changeswith temperature and decreases if the temperature increases. In this case, theelectronic 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 inthe time-of-flight measurements[Karl 1991]. This indicate that the mobility can bedescribed within the Drude theory[Ashcroft 1976]. However, the classical Drudetheory 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 areassociated with the degree of purity in the molecular crystals which is a drawbackdue to high cost and complexity of fabrication. However, using vaccum sublimationtechniques, it is possible to synthesize molecular crystals with a very high degree ofpurity[Karl 1991, Warta 1985b].

The charge mobility is not affected only by disorder or temperature. There existseveral other parameters that would change the mobility. One major factor is themolecular packing parameter[Sancho-García 2010]. The anisotropy which exists inthe charge transport of the single crystals shows that the efficiency of the transportis crucially dependent on the positions of the molecules that are interacting witheach other which in turn is related to crystal packing. Hence, the mobility canchange depending on the direction in which it is measured[Sancho-García 2010].For instance, the mobility anisotropy for Pentacene single crystal in contact with anelectrod array is measured experimentally and the mobility (within the herringbonelayer) is found to vary between 2.3 and 0.7 cm2.V −1.s−1 as a function of polarangle[Lee 2006]. Applying an external pressure can also influence the transportprocess that will reduce the adjacent intermolecular distances and therefore increasethe mobility[Chandrasekhar 2001].

In addition to the discussed parameters above, it should also be noted that inthe absence of chemical and physical defects, the transport depends on how theelectronic 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 appliedelectric field, Eq.1.1. This quantity in fact determines how fast a device or a circuitis 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 mostcommon 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 thicknesswith a laser pulse. The film is sandwiched between two electrodes. Irradiationproduces charges at the proximity of one electrode. When an electric fieldis applied the charges move. The current at the second electrode is then

4 Chapter 1. Introduction

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

• Field Effect Transistor (FET) ConfigurationIn 1998, Horowitz[Horowitz 1998], showed that the current-voltage (I − V )

expressions derived for the inorganic transistors can also be applied to organinctransistors (OFET). In this method, the charge carrier mobility is measuredin 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 andorder of the material while other techniques measure the mobility in a microscopicdistances which usually is independent or less dependent on these characteristics.For instance, TOF measurements clearly show how the mobility changes due to thestructural defects that are present in the material. In FET configuration measure-ments, the contact resistance at metal/organic interfaces is one of the parametersthat 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 andthe deformations in the lattice, result in quasiparticles of electrons surrounded byclouds 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, "largepolaron" (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 showedby 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 polaronin molecular crystals. Thereafter, an enormous amount of research have been carriedout on polaron properties in different systems in various situations.

The polaron concept is of interest not only because it gives a picture of physicalproperties of charge carriers in polarizable solids but also because it brings in aninteresting area of theoretical modelling of systems in which a fermion (electronor hole) is interacting with a scalar bosonic field (phonons). One motive to studythe polaron motion in molecular crystals is that it is beleived to play a crucialrole in how charge is transported in materials and thus how the organic electronicdevices function. In undoped materials, polarons are the predominant excitationsresponsible for the charge transport process.

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 andunderstanding the physical processes occuring during charge transport. Exploringthe polaronic transport is of crucial importance since understanding the fundamentalaspects of the transport process ultimately determines how a device will operate. Inthis thesis, one of the developing theories for the charge dynamics, mainly polarondynamics, in organic molecular crystals is studied theoretically.

The calculations are done according to Holstein-Peierls model in one dimensionaland two dimensional molecular crystal systems. In this model, the intra- and inter-molecular lattice vibrations are treated classically and the electronic part is treatedquantum 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 thecode used for nummerical calculations. It includes the methods used for geometryoptimization and the dynamics (charge transport). The disorder and the temper-ature effects are also accounted for. Finally, Chapter 4 glances at the particularresearch topics covered in the supplemented papers.

Chapter 2

Theory and Methodology

Contents2.1 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-PeierlsModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Charge transport phenomena in molecular crystals are crucially depen-dent on how ordered the system is and at which temperature the system isoperating. 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 alwaysbeen a necessitiy along with experiments. The models in which the charge transportis studied theoretically are either adiabatic band models or nonadiabatic hoppingmodels.

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 ofa Bloch wave or becomes localized as a result of the interaction with electronic ornuclear subsystems of the lattice. The strength of this interaction is characterizedby a parameter named "transfer integral", Jmn, between two sites m and n.

