Modeling organic diodes 

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Electronic polarization

When the charge localization effect is pronounced, the transport is described by the hopping between localized states. In this case, thermally activated transport (d =dT > 0) is expected because the lattice vibration helps localized charges to overcome potential barriers [Figure 2.6(b)]. Generally, we consider two origins of localization; polarization and disorder.
Electronic polarization is a local deformation of the electron density due to the additional charge staying at this location [48]. The consequence is an immediate relaxation of that particular state by the induced potential that in turn localizes the staying charge, which is sometimes called the self-trapping effect. Once this charge moves to another molecule (or a segment of molecule), this effect takes place again. Therefore, we can visualize this type of charge transport by the movement of charge with its accompanying polarization, and these two are together called a ‘polaron’. Polaronic transport is a distinctive character of all organic semiconductors. It is again due to small transport bandwidth that results in residence time higher than polarization time, and also due to small dielectric constant in organic materials that intensifies the localizing potential.

Multiple trapping and release (MTR)

The second mechanism of localization is induced by energetic and geometrical dis-order. When the degree of disorder is not significant (compared to the transport bandwidth), we consider ‘weak localization’, and the energy diagram is represented by extended states (band) plus a localized-state band tail that acts as charge trap-ping sites. This picture is widely-used in conventional model for amorphous silicon (a-Si) [49, 50]. For organic semiconductors, vacuum-evaporated small-molecules can be generally well described by this model. They can form relatively well-ordered small domains with different orientation between domains, which can be regarded as poly-crystalline phase. The main origin of localized tail states in a-Si are dangling bonds, and that in polycrystalline organic semiconductors can generally be attributed to the grain boundaries [51]. In this energetic situation, the charge-transport is based on the multiple trapping and release (MTR) process illustrated in Figure 2.7. Here we only consider electron transport at LUMO level. The mobility edge (ME) is defined as the energy above which the states are delocalized, thus the mobility increases dramatically. Charge carriers tend to be trapped into the localized states and be thermally released up to the band states where an effective transport can take place. The expression for mobility is dependent on the energetic distribution of the trap states. Exponential density-of-states (DOS) N(E) for trap states are expressed by N(E) = Et exp Et ; (2.2) Nt Ec E.
where Nt is the total trap states, Et the trap distribution energy, and Ec the conduc-tion band edge (equivalent to ME). A detailed analytical formulation for obtaining trapped and free charge carrier density will be given in Section 4.1.5. Here, for discussion purpose, we simply present the expression of the effective mobility MTR determined by the trapped-to-total carrier density, MTR = 0 nf + nt ’ 0 Nt exp kBT ; (2.3) nf Nc Ec EF.

Gaussian disorder model (GDM)

In a system where the degree of disorder is high, such as non-crystalline polymeric films, ‘strong localization’ results in absence of extended band states. Therefore, all electronic states have to be regarded as localized states, and their energetic distribu-tion is described by Gaussian distribution where N0 is the total density of transport sites and is the half-width of Gaussian DOS. In the Gaussian disorder model (GDM), the charge transport is based on the thermally assisted tunneling events between localized states, which is generally called hopping transport. There is no analytical solution for charge transport in the GDM.
In 1993, Bälsser reported on a Monte Carlo simulation study for the charge transport in the GDM [54], where hopping rate is assumed to be Miller-Abraham type [55]: where 0 is the attempt frequency, Rij the geometrical distance between sites, the localization length, Ei the energy of the initial site, and Ej the energy of the arrival site. An illustration of the hopping charge transport related to Equation (2.5) is shown in Figure 2.8. Electrons at high-energy states tend to make downward hops. However, below a certain energy level, there is difficulty in finding any low-energy site with small distance Rij. At this moment, they can make an upward hop toward a nearest site with the aide of thermal energy. This type of continuous process defines the transport energy (TE) around which the hopping transport proceeds [56]. Because the electrons below the TE need to be thermally elevated for efficient hopping transport, the transport near and below this TE resembles the MTR process, with the TE analogous to the ME. From the Monte Carlo simulation in [54], an empirical expression for the GDM mobility GDM can be written as GDM = 1 exp  » 3kBT 2 # exp (C  » kBT 2 # pF ) ; (2.6) where 1 is the mobility prefactor (zero field, high temperature limit), C an empirical constant, the variance of geometrical disorder, and F the strength of electric field. Therefore, the major characteristics of this model are the non Arrhenius temperature behavior and the dependence of the mobility on the electric field.

