A drift-diffusion simulation model for organic field effect transistors: on the importance of the Gaussian density of states and traps

The nature of charge transport in organic materials depends on several important aspects, such as the description of the density of states, and the charge mobility model. Therefore specific models describing electronic properties of organic semiconductors must be considered. We have used an organic based drift-diffusion model for the electrical characterization of organic field effect transistors (OFETs) utilizing either small molecules or polymers. Furthermore, the effect of interface traps, bulk traps, and fixed charges on transistor characteristics are included and investigated. Finally, simulation results are compared to experimental measurements, and conclusions are drawn out in terms of transistor performance parameters including threshold voltages, and field-dependent mobilities.


Introduction
Organic materials and devices have been at the forefront of the research community in recent years due to their flexible properties, as well as the large scale roll-to-roll cheaper fabrication procedures. Eventhough, inorganic semiconductors have been (and still are to a large extent) the go-to materials for today's electronic industry, research have come a long way in improving performance measures of organic based devices since their first introduction. Such devices include organicsolar cells (OSCs) [1][2][3], and light emitting diodes (OLEDs) [4][5][6][7]. The latter one especially made a breakthrough on the commercial scale for display applications (i.e. mobile phones and the recently introduced OLED TVs). Organic field-effect transistors (OFETs) (the main focus of this work) have also been (and still are) investigated heavily [8][9][10][11]. In a recent review published by Paterson and co-workers [12], the authors provide a graphic showing the progress acheived in OFET mobilities over the past 30 years. According to this graphic, both p-and n-type mobilities have reached up to 20 cm 2 V −1 s −1 and 10 cm 2 V −1 s −1 , respectively. This definitely spells significant improvements, but they go on to say that these values do not paint the entire picture and in reality there is what they refer to as a 'mobility hype'. This hype points towards that there has been an overestimation in OFET mobilities, as a consequence of extracting mobilities from characteristics (i.e. transfer and output) that show non-ideal behaviour. This non-ideality is depicted as a 'double slope', leading to overestimated mobilities from the steeper portion of the transfer characteristics. Contact resistance is identified as the main S Supplementary material for this article is available online (Some figures may appear in colour only in the online journal) Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. cause for the resulting non-idealities [13][14][15][16]. Nevertheless, even with the presence of this mobility hype, there is no denying the fact that improvements in OFET mobilities have been made [12]. Advancements in OFETs is also aided with the different types of materials available to choose from [17], for both hole and electron conduction. This also, theoretically, allows organic semiconductors the possibility to enter the digital market (i.e. complementary technologies) [18][19][20]. More recently, research has gone into investigating blend based OFETs, incorporating both small molecules and polymers. This is particularly interesting as we can take advantage of what both types of materials have to offer, the high mobility of small molecules and the easier processability of polymers. Mobilities ranging up to 13 cm 2 V −1 s −1 have been reported [21][22][23][24]. There is not alot available on blends in the literature to date, but with the the early promise blends have shown, more interest will certainly focus on that front.
The working principle of a FET is always the same, regardless of the active material used for the channel region. However, the underlying physics governing charge transport is different between organic and inorganic materials. For the latter, charge transport occurs via coherent Bloch waves. In contrast, for the former, it is through hopping from one site to another within a varying energetic landscape, owing to individual molecules being bound together through weak Van der Waals interactions. Therefore, charge transport occurs through hopping between localized states. This hopping mechanism is appropriately described by the Gaussian disorder model (GDM) [25]. These energetic sites follow a random distribution which is described by a Gaussian function in the vicinity of an energy level of interest (i.e. highest or lowest occupied molecular orbitals (HOMO or LUMO) respectively). The width of this function is referred to as the energetic disorder. Such disorder is found in both small molecules and polymers, and is more pronounced in the latter. Multiple theoretical models have been proposed to describe the hopping of charges, namely: (1) nearest neighbour hopping (NNH), (2) variable range hopping (VRH), (3) the transport energy model (TE), and (4) multiple trap and release model (MTR). Motion of charges described by any of these models does not only depend on the electric field but also the temperature. A more detailed overview can be found here [26].
This hopping mechanism is one of several forms of transport used to explain the movement of charges in organic materials. These also include band and polaron transport. Two important requirements need to be achieved for band transport to occur, which are the growth of high quality crystals, and the observation of a negative temperature dependence of the mobility. This dependence has been previously observed in Rubrene FETs by Podzorov et al [27,28]. The review by Hasegawa and Takeya [29] further highlights the significance of fabricating high quality single crystals to achieve high charge mobilities. On the other hand, polaron transport accounts for polarizations, which is a common occurence in organic materials. If a charge is present on a site for a long enough time, it starts to polarize the surrounding environment. The initial charge and the resulting polarization then propagate (together) within the material known as a polaron. As Horowitz thoroughly explains in [30], there can be different types of polarizations (i.e. electronic, molecular, and lattice) depending on the residence time of a charge on a specific site. A comparison between the different charge transport theories have been addressed, for example see Stallinga [31] and Troisi [32].
Charge transport in organic semiconductors is a much more complex process, therefore careful considerations need to be taken into account for the description of the sub-models used and their parameterizations in any drift-diffusion (DD) simulation. That includes a correct description of the density of states (DOS), the mobiltiy model, and the effect of charge trapping. More importantly, a connection between these models and what actually occurs within organic materials needs to be established, to justify their usability. This allows for the appropriate characterization of organic based devices and subsequent comparison and validation with experimental measurements. The paper is outlined as follows: in section 2, we present theory behind the DD simulation and go through the sub-models used for our investigations. Section 3 will present the results, extracting all important performance measures (i.e. mobilites, threshold voltages) and drawing out a comparison with experimental measurements. Finally, in section 4, we end with our concluding remarks.