Jmn =⟨ϕm | Hel | ϕn

⟩(2.1)

8 Chapter 2. Theory and Methodology

in which ϕm and ϕn are the molecular orbitals of two isolated molecules, m andn, and Hel is the one electron Hamiltonian of the crystal. The transfer integralvalue affects the width of the electronic energy bands in the solid. In broad bandmaterials, the charge carriers are delocalized and move adiabatically as a Blochwave. The delocalization of a charge carrier in a solid is always accompanied byan energy gain. Accordingly, the localization of the charge demands energy. Thisenergy is called localization energy, Eloc > 0, which is a magnitude with a positivesign.

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

In addition, during the localization process, the charge carrier may form localbonds 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 dueto the interaction of the charge and the intra-molecular vibrations). This processwill 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 threefactors[Silinsh 1994].

δEloc + δEpol + δEb = δE (2.2)

The delocalization occurs when δEpol and δEb are small and δE > 0. This canhappen if the transfer integral value is large and the bands are broad. The largerthe polarization gets, the narrower the bandwidths get, hence eventually results inlocalization. If the contribution of the polarization comes from the interaction withthe 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 otherwords when the local electron-phonon coupling is neglected, the simple band modelbased is a proper choice. In Bloch theorem description, the state of the system isdescribed by

|Ψk(r)⟩

= fN∑

n

exp(ik.rn)|ϕk(r− rn)⟩

(2.3)

in which k is the wave vector of the carrier and |ϕ(r − rn)⟩≡ |ϕn

⟩are the basis

functions which for instance can be molecular orbitals centered at site n. Theprefactor fN 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 =∑

n

εnc†ncn +

∑

n,m 6=nJnmc

†ncm (2.4)

2.2. Charge Transport Models 9

c†n and cn are the annihilation and creation operators, respectively. The diagonalelements 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 thetransfer integrals, eq.2.1. Note that the Coloumb interactions between the excesscharges are neglected in the tight-binding model.The spatial overlap between theelectronic states of the adjacent molecules, Sm,n=m±1 =< ϕm|ϕn >, has to be alsotaken into account.

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

Hel | Ψk

⟩= Ek | Ψk

⟩(2.5)

and multiplying it with⟨Ψk | from the left side. Then

Ek =< Ψk|Hel|Ψk >

< Ψk|Ψk >=ε+ 2J ′cos(ka)− 2J ′′sin(ka)

1 + 2S′cos(ka)− 2S′′sin(ka)(2.6)

in which the transfer integral value, J = J ′+iJ ′′ and the spatial overlap S = S′+iS′′

are complex values to maintain generality of the relation 2.6. It has to be notedthat 2.6 is derived for a 1D system. In the case of 2D or higher dimensions, thetransfer integral and spatial overlaps will have more components to be included inthe 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, leadingto transitions between Bloch states changing the wave vector from k to k′. Themobility can then be predicted from Drude theory[Ashcroft 1976].

In summary, there are several parameters needed to be taken into account fordetermining the nature of the band model. The parameter Jnm must be sufficientlylarge in addition to translational symmetry (periodicity) which would provide uswith the possibility to harness Bloch wave functions for describing the motion of adelocalized carrier[Silinish 1980].

It should be noted that the band model and the Drude theory hold if the chargecarrier does not undergo strong interactions with phonons or scattering due to theimpurities in the lattice. This is true since Drude theory is based on the assumptionof weak scattering. This implies that the band width of the material should be largerthan the change in the energy due to the scattering. The band model, discussedabove, in combination with this condition, implies that the mobility has to havea lower limit of µ > ea2

2~ with a as the lattice constant. Considering the fact thatmolecular crystals have a typical intermolecular distance of a = 0.3-0.4 nm, incomibination with the band width condition, yields to the conclusion that the bandmodel, in the case of narrow band materials, can be applied only when the mobilityis higher than 1 cm2.V −1.s−1[Grozema 2008].


2.2.2 Electron Transport in the Polaron Model

The next step forward, is to consider a more general model than the simple bandmodel 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 islarger than the lattice spacing (large polaron). This case was first studied byFröhlich[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 theself-induce localization caused by an excess charge is of the same order of the latticeconstant. In the following section the way to treat the system in these two cases willbe discussed. To summarize, one can refer to Fig. 2.1 depicting the temperaturedependence of the mobility as predictied by the Holstein polaron model for limit-ing cases of weak electron-phonon coupling (g2 � 1) and strong electron-phononcoupling (g2 � 1) where g2 is the coupling strength. In the Holstein paper, theelectron-phonon coupling constant is denoted by A. In fact A is a parameter whichaccounts for the energy gain due to the polaron formation. The parameters g andA are linked by the relation[Zoli 2000]

g =dA2~2Mω0

(2.7)

d being the system dimension and M the reduced molecular mass and ω0 denotesthe intra-molecular vibrational frequency.