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Equivalent circuit modeling

Our particular interest in the thesis is to use impedance spectroscopy for equivalent circuit modeling of two-terminal organic devices (organic diodes). Impedance spec-troscopy is a well established technique, by which we measure complex impedance of a given electrical or electrochemical system while sweeping the ac small-signal frequency usually at a fixed dc voltage [65]. Because every circuit element responds differently to the frequency ! (Figure 3.1), a full spectrum of impedance spectroscopy provides the information on which elements exist in the system and how they are connected to each other. In other words, one can extract a realistic equivalent circuit of a two-terminal device under analysis by measuring its total impedance over a broad frequency range. This whole process from the measurement to the extraction of an equivalent circuit with the exact circuit parameters (R, C, L, etc.) is to be called impedance modeling, impedance fitting, or equivalent circuit modeling. More advanced analysis involves assigning physical meaning to particular circuit elements and monitoring the varia-tion of circuit type and parameters upon different measurement or biasing conditions. In the literature, there are many reports showing practical applications of impedance spectroscopy to the analysis of organic devices [66–70].

Table of contents :

1 Introduction 
1.1 Organic electronics
1.2 Motivation
1.3 Thesis overview
2 Fundamentals 
2.1 Organic semiconductors
2.1.1 -conjugated molecules
2.1.2 Organic molecular solids
2.1.3 Material classification
2.2 Charge-carrier transport
2.2.1 Band transport
2.2.2 Electronic polarization
2.2.3 Multiple trapping and release (MTR)
2.2.4 Gaussian disorder model (GDM)
2.3 Charge-carrier injection
2.3.1 Barrier formation
2.3.2 Non-ideal effects
3 Methods 
3.1 Impedance spectroscopy
3.1.1 Generalities
3.1.2 Equivalent circuit modeling
3.1.3 Typical impedance spectra
3.2 Analytical modeling
3.2.1 Physical point of view: Goals and limitations
3.2.2 Device models for circuit simulation
3.3 Numerical simulation
3.3.1 How a device simulator works
3.3.2 Defining a mesh
3.3.3 Use in device modeling
3.4 Experimental investigation
4 Modeling organic diodes 
4.1 Vertical rectifying diodes
4.1.1 Metal-insulator-metal (MIM) model
4.1.2 Frequency-dependent impedance analysis
4.1.3 Dielectric properties of pentacene
4.1.4 Modeling the low-voltage regime
4.1.5 Trap-induced ideality factor
4.2 Vertical symmetrical diodes
4.2.1 Film and device properties
4.2.2 Equivalent circuit model
4.2.3 Circuit parameters during relaxation
4.3 Gap-type diodes
4.3.1 Current in organic gap-type diodes
4.3.2 Compact modeling approach for gap-type diodes
5 Modeling organic transistors 
5.1.1 Current-voltage characteristics
5.1.2 Operation mechanism
5.1.3 Effect of the channel dimension
5.2 OFETs
5.2.1 Staggered versus coplanar structure
5.2.2 Coplanar OFET model: Charge distribution
5.2.3 Coplanar OFET model: Contact resistance
5.2.4 Coplanar OFET model: Experimental analysis
5.2.5 Compact modeling approach for OFETs
6 Conclusion and outlook 
A Publications
B Activities
C Experimental details


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