The drift-diffusion model
The drift-diffusion model involves solving a set of equations, within the finite element method using the the commercial software TiberCad. This DD model has been previously used investigating both inorganic and organic based devices for different applications [33][34][35][36][37][38][39][40]. Furthermore, it has also been used for the investigation of charge carrier mobility unbalance in organic solar cells [41], as well as alongside circuit level modelling for identifying low conductive regions in organic thin films as a result of charge carrier trapping [42]. In this study we start by creating a structure, then a corresponding mesh is generated consisting of many elements, for which the DD equations are applied to and solved. We will be considering a 2-dimensional (2D) bottom gate-bottom contact (BG-BC) structure. Both the 2D device and the meshed representation are shown in figure 1.
The main equations of the DD model are the Poisson and continuity equations given by: where ε is the permittivity, e the electronic charge, ϕ the electric potential, n and p the electron and hole mobile charge densities, N D and N A the donor and acceptor concentrations respectively. The terms p t , n t represent trap densities, and finally ρ is the charge density. In (2) and (3), µ n and µ p are the charge mobilites, G and R are the generation and recombination terms. In our transistor analysis both generation and recombination will not be considered. The presented work focuses on unipolar devices, hence the reasoning behind not considering the latter. We start by sweeping on the gate voltage, followed by a sweep on the drain voltage at each gate. During this process the DD equations are solved self consistently and the drain current calculated. In the following subsections, we will go through (in more detail) the different physical models used for our transitor analysis.

Gaussian density of states (GDOS)
As it was stated before, charge transport in organic semiconductors relies on a a hopping mechanism through a random distribution of localized states. In order to have an appropriate description for this charge hopping, the DOS is best described by a Gaussian function. This function is expressed as where N 0 is the site density, E the energy, and σ the energetic disorder respectively. Energetic disorder in organic semiconductors typically lies in the range between 30 meV and 100 meV [43,44], and it relies on the morphology of the material. That is why polymers suffer from higher disorder compared to small molecules. Going from low to high disorder leads to an increase in the random distribution of sites further away from the transport level, as a consequence some of these sites might act as trap states affecting overall charge transport. Moreover, it has been shown that charge mobilites depend on the applied field, carrier density, and temperature (more on this will be discussed in sub-section (C)). The dependence of the mobility on the latter two is affected by the DOS. Therefore, it is crucial to have a correct description of the DOS and the appropriate amount of disorder depending on the material chosen.