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

In the other extreme limit of strong local coupling, the temperature dependenceis 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 temperatureincreases to high values, T > T2, the thermal energy overcomes the polaron energyresulting in polaron dissociation, hence the residual electron is scattered by thermalphonons 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 interactionwith the phonons. The problem can then be treated using the known perturbationtheories such as Rayleigh-Schrödinger perturbation theory, Brillouin-Wigner per-turbation theory[Lowdin 1964] or other various advanced methods[Mahan 2000]. Asimple but yet sufficient result achieved using Rayleigh-Schrödinger theory is thatthe electron-phonon coupling impacts the effective mass of the carrier[Mahan 2000].

m∗ =m∗

1− α/6 (2.8)


T1 T2

Figure 2.1: Temperature dependence of the mobility for the limiting cases ofweak and strong electron-phonon couplings (g2) as predicted by Holstein polaronmodel[Coropceanu 2007].

α = ∆E/~ω is the polaron stabilization energy due to lattice deformation. Weakelectron-phonon coupling implies α < 6. Delocalized charges can then be describedby a semiclassical model and a renormalized effective mass, m∗. The semiclassicaltheories were first introduced in 1950s, discussing the idea that the carrier movingin an ionic crystal, carries a polarization cloud (Fröhlich polaron) with itself givingrise to a slight increase in the effective mass. Since the polarity in organic crystals issmall, the Frölich polaron theory has not been applied for them, however the theoryhas recently been revived for describing the transport phenomena in the interfacebetween 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, Vel is then considered as a small perturbation, i.e., the molecularcrystal is pictured by a collection of isolated molecules. Vel is the electronic couplingdefining the interaction between the neighbouring localized states. The transformed


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

tion. eS is a unitary operator defined as

S = −g∑

n

(b†n − bn

)c†ncn (2.9)

in which bn and cn are 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 Vel with its thermalaverage,

⟨Vel

⟩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 electroniccoupling interaction

Vel = Je−2g2

(Nω+ 1

2

)∑

n

c†ncn+1 (2.10)

with Nω = [exp(~ω/kBT )− 1]−1. Comparing 2.10 with Vel = −J∑n c†ncn+1, shows

that the effect of the phonons on delocalized carriers is to reduce the effective hop-

ping integral by a factor of e−2g2

(Nω+ 1

2

). This temperature dependent reduction

factor causes a reduction in band width and also the inverse effective mass. Thisimplies that the effect of phonons is similar to that of a large polaron theory. Itis 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 increasingtemperature 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 themolecular units vary from one site to the next implies that the polarization energyand consequently the on-site energy in equation 2.4 to have static disorder. Inaddition, the orientation of adjacent molecules causes a static disorder in the transferintegral, eq.2.1. For disorder large enough to cause localization the charge carrierthen hops between the neighbouring sites while the phonon occupation number of thetwo sites will change. Generally, two different modes of hopping are distinguished:(i) phonon-assisted hopping without polaronic effects and (ii) polaronic hoppingaccompanied by a lattice deformation (small polaron)[Grozema 2008]. Although thiskind of transport is outside of the scope of the work we have done which in generaldeals with ordered systems (molecular crystals), it is worthwhile to be explained.

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


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 magnitudeof the transfer integral. Rif denotes the spacial seperation between the initial andthe final sites; α is called the decay factor taking into account how much the chargetransfer integral decays with distance and the whole exponential term accountsfor the decrease of the electronic coupling with distance. εi and εj are the siteenergies and the second exponential term is just the Boltzman factor for an upwardjump in energy which will be equal to 1 for a downward jump. Equation 2.2.4demonstartes that the hopping rate is determined by the competition in betweenthese two exponential factors as Mott discusses[Mott 1979].

This model was originally developed to describe the charge transport mechanismin doped inorganic materials[Miller 1960] but has recently been applied to organicmaterials 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 modeldescribed by semic-classical Marcus theory of electron transfer rates[Marcus 1993].In this case, the charge carrier is assumed to couple to harmonic nuclear vibrationsin the lattice. In other words, the precise form of the hopping rate expression in thiscase 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 (ωmis the intermolecular vibration frequency), Marcus theory defines the hopping rateexpression as

κif =2π|J2

if |~

√1

4πλreorgkBTexp

[− (εf − εi + λreorg)

2

4λreorgkBT

](2.11)

The first term in the above equation 2.11,2π|J2

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 stateto a charged state and vice versa. Note that equation 2.11 first increases withthe magnitude of ∆G◦ (normal region) for a negatice driving force, and gains itsmaximum when ∆G◦ = εf − εi = −λreorg. In the case where ∆G◦ < −λreorg, thehopping rate decreases with ∆G◦ decreasing. This is the so-called Marcus invertedregion which is totally absent in Miller-Abrahams formalism.

