Download A Theoretical Study of Charge Transport in Molecular Crystals Elham Mozafari

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
Linköping 2012
c
Elham
Mozafari
ISBN: 978-91-7519-731-9
ISSN 0280-7971
Printed by LiU-Tryck, Linköping, Sweden, 2012
A Theoretical Study of Charge Transport in Molecular
Crystals
Abstract: The main objective of this thesis is to provide a deeper understanding of the charge transport phenomena occuring 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 inter- (Peierls) molecular interactions, in particular the electron-phonon interactions.
First, the stability of the polaron in an ordered two dimensional molecular lattice
with an excess charge is studied using Resilient backPropagation, RPROP, algorithm. 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 frequencies, transfer intergral and electric field. We found that the polaron formation
energies 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 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 considered. 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 critical temperature, the localized polaron becomes delocalized.
All this research work is coded in MATLAB software , allowing us to run the calculations, 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 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 Khajavizadeh, 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.
Contents
1 Introduction
1.1 A brief introduction to charge transport in
crystals . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Mobility . . . . . . . . . . . . . . . . . . . . .
1.1.2 Charge carrier mobility measurement .
1.1.3 Polaron Concept . . . . . . . . . . . . . . .
1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . .
1
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2 Theory and Methodology
2.1 Charge Carrier Localization and Delocalization . . . . . .
2.2 Charge Transport Models . . . . . . . . . . . . . . . . . . . . .
2.2.1 Delocalized Transport in Simple Electronic Band .
2.2.2 Electron Transport in the Polaron Model . . . . . .
2.2.3 Weak Electron-Phonon Coupling . . . . . . . . . . . .
2.2.4 Hopping Transport for Localized Carriers in Disordered Materials . . . . . . . . . . . . . . . . . . . . . .
2.3 Transport in the Presence of Nonlocal Electron-Phonon
Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Charge Carrier Dynamics in the Holstein-Peierls
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Computational Details
3.1 Model Systems . . . . . . . . . . . . . . . . . . . . .
3.1.1 One Dimensional Molecular Chain . . .
3.1.2 Two Dimensional Molecular Lattice . .
3.2 Geometry Optimization and Polaron Stability
3.3 Polaron Dynamics . . . . . . . . . . . . . . . . . .
3.3.1 Dynamics in the Presence of Disorder .
3.3.2 Temperature Impact on Dynamics . . . .
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4 Comments on Papers
4.1 Paper One . . . . . . . . .
4.1.1 Overview . . . . .
4.1.2 My Contribution
4.2 Paper Two . . . . . . . .
4.2.1 Overview . . . . .
4.2.2 My Contribution
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Bibliography
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5 Papers
43
Chapter 1
Introduction
Contents
1.1
1.2
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
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . .
5
olecular crystals have been of great interest during the past few
decades 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 semiconductors. 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.
M
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
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 localized state to another. This process can be described within the standard perturbation 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
of cm2 · V −1 · s−1 . For instance, the highest mobility reported for tetracene and
pentacene 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 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 temperatures, 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 interactions 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 voltage. 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
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 measurements, 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 induce 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 surrounding 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 detailed book of Pekar in 1951[Pekar 1963]. Landau and Pekar investigated the selfenergy 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.
1.2. Thesis Outline
1.2
5
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 intermolecular 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 summerized 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 temperature effects are also accounted for. Finally, Chapter 4 glances at the particular
research topics covered in the supplemented papers.
Chapter 2
Theory and Methodology
Contents
2.1
Charge Carrier Localization and Delocalization .
7
2.2
Charge Transport Models . . . . . . . . . . . . . . . . .
8
2.3
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 Disordered Materials . . . . . . . . . . . . . . . . . . . . . . . .
12
Transport in the Presence of Nonlocal ElectronPhonon Coupling . . . . . . . . . . . . . . . . . . . . . . .
14
2.3.1
Charge Carrier Dynamics in the Holstein-Peierls
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
harge transport phenomena in molecular crystals are crucially dependent on how ordered the system is and at which temperature the system is
operating. Inter- and intra-molecular interactions, in particular the transfer 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.