The trap model
Charge carrier trapping in organic semiconductors is one of the major factors affecting device performance. Depending on the energy associated with a trap state (i.e. how far it is inside the band gap), this can have different effects on device operations such as degrading mobilites [27,45,46], causing hysteresis [47], and leading to threshold voltage shifts [48,49]. Stability over extended periods of time is also a major issue in organic devices, more specifically the ones incorporating n-type materials as the active region. This is one of the reasons why complementary logic based on organics has been proven difficult to achieve. It is imperative to include traps in any DD simulation, and their effects could be seen from the transfer characteristics of an OFET. Traps in our DD simulations are modelled as follows: (i) A neutral trap that becomes negatively charged when an electron is trapped (ii) A neutral trap that becomes positively charged when an hole is trapped (iii) Fixed charges where the density is simply defined. They can be either positive or negative.
In (5) and (6), N t denotes the density of traps, E tn and E tp the electron or hole trap energy level, E F,n and E F,p are the quasifermi energies, k b the Boltzmann constant, and T the temperature respectively. The trap level can be taken with reference to the conduction or valence bands (i.e. E c or E v ). The above mentioned models (and their subsequent effect on charge transport) have been used extensively for the investigation of inorganic materials and devices. In order to justify their usage towards investigating charge transport in organics, they must be able to represent the physical meaning of certain phenomena (or events) that occur within organic materials and devices. One of these events is the presence of dipoles, more specially interface dipoles. Dipoles exist as a result of inhomogeneous distribution of charges at a surface [50]. Such dipoles can occur at organic-organic [51,52], organic-metal interfaces [53][54][55], and even with dielectric interfaces [56]. Infact, self-assembled monolayers (SAMs) have been previously used to intentionally induce interface dipoles between the organic and dielectric materials, to influence the electrical characteristics of OFETs. As a result, controlling of charge densities leading improved field effect mobilities, as well as threshold voltage shifts have been observed [57][58][59]. A presence of a dipole causes a shift in the potential between two different interfaces (or surfaces). This shift is expressed as where q is the electronic charge, m d is the electric dipole moment (given in units of debye and depends on the distance between the positive and negative charges), θ is the angle of the dipole with respect to the surface, A d is the area of the surface charges, and lastly ε is the dipole layer's dielectric constant. Similarly, fixed charges represent a surface area of charges present at an interface and through coulomb attraction, charges of the opposite sign are situated on the other side. Hence, this would be expressed as Both (7) and (8) are almost identical with the exception of the angle θ, allowing us to use a model of fixed charges to represent the effect of dipoles in our simulations. This assumption is valid if the distance between the separate charges is within a fraction of a nanometer. In the following section, we show how the fixed charges does indeed affect the threshold voltage and subsequently the field-effect mobility, therefore allowing us to account for interface dipole effects. A similar connection can be achieved in regards to the modelling of trap states in our DD simulations. It is already quite obvious that the mentioned trap models can be used to describe defects withing organic materials. Such defects can be structural, energetic, or a combination both. However, these trap models can also be used to account for redox states. Organic materials have been shown to go through redox reactions [60]. These reaction involves transfer of electrons from (i.e. oxidation) or to (i.e. reduction) the host material. Organic electrochemical transistors (OECTs) that rely on transitioning between redox states, have been under investigation for biological applications [61]. According to a recent review on OECTs by Zeglio and Inganäs [62], changes in redox states of organic materials are associated with energy levels formed inside the band gap. These intermediate levels act as doping or de-doping centers when the device is operated under an external potential. This process can also be interpreted as charge carrier trapping and de-trapping, and our DD trap models can be used to account for these effects as well.
Therefore, if simulated traps are intended to represent defects, then (5) and (6) stand. On the other hand, if they were to represent redox reactions, in this case both E tn and E tp in (5) and (6) are simply replaced with E redox . Furthermore, N t would then represent the molecules (i.e. they are either reduced or oxidized). This effectively means that N t has to be equal to N 0 from (4). Nevertheless, it might be possible that N t be somewhat higher or lower. Equations (5) and (6) represent a classical description of the density. Conversely, in organics the description of the density is of a quantum mechanical nature. This is linked to the delocalization of the wavefunction, where molecules can be charged by a fractional charge (i.e. only a fraction of the quantum state is localized on a specific molecule). As a result, the density N t can have values which are both larger or smaller than the value of the molecular density N 0 . In our simulation process, we will start by defining single trap levels, but if needed it is also possible to define a distribution of trap states with the Gaussian function from (4).