It is worthwhile recalling that both Marcus and Miller-Abrahams theories are thetwo 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 isapplied for weak electron-phonon coupling at low temperatures while in contrast theformer one is valid for large electron-phonon coupling values at high temperatures.


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 aresult of the variation in site energies from one site to the next and also chargetransfer 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 consideredto exhibit a Gaussian distribution with adjustable width in order to achieve abetter agreement between theory and experiment on charge transport. Differentdistributions can be used, they can either be spatially correlated[Gartstein 1995,Dunlap 1996] or uncorrelated[Pasveer 2005, Coehoorn 2005]. Different approacheshave also been used to study the charge transport in disordered materials based onMiller-Abrahams hopping rates, namely the analytical effective medium approach(EMA)[Fishchuk 2001], the master equation approach[Pasveer 2005], or by MonteCarlo simulations[Hilt 1998, Martin 2003, Kreouzis 2006, Olivier 2006]. There hasalso been several theoretical studies on polaronic hopping transport in recentyears for variety of materials such as polymers[Kreouzis 2006, Athanasopoulos 2007,Jakobsson 2012] and π-stack molecular materials[Kirkpatrick 2007].

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

2.3 Transport in the Presence of NonlocalElectron-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 byphonons.

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

2.3. Transport in the Presence of Nonlocal Electron-PhononCoupling 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] inwhich the inter-site vibrations are considered and have been investigated by manyauthors, amongst all most notably by Munn et al.[Munn 1985] and also Zhao etal.[Zhao 1994]. Hannewald et al.[Hannewald 2004] in their study generalized theHolstein 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 exploredthe possibility of seperating intra- and inter-molecular vibrations. They performedseveral Raman spectroscopies on pentacene polymorphs and analyzed their resultswith computations in order to check the inter-molecular vibrational modes effect onthe intra-molecular modes. They showed modes above 200 cm−1 have a 100% intra-molecular characteristics. In this case the inter-molecular coupling can be neglecteddue to the very high frequency of modes whereas most modes in the intermediaterange between 60 cm−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 mentionedbefore. On the one hand, the small polaron picture and the band model are usefulas long as the charge is delocalized which is not the case in the presence of thermaldisorder that produces localization. On the other hand, if the average intermolecularcoupling is stronger than both local and nonlocal couplings, the simple hoppingtheories are also of no use. However, it is still possible to study the charge carrierdynamics using a simplified model system. The total Hamiltonian which considersboth intra- (local) and inter- (nonlocal) molecular interactions for a one dimensionalsystem with N molecules (note that each molecule represents a single "site" inHolstein model) can be expressed as

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

with Hel,intra being the diagonal elements of H (Holstein model plus disorder con-siderations) defined as

Hel,intra =

N∑

n=1

(εn +Aun)c†ncn (2.13)

and the off-diagonal terms (SSH model)

Hel,inter = −N∑

n=1

(J0 + α(vn+1 − vn)

)(c†n+1cn + c†ncn+1

)(2.14)

un and vn 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 Adenotes the coupling strength between a single internal phonon and the electronicsystem. J0 is the transfer integral value (assumed to be the same for all sites) andα is the inter-molecular electron-phonon coupling.


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∑

n=1

u2n +

m

2

N∑

n=1

un2 (2.15)

Hlatt,inter =K2

2

N∑

n=1

v2n +

M

2

N∑

n=1

vn2 (2.16)

The force constants K1 and K2 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 extrenalelectric field which can be introduced in the system by a vector potential definedas Λ(t) = −cEt[Kuwabara 1991, Ono 1990]. The effect of the field is denoted by aphase factor, exp(iγΛ(t)), included in the inter-molecular transfer integral.

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

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

2.3.1 Charge Carrier Dynamics in the Holstein-PeierlsModel

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 dependentSchrödinger equation (TDSE).

i~∂Ψ(t)

∂t= HelΨ(t) (2.18)

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

Mrn = −∇rnEtot (2.19)

The total energy of the system can be expressed as

Etot = 〈Ψ|H|Ψ〉 (2.20)

|Ψ〉 is the total wavefunction consisting of all molecular orbital wavefunctions, ψk.For a 1D system described via the Hamiltonian 2.12, the Newton’s equations of

motion for intra-molecular and inter-molecular vibrations are then written as

mun = −K1un −Aρn,n(t) (2.21)

2.3. Transport in the Presence of Nonlocal Electron-PhononCoupling 17

Mvn = −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 approximationare defined as