C
2.1
Charge Carrier Localization and Delocalization
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 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 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 formation 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 when δEpol and δEb are small and δE > 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
2.2.1
Charge Transport Models
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
X
|Ψk (r) = fN
exp(ik.rn )|ϕk (r − rn )
(2.3)
n
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. The
prefactor fN is just a normalization constant.
To achieve the band structure one should be able to introduce a proper Hamiltonian. Considering tight-binding model, the electronic Hamiltonian is written as
X
X
Hel =
εn c†n cn +
Jnm c†n cm
(2.4)
n
n,m6=n
2.2. Charge Transport Models
9
c†n and cn 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 with Ψk | from the left side. Then
Ek =
< Ψk |Hel |Ψk >
ε + 2J 0 cos(ka) − 2J 00 sin(ka)
=
< Ψk |Ψk >
1 + 2S 0 cos(ka) − 2S 00 sin(ka)
(2.6)
in which the transfer integral value, J = J 0 +iJ 00 and the spatial overlap S = S 0 +iS 00
are complex values to maintain generality of the relation 2.6. It has to be noted
that 2.6 is 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 transport 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 k 0 . 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 parameter Jnm must 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
2
a lower limit of µ > ea
2~ with a 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 1 cm2 .V −1 .s−1 [Grozema 2008].
10
2.2.2
Chapter 2. Theory and Methodology
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 intramolecular coupling can be devided into two general cases. (i) When the electronphonon 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 investigated 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. 2.1 depicting the temperature
dependence of the mobility as predictied by the Holstein polaron model for limiting cases of weak electron-phonon coupling (g 2 1) and strong electron-phonon
coupling (g 2 1) where g 2 is the coupling strength. In the Holstein paper, the
electron-phonon coupling constant is denoted by A. In fact A is a parameter which
accounts for the energy gain due to the polaron formation. The parameters g and
A are linked by the relation[Zoli 2000]
g=
dA2 ~
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 component 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 perturbation, 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 perturbation 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].
m̄∗ =
m∗
1 − α/6
(2.8)
2.2. Charge Transport Models
T1
11
T2
Figure 2.1: Temperature dependence of the mobility for the limiting cases of
weak and strong electron-phonon couplings (g 2 ) as predicted by Holstein polaron
model[Coropceanu 2007].
α = ∆E/~ω is the polaron stabilization energy due to lattice deformation. Weak
electron-phonon coupling implies α < 6. Delocalized charges can then be described
by a semiclassical model and a renormalized effective mass, m̄∗ . 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. Under this assumption, Vel is then considered as a small perturbation, i.e., the molecular
crystal is pictured by a collection of isolated molecules. Vel is the electronic coupling
defining the interaction between the neighbouring localized states. The transformed
12
Chapter 2. Theory and Methodology
coupling interaction, Ṽel = eS Vel e−S , will then carry the electron-phonon interaction. eS is a unitary operator defined as
S = −g
X
n
b†n − bn c†n cn
(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 Dprocesses
are not important), one can then substitute Ṽel with its thermal
E
average, Ṽel , hence the bosonic component can be neglected[Troisi 2010]. It
T
should be kept in mind that this method can be applied if only the carrier is delocalized. The following relation can then be derived for the transformed electronic
coupling interaction
−2g 2 Nω + 12 X †
Ṽel = Je
cn cn+1
(2.10)
n
P
with Nω = [exp(~ω/kB T ) − 1]−1 . Comparing 2.10 with Vel = −J n c†n cn+1 , shows
that the effect of the phonons on delocalized
carriers is to reduce the effective hop
−2g 2 Nω + 1
2
ping integral by a factor of e
. 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 simple 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 Disordered 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
2.2. Charge Transport Models
13
hopping rate is ususally expressed as
κif = κ0 exp(−2αRif )

exp
−

εf −εi
kB T
1

εf > εi 
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 deformation 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 temperature regimes[Jortner 1976]. In the limit of high temperatures, kB T ~ωm (ωm
is the intermolecular vibration frequency), Marcus theory defines the hopping rate
expression as
s
"
#
2|
2π|Jif
(εf − εi + λreorg )2
1
κif =
exp −
(2.11)
~
4πλreorg kB T
4λreorg kB T
2π|J 2 |
The first term in the above equation 2.11, ~ 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◦ = εf − 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 perturbation 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.