ρn,m(t) =∑

k

Ψnk(t)Ψ∗mk(t) (2.23)

Chapter 3

Computational Details

Contents3.1 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

To this date, the best model to describe polaron motion in molecularcrystals is most likely the Holstein model[Holstein 1959], although itonly considers local electron-phonon coupling. In order to develop thismodel, nonlocal electron-phonon coupling should also be added, Peierls

coupling[Munn 1985, Zhao 1994], stating that considering lattice contribution willenhance the hopping behaviour. The model is descibed in detail in the previ-ous chapter. This chapter deals with the calculations that have been done usingHolstein-Peierls model Hamiltonian in a one dimensional and a two dimensionalsystem to describe polaron dynamics in the presence of some important factors suchas 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 ofthe dimensionality on the polaron characteristics. This was later also studied byKalosakas et al [Kalosakas 1998].

3.1.1 One Dimensional Molecular Chain

The considered system for a set of calculations in a one dimensional chain ofmolecules is represented in Fig.3.1 in which the intra-molecular displacements, uisand the inter-molecular ones, vis are shown. To be more didactic, the moleculesare schematically demonstrated by big circles in which their constituents, atoms,are shown by small circles. The gray colored circles show the system before the

20 Chapter 3. Computational Details

i-1 i i+1

Equilibrium

Displaced

vi

ui0

ui

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 alattice deformation resulting in vibrations of sites. They then adjust themselves toa new position, shown by dashed circles in Fig.3.1. After the carrier is passed, themolecules go back to equilibrium positions of a neutral system (The dashed circlesare 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 perhapsconsider a system in higher dimensions than one. Generally, molecular crystals arehighly anisotropic. They exhibit a strong in-plane electronic overlap whereas theoverlap in the perpendicular direction to these planes is weaker, see Fig.1.1. Itsounds then logical to restrict the study to a two dimensional system, Fig. 3.2. Inthis model, each molecular site is represented by two indices (i, j), i (x-direction)and j (y-direction). The intra- and inter-molecular displacements, ui,js and vi,js(consisting of two components vxi,j and vyi,j), are also distinguished by two indices,in this case. The same thing as Fig. 3.1 occurs when a charge carrier arrives at asite. Both intra- and inter-molecular distances vibrate back and forth till the chargepasses. However, in the dynamics section, Sec. 3.3, it will be shown that the latticedeformation and the charge density are coupled, hence one follows the other.

The electronic Hamiltonian will in this case contain some extra terms taking the

3.1. Model Systems 21

i i+1i-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 =∑

i,j

(εi,j +Aui,j)c†i,j ci,j

+∑

i,j

(Jxi+1,j;i,j c†i+1,j ci,j +H.C.)

+∑

i,j

(Jyi,j+1;i,j c†i,j+1ci,j +H.C.) (3.1)

with

Jxi+1,j;i,j = Jx0 − α(vxi+1,j − vxi,j)eiγΛx(t) (3.2)

and

Jyi,j+1;i,j = Jy0 − α(vyi,j+1 − vyi,j)e

iγΛy(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


both directions will be

Hlatt =K1

2

∑

i,j

(ui,j)2 +

m

2

∑

i,j

(ui,j)2

+K2

2

∑

i,j

(vxi+1,j − vxi,j)2

+K2

2

∑

i,j

(vyi,j+1 − vyi,j)

2

+M

2

∑

i,j

[(vxi,j)2 + (vyi,j)

2] (3.4)

With these in mind, it is then possible to solve Newtonian equations of motion

mui,j(t) = −K1ui,j(t)−Aρi,j;i.j(t) (3.5)

and

Mvxi,j(t) = −K2(2vxi,j(t)− vxi+1,j(t)− vxi−1,j(t))

− αe−iγΛx(t)(ρi,j;i−1,j(t)− ρi+1,j;i,j)(t))

− αeiγΛx(t)(ρi−1,j;i,j(t)− ρi,j;i+1,j(t)) (3.6)

Mvyi,j(t) = −K2(2vyi,j(t)− vyi,j+1(t)− vyi,j−1(t))

− α(ρi,j;i,j−1(t)− ρi,j+1;i,j(t)

+ ρi,j−1;i,j(t)− ρi,j;i,j+1(t)) (3.7)

along with TDSE, eq. 2.18, simultaneously to achieve the dynamical behaviourpresented 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 Braunin 1992[Riedmiller 1993]. RPROP is an efficient algorithm in which a weight stepis directly adapted based on local gradient information and hence the adaptationprocess 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 accountin order to perform minimization and get the optimum value. In other words, thesize of the step used to update the values is defined by the signs rather than themagnitude of the derivatives.

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

3.2. Geometry Optimization and Polaron Stability 23

02

46

810

0

2

4

6

8

100

0.1

0.2

0.3

0.4

0.5

nx

ny

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 (nx and ny are thenumber of sites in x 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 thesystem with molecules in their equilibrium geometries at their equilibrim positionsin the lattice and the energy of the system in its new relaxed configuration withmolecules at their new equilibrium geometries and positions when an excess chargeis introduced into the system. The ground state energy of the neutral structure inthe 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 thecharged system is obtained by RPROP, E±p for an added hole (+) or electron (−).For a 2D lattice

Ep = 2(Jx0 + Jy0 )− E±p (3.8)

For the polaron to be stable, Ep has to be negative (Ep < 0) if Jx,y0 > 0. It is shownthat in the frame of Holstein model the polaron is always defined as the groundstate of the lattice with the additional charge. The solutions in this case can cover acontinuous transition from a small polaron (Ep � J where the polaron is localizedmostly on a single site) to a large polaron (Ep ∼ J where the polaron is extended overseveral sites)[Emin 1975, Holstein 1959]. Taking the lattice role (Peierls electron-phonon interaction) also into account will result in a slightly less localized polaronbut increases the stability[Mozafari 2012]. One can then conclude that both intra-and inter-molecular electron-phonon couplings, A and α, play important roles inpolaron stability in a way that increasing A and α, eq.3.1 will enhances the polaron


formation energy. The choice of parameters will affect the polaron stability as wellas its shape which is discussed in detail in Paper I. A typical ground state molecularcharge density of a polaron is depicted in Fig.3.3. As can be seen, the charge iscentered on a single site. Polaron stabilities which lie in the range of 50−100meV areof more ineterst. They are in agreement with the stabilities reported for Pentaceneand Rubrene, 55meV [Kera 2009] and 78meV [Duhm 2012] respectively.

3.3 Polaron Dynamics

In order to do the dynamical simulations and solve the differential equations , eq.3.5, 3.6, 3.7 along with eq. 2.18 in MATLAB, an ODE solver is harnessed whichis implemented in the software. There exist several classes of ordinary differentialequation, ode (ode15, ode23, ode45) solvers that can be used to solve differentialequations of different orders numerically. Apart from ode15, the others use the verywell-known Runge-Kutta algorithm with varying time step. ode23 uses the secondand the third order formulas whereas ode45 takes the forth and the fifth formulasinto account. From the accuracy part of view, both ode23 and ode45 are quiteequivalently accurate, albeit ode23 needs more time steps though each time step iscalculated 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 systemconsisting of 20 sites is demonstrated. The calculations are carried out consideringan adiabatic approximation in which the wavefunction of the charge is assumed tobe a single eigenstate (the lowest LUMO) of the total Hamiltonian in eq.2.12 and theevolution 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 transportprocess and only this state changes its position with time.

After the electric field is applied, it takes a while (about 250fs) for the chargeto start moving but then it moves with a constant velocity of about 25 Å/ps. Thisvelocity can be calculated from the molecular charge denisty (panel (a)) by countingthe number of the sites that has been travelled during the simulation time. Besidethe 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 conditionswhich makes the polaron to have a circular motion in a way that it would appearon the first site (i = 1) when reaches to the end of the system (i = 20) which is notshown in Fig.3.4. Note that both intra-molecular displacements and inter-moleculardistances are following the localized moving charge which proves that the movingpolaron exhibits a Holstein-Peierls polaron nature.

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

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

oscillatory behaviour have a frequency expressed as√

K1m .

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

3.3. Polaron Dynamics 25

T1

T2

T3

Figure 3.4: Polaron dynamics in a one dimensional system: (a) molecular chargedistribution, (b) intra-molecular displacement, ui, (c) inter-molecular bond lengthin the x direction, vxi+1−vxi for A=1.5 eV/Å, K1=10.0 eV/Å2, α=0.5 eV/Å, K2=1.5eV/Å2, J0=0.05 eV and E0x=2.0 mV/Å.


in the opposite direction of the polaron motion. In summary, when the polaron issituated on site i, the bond length between sites i− 1 and i (in this case where thepolaron 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 sites iand i + 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 frequency√

K2M .

It is important to get a view over the polaron transport phenomena in oursystem. In Fig.3.4 (a), it can be seen that the charge is localized on two site inthe beginning. It starts to move due to the applied force of electric field. One candescribe the motion as an adiabatic process in which at each instant of time thecharge goes from being centered on a single molecule (T1) to being shared equallybetween two neighbouring molecules (T2) and then drifts and gets centered on thenext molecule (T3). More details on this issue and how the transport is affected bychanging different parameters can be found in Paper II in which the more generalcase of a two dimensional molecular lattice is studied.