14
Chapter 2. Theory and Methodology
Localized states
+
Electron-phonon coupling,
and
Reorganization energy
Hopping transport,
Miller-Abrahams theory
Delocalized states
+
Electron-phonon coupling,
and
Reorganization energy
Band Transport,
Drude theory
Marcus theory
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 demanding 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] and π-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
Electron-Phonon Coupling
of
Nonlocal
The transport models discussed up to now lack an important component when applied 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 electron-
2.3. Transport in the Presence of Nonlocal Electron-Phonon
Coupling
15
phonon coupling acting purely on-site of the electronic excitation. Considering nonlocal 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 closerun.
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 200 cm−1 have a 100% intramolecular characteristics. In this case the inter-molecular coupling can be neglected
due to the very high frequency of modes whereas most modes in the intermediate
range between 60 cm−1 to 200 cm−1 possess a significant mixing of intra- and intermolecular 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)
with Hel,intra being the diagonal elements of H (Holstein model plus disorder considerations) defined as
Hel,intra =
N
X
(εn + Aun )ĉ†n ĉn
(2.13)
n=1
and the off-diagonal terms (SSH model)
Hel,inter = −
N X
n=1
J0 + α(vn+1 − vn )
ĉ†n+1 ĉn + ĉ†n ĉn+1
(2.14)
un and vn are the intra- and inter-molecular displacements respectively. εn is onsite 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. J0 is the transfer integral value (assumed to be the same for all sites) and
α is the inter-molecular electron-phonon coupling.
16
Chapter 2. Theory and Methodology
To consider the role of the lattice (Hlatt ) in semi-classical treatment adopted during this thesis, the phonon system is devided into two seperate harmonic oscillators,
one for intra- and the other for inter-molecular vibrations.
Hlatt,intra =
N
N
K1 X 2 m X 2
un +
u˙n
2
2
(2.15)
N
N
K2 X 2 M X 2
vn +
v˙n
2
2
(2.16)
n=1
Hlatt,inter =
n=1
n=1
n=1
The force constants K1 and K2 and also the masses m and M refer to the intraand 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
as Λ(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)
with γ ≡ 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)
= Hˆel Ψ(t)
∂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 r¨n = −∇rn Etot
(2.19)
The total energy of the system can be expressed as
Etot = hΨ|Ĥ|Ψi
(2.20)
|Ψi 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
mu¨n = −K1 un − Aρn,n (t)
(2.21)
2.3. Transport in the Presence of Nonlocal Electron-Phonon
Coupling
17
M v¨n = −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
X
ρn,m (t) =
Ψnk (t)Ψ∗mk (t)
(2.23)
k
Chapter 3
Computational Details
Contents
3.1
Model Systems . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.1.1
One Dimensional Molecular Chain . . . . . . . . . . . .
3.1.2
Two Dimensional Molecular Lattice . . . . . . . . . .
19
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
o this date, the best model to describe polaron motion in molecular
crystals 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 previous 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.
T
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.3.1 in which the intra-molecular displacements, ui s
and the inter-molecular ones, vi s 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
20
Chapter 3. Computational Details
ui0
i-1
i+1
i
Equilibrium
Displaced
vi
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 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)
and j (y-direction). The intra- and inter-molecular displacements, ui,j s and vi,j s
x and v y ), are also distinguished by two indices,
(consisting of two components vi,j
i,j
in this case. The same thing as Fig. 3.1 occurs 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.
The electronic Hamiltonian will in this case contain some extra terms taking the
3.1. Model Systems
21
i-1
i
i+1
j+1
j
y
j-1
x
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 )ĉ†i,j ĉi,j
X
x
(Ji+1,j;i,j
ĉ†i+1,j ĉi,j + H.C.)
i,j
+
X
y
(Ji,j+1;i,j
ĉ†i,j+1 ĉi,j + H.C.)
(3.1)
i,j
with
x
x
x
Ji+1,j;i,j
= J0x − α(vi+1,j
− vi,j
)eiγΛx (t)
(3.2)
y
y
y
Ji,j+1;i,j
= J0y − α(vi,j+1
− vi,j
)eiγΛy (t)
(3.3)
and
The parameters have the same definition as introduced in Sec.2.3. The lattice distribution in the extended total Hamiltonian, eq.2.12, to include all components in
22
Chapter 3. Computational Details
both directions will be
Hlatt =
K1 X
mX
(ui,j )2 +
(u̇i,j )2
2
2
i,j
i,j
K2 X x
x 2
+
(vi+1,j − vi,j
)
2
i,j
K2 X y
y 2
+
(vi,j+1 − vi,j
)
2
i,j
MX x 2
y 2
+
[(v̇i,j ) + (v̇i,j
) ]
2
(3.4)
i,j
With these in mind, it is then possible to solve Newtonian equations of motion
müi,j (t) = −K1 ui,j (t) − Aρi,j;i.j (t)
(3.5)
and
x
x
x
x
M v̈i,j
(t) = −K2 (2vi,j
(t) − vi+1,j
(t) − vi−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)
y
y
y
y
M v̈i,j
(t) = −K2 (2vi,j
(t) − vi,j+1
(t) − vi,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 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 molecular lattice, the initial geometry is always optimized using Resilient backPROPogation 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 inspite 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,
Molecular Charge Density
3.2. Geometry Optimization and Polaron Stability
23
0.5
0.4
0.3
0.2
0.1
0
10
8
10
6
ny
8
6
4
4
2
0
2
0
nx
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 the
number of sites in x and y direction, respectively).
Marcus 1956, Emin 1975, Holstein 1959]. The polaron formation energy, Ep , is expressed 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, Ep± for an added hole (+) or electron (−).
For a 2D lattice
Ep = 2(J0x + J0y ) − Ep±
(3.8)
For the polaron to be stable, Ep has 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 electronphonon interaction) also into account will result in a slightly less localized polaron
but increases the stability[Mozafari 2012]. One can then conclude that both intraand inter-molecular electron-phonon couplings, A and α, play important roles in
polaron stability in a way that increasing A and α, eq.3.1 will enhances the polaron
24
Chapter 3. Computational Details
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 of 50−100meV are
of more ineterst. They are in agreement with the stabilities reported for Pentacene
and 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 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 of 20 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.2.12 and 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 (about 250f 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, ui s (panel (b)) and intermolecular 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
and Hlatt,inter . The former deals with the local vibrations,
ui s, Fig.3.4 (b). This
q
oscillatory behaviour have a frequency expressed as Km1 .
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
3.3. Polaron Dynamics
25
T3
T1
T2
Figure 3.4: Polaron dynamics in a one dimensional system: (a) molecular charge
distribution, (b) intra-molecular displacement, ui , (c) inter-molecular bond length
x − v x for A=1.5 eV/Å, K =10.0 eV/Å2 , α=0.5 eV/Å, K =1.5
in the x direction, vi+1
1
2
i
eV/Å2 , J0 =0.05 eV and E0x =2.0 mV/Å.
26
Chapter 3. Computational Details
in the opposite direction of the polaron motion. In summary, when the polaron is
situated on site i, 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 sites i
and i + 1 (red color in panel (b) and (c)). It should be noted that the oscillatory
q
behaviour follows the classical spring-mass-spring oscillator with a frequency
K2
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 values K1 and K2 and 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 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 force
and for values above 1.7 eV /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 35 meV to 64 meV . The
range 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 that
increasing α will result in a more extended polaron whereas increasing A makes
x,y
the polaron more localized. finally, by varrying the transfer integral value, Ji,j
,
we observed that the polaron is stable for the values in the range of 40-80 meV
x,y
corresponding to the formation energies of 78 meV at Ji,j
= 40 meV and 44 meV
x,y
x,y
at Ji,j = 80 meV . For values of transfer integral below Ji,j
= 40 meV , the polaron
x,y
is immobile while for values larger than Ji,j
= 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 3.8 mV /Å.
For higher field strengths the polaron becomes unstable and dessociates into a band
state.