The dynamic calculations are not limited to only one set of parameters. Wehave performed several calculations using different parameters most importantly theintra- and inter-molecular elecron-phonon couplings, A and α, respectively. Fixingthe values K1 and K2 and the masses, m andM , the effects of the other parametersincluding the electric field strength is studied individually. To summerize, we foundthat the polaron remains localized and moves with a constant velocity for values ofA between 1.2-1.7 eV/Å. These values correspond to the polaron formation energiesof 25 meV 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 forceand for values above 1.7 eV/Å, the polaron is immobile. The same trend is observedfor varrying the inter-molecular coupling strength, α. In this case, the polaronformation energies lie in a slightly higher range between 35 meV to 64 meV . Therange of α values for which we have a moving polaron is limited to 0.5 eV/Å, up to(and including) 0.6 eV/Å. The values of the polaron formation energies show thatincreasing α will result in a more extended polaron whereas increasing A makesthe polaron more localized. finally, by varrying the transfer integral value, Jx,yi,j ,we observed that the polaron is stable for the values in the range of 40-80 meVcorresponding to the formation energies of 78 meV at Jx,yi,j = 40 meV and 44 meVat Jx,yi,j = 80 meV . For values of transfer integral below Jx,yi,j = 40 meV , the polaronis immobile while for values larger than Jx,yi,j = 80 meV , it becomes a band state.

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


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 impuritiesinto the system either willingly (doping) or unwillingly (defects). The very firstmodel to consider disorder was introduced by P. W. Anderson[Anderson 1958] inwhich 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 oncethe disorder exceeds than a critical value of (W/B)crit (B is the bandwidth), thesolutions 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 thenext by just exchanging energy with lattice phonons. However, it should be notedthat the transition between extended and localized states has only been observedin three dimensional lattices. In lower dimensions, any non-zero value of disorderwill result in a localization. More detailed description of what happens in a twodimensional 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 adiabatictransport and also nonadiabatic transport. At low temperatures, mobilities ashigh as a few hundred cm2V −1s can be obtained via time-of-flight experimentalmeasurements[Karl 1991]. A value up to 300cm2V −1s is achieved for hole mobil-ity in Naphthalene at T = 10K.[Warta 1985a]. In general, a room temperaturemobility in the range of 1 − 50cm2V −1s is obtainable for -acene family molecularcrystals[Karl 1991, Jurchescu 2004], ruberen[Podzorov 2003] or perylene[Karl 1999].This values are indicating a band-like transport for which the mobility decreaseswith increasing temperature. However, in the systems with localized electronicstates in which the charge carriers should overcome a potential in order to pass theenergy barriers, the mobility will increase with temperature enhancements. Themaximum mobility can be observed in highly ordered molecular materials such aspentacene[Nelson 1998] to lie between temperatures from 200K 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] whichcan 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 transportprocess is temperature activated. Above this critical temperature, the role of thetraps become less important whereas the lattice phonons become dominant in thetransport process which is not the case in our studies. Our system is an orderedmolecular 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 variance

⟨Rintrai,j (t)Rintrai′,j′ (t′)

⟩= 2kBTmλδi,j;i′,j′δ(t − t′) and


⟨Rinteri,j (t)Rinteri′,j′ (t′)

⟩= 2kBTmλδi,j;i′,j′δ(t− t′). These forces are added to the New-

tonian equations of motion, Eq.2.21 and 2.22. In order to keep the temperatureconstant at its initial value, it is necessary to introduce a damping factor, λ. Thelattice dynamics will then be goverened by the following expressions (in a 1D case).

mui = −K1ui −Aρi,i(t)−mλui(t) +Rintrai (t) (3.9)

and

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

− ρi+1,i(t))−Mλvi(t) +Rinteri (t) (3.10)

These equations are no longer ordinary differential equations, they are stochasticaldifferential equations (SDE). It is then important to find a proper integrator forsolving SDEs. One way is to use the Langevin dynamics. In our calculations, anintegrator called BBK[Brunger 1984, Izaguirre 2001] is used. The method is alsocalled 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 theintra-molecular, ui or the inter-molecular, vi, displacements. Accordingly, X andX will represent the velocities and accelerations, respectively. To simplify more,the site indices (i) are also removed. M can demonstrate either of the intra- orinter-molecular oscillators’ masses, m or M . The algorithm will then behalf a kick

X+ 12 = (1− 1

2λ∆t)Xn +

1

2M−1∆t(Fn +Rn) (3.11)

driftXn+1 = Xn + ∆tXn+ 1

2 (3.12)

half a kick

Xn+1 =Xn+ 1

2 + 12M

−1∆t(Fn+1 +Rn+1)

(1 + 12λ∆t)

(3.13)

where Rn =√

2λkBT∆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 + 1

2

means the time has moved forward for half a time step, t → t + ∆t2 . Fn is the

intra-molecular, eq.3.9 or 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 eachinstant of time can be expressed as[Ono 1990]

ψ(n, t+ ∆t) =∑

l

[∑

m

φ∗l (m)ψ(m, t)

]e

(−iεl∆t/~

)φl(n) (3.14)


where φl(m) and εl are the instantaneous eigenfunctions and eigenvalues of theelectronic part of the Hamiltonian, Hel, at time t.

Solving equations, eq.3.9, 3.10 and 3.14, simultaneously, we were able to studythe dynamics of the polaron in a 1D system. In the following the results for the 1Dcase is shown and discussed.

(a)

(b)

(c)

Figure 3.5: Polaron dynamics in a one dimensional system: (a) molecular chargedistribution, (b) intra-molecular displacement, ui, (c) inter-molecular bond lengthin the x direction, vxi+1−vxi for A=1.5 eV/Å, K1=10.0 eV/Å2, α=0.5 eV/Å, K2=1.5eV/Å2, J0=0.05 eV, λ=105 eVas/Å, and E0x=0.0 mV/Å, at T = 200 K.


(a)

(b)

(c)

Figure 3.6: Polaron dynamics in a one dimensional system: (a) molecular chargedistribution, (b) intra-molecular displacement, ui, (c) inter-molecular bond lengthin the x direction, vxi+1−vxi for A=1.5 eV/Å, K1=10.0 eV/Å2, α=0.5 eV/Å, K2=1.5eV/Å2, J0=0.05 eV, λ=105 eVas/Å, and E0x=0.0 mV/Å, at T = 300 K.


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. Theresults are obtained for the temperature range from 50 K to 300 K for the sameparameter set used in Fig.3.4 and the electric field is set to zero.

The calculations to trace the behaviour of the polaron are performed for 5 ps.Fig. 3.5 (panel (a)), demonstrates the molecular charge density at 200 K. It canbe 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 correspondingintra-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 polaronis still clearly visible.

The total vibrational energy of the lattice is achieved by∑

i

((mui

2/2) +

(Mvi2/2)

). At thermal equilibrium, this energy is equal to the inner energy of

the lattice,∑NkBT/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 300 K.As can be seen from the molecular charge density, panel (a), the initial localizedpolaron becomes delocalized at this temperature. One can conclude then that thereexist a critical temperature for which the localized polaron destabilizes. In our caseit should lie between 200 K and 300 K. Running more calculations for this range oftemperatures, we found that the critical temperature for this system is around 250K. We also observed that in each calculation above the critical temperature, it willtake a while for the localized polaron to evolve into a delocalized state depending onthe magnitude of the temperature. As the temperature increases, this time becomesshorter.

Chapter 4

Comments on Papers

Contents4.1 Paper One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1.2 My Contribution . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Paper Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.2 My Contribution . . . . . . . . . . . . . . . . . . . . . . . 34

The intorduction given in this thesis, is just a brief theoretical expla-nation of what is presented and published in the following papers. Insummary, the subject deals with the charge transport phenomena inthe 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 using thismodel for a wide range of intra- and inter-molecular parameters in order to obtaina detailed description of polaron formation energies and stabilities in a system withan excess charge but no external force.

4.1.2 My Contribution

I wrote the code getting help from Magnus Boman, performed all the calculationsand 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 isstudied in a two dimensional molecular lattice with and without disorder. Both

34 Chapter 4. Comments on Papers

intra- and inter-molecular electron-phonon couplings are cosidered in the modeland the paper describes the dynamics of the charge carrier. In this study onlythe dynamically stable polaron solutions are considered for the dynamics studies.We found that the parameter space in which the polaron can move adiabatically isquite confined. Increasing the on-site electron-phonon coupling, A, will result in amore localized polaron whereas enhancing the inter-molecular one, α, will reducethis effect and increases the width of the polaron. We observed that for a largevalue of electron-phonon coupling and a weak inter-molecular electron interactionthe polaron is very much localized and immobile whereas for small electron-phononcoupling and a strong inter-molecular electron interaction, is dynamically unstableand dissociates into a band state decoupled form the lattice. Adding disorder to thesystem 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 partsof the introduction, the methodology section and took part in writing the discussionsection.

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a theoretical study of charge transport in molecular crystals

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