3.3. Polaron Dynamics
3.3.1
27
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 different 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 mobility in Naphthalene at T = 10K.[Warta 1985a]. In general, a room temperature
mobility in the range of 1 − 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 from 200K up to room temperature. 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 temperatures 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 random
forces,
Rn (t)[Berendsen 1984,
Wen 2009, Ribeiro
intra
2011] with zero mean 0value,
0
Rn (t) = 0, and the variance Ri,j
(t)Riintra
= 2kB T mλδi,j;i0 ,j 0 δ(t − t ) and
0 ,j 0 (t )
28
Chapter 3. Computational Details
inter (t)Rinter (t0 ) = 2k T mλδ
0
Ri,j
B
i,j;i0 ,j 0 δ(t − t ). These forces are added to the Newi0 ,j 0
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).
müi = −K1 ui − Aρi,i (t) − mλu̇i (t) + Riintra (t)
(3.9)
and
M v¨i = −K2 (2vi − vi+1 − vi−1 ) − 2αe(iγΛ(t)) (ρi,i−1 (t)
− ρi+1,i (t)) − M λv̇i (t) + Riinter (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, ui or the inter-molecular, vi , displacements. Accordingly, Ẋ and
Ẍ 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
1
1
1
Ẋ + 2 = (1 − λ∆t)Ẋ n + M−1 ∆t(F n + Rn )
2
2
(3.11)
drift
1
X n+1 = X n + ∆tẊ n+ 2
(3.12)
half a kick
1
Ẋ n+1 =
Ẋ n+ 2 + 12 M−1 ∆t(F n+1 + Rn+1 )
(1 + 12 λ∆t)
(3.13)
q
2λkB T
1/2 n
where Rn =
Z with Z n being a vector of independent Gaussian
∆t M
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
n
means the time has moved forward for half a time step, t → t + ∆t
2 . F 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 each
instant of time can be expressed as[Ono 1990]
XX
∗
ψ(n, t + ∆t) =
φl (m)ψ(m, t) e −iεl ∆t/~ φl (n)
(3.14)
l
m
3.3. Polaron Dynamics
29
where φl (m) and εl are the instantaneous eigenfunctions and eigenvalues of the
electronic part of the Hamiltonian, Hel , at time t.
Solving equations, eq.3.9, 3.10 and 3.14, 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)
Figure 3.5: Polaron dynamics in a one dimensional system: (a) molecular charge
distribution, (b) intra-molecular displacement, ui , (c) inter-molecular bond length
x − v x for A=1.5 eV/Å, K =10.0 eV/Å2 , α=0.5 eV/Å, K =1.5
in the x direction, vi+1
1
2
i
eV/Å2 , J0 =0.05 eV, λ=105 eVas/Å, and E0x =0.0 mV/Å, at T = 200 K.
30
Chapter 3. Computational Details
(a)
(b)
(c)
Figure 3.6: Polaron dynamics in a one dimensional system: (a) molecular charge
distribution, (b) intra-molecular displacement, ui , (c) inter-molecular bond length
x − v x for A=1.5 eV/Å, K =10.0 eV/Å2 , α=0.5 eV/Å, K =1.5
in the x direction, vi+1
1
2
i
2
eV/Å , J0 =0.05 eV, λ=105 eVas/Å, and E0x =0.0 mV/Å, at T = 300 K.
3.3. Polaron Dynamics
31
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 50 K to 300 K for the same
parameter 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 can
be seen that the polaron is localized and has a diffusive motion due to the fluctuations 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.
P
2
The total
vibrational energy of the lattice is achieved by
i (mu̇i /2) +
2
(M v˙i /2) . At thermal equilibrium, this energy is equal to the inner energy of
P
the lattice,
N kB T /2 (N ≡ degrees of freedom). It is obvious that if the temperature increases, the random forces will also increase. The larger the forces become,
the higher the amplitude of the lattice vibrations become. Therefore, the total vibrational 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 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 200 K 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.
Chapter 4
Comments on Papers
Contents
4.1
4.2
Paper One . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.1.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.1.2
My Contribution . . . . . . . . . . . . . . . . . . . . . . .
33
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.2.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.2.2
My Contribution . . . . . . . . . . . . . . . . . . . . . . .
34
Paper Two
he intorduction given in this thesis, is just a brief theoretical explanation 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 temperature is studied on the charge dynamics.
T
4.1
4.1.1
Paper One
Overview
The stability of polarons in a two dimensional molecular crystal is studied applying the semiclassical Holstein-Peierls model. Calculations are performed using 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
4.2.1
Paper Two
Overview
Harnessing the semiclassical Holstein-Peierls hamiltonian, the charge transport is
studied 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 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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