The mobility model
Describing how charges move within a transistor channel is an integral part of any DD simulation, and there are multiple models available in the TiberCad tool to choose from. Specific to organic semiconductors there are two models that are usually used for such investigations. We will present an overview on both models, and how the decision is taken on which one to use moving forward with the simulations.  [63,64], is obtained as a parametrical fit of numerical solutions of a system of master equations implementing the Miller-Abrahams [65] hopping model between localized states with a Gaussian energy distribution. The model takes into account the dependence of the mobility on the temperature, charge density, and electric field: The Pasveer mobility has been used to fit experimental results for different semiconducting polymers [37,38,66]. Nevertheless, the authors also state that the above parameterization starts to breakdown at high electric field, hence a cutoff has been defined where beyond it the breakdown occurs [38], and is given by 2.3.2. The Poole-Frenkel mobility. According to Pasveer, at high electric field and beyond the cut-off mentioned above, the mobility starts to follow the Poole-Frenkel law (PF) [67]. The PF mobility model is a widely known and used one in the literature for investigating charge transport in organic semiconductors [26,[68][69][70]. According to PF, the mobility is expressed as: where µ 0 is the zero-field mobility, ∆ the zero-field activation energy, γ the Poole-Frenkel constant [71], and F the electric field respectively. This relation specifies that charges can gain the required energy from the applied electric field to overcome potential barriers, as they hop from one site to another, equal to a certain activation energy that is otherwise not provided (or insufficient) thermally. Furthermore, as the mobility depends on both the temperature and applied field, the dependence on the temperature vanishes at the point of F = (∆/γ) 2 . At this point, the applied field is equal to the activation energy so that at different temperatures the mobility will always reach the same value [68].
Deciding on which mobility model to use for our DD simulations, depends on several aspects. The PF model better suits small molecules, while the hopping mobility is used for polymers. The numerical solution presented by Pasveer was mainly based on investigating polymers. Furthermore, if the DD model is used to investigate polymer based OFETs, we must first find out where we lie with respect to the cut-off defined by (10). If the electric field is lower then the hopping mobility is chosen, otherwise the PF is then the one used.

Simulation related remarks
In this sub-section we will point out some simulation related remarks. As mentioned earlier, a BG-BC transistor architecture will be considered. Looking back to figure 1(b), the device is split up into two major regions, the oxide and the organic. The organic region itself is further split up into two, where the top part is the bulk, and the bottom part used to to model interface effects. In regards to the contacts, they are all defined as physical lines (i.e. boundaries) of the device. Bulk representation of the contacts are not taken into account, as they effectively do not change anything in terms of currentvoltage characteristics. With this exclusion we are able to gain improvements on simulations times.
In regards to the dimensions, we construct our 2D meshed representation based on the following device dimension labels: the oxide thickness (Ox th ), the channel length (Org ch_l ), the channel thickness (Org ch_th ), the lateral dimension of the organic near the contacts (Org lat ), the contact length (Con l ), and lastly the contact thickness (Con th ). Only the Ox th and Org ch_l are changed with respect to experimental values, while the rest are pre-defined and always fixed. A summary of these dimension labels are given in table 1 (also shown in figure 1(b)).
Org ch_l was chosen to be 10 nm, as it is well known that charge transport usually occurs within the first few monolayers just above the oxide-organic interface [72,73]. Furthermore, we did not observe any change in the current-voltage characteristics going beyond this value 1 . For the sake of being consistent, Org lat was chosen to be the same as Org ch_l . Similarly, Con th was also fixed at 100 nm, as increasing the thickness simply increased simulation times without any observed changes in the characteristics. The PF mobility model has been chosen for our simulations, as we will be presenting results on Pentacene (i.e. small molecule) based OFETs (see section 3). Lastly, both the source and drain are modelled as Schottky contacts (typical for all organic based devices), unless there has been mention of doping near the source/drain. In the latter case the contacts would then be modelled as ohmic.

Simulation results
Here, we present a comparison between the DD simulation and experimental results for a BG-BC Pentacene transistor from [74]. Per the fabricated transistor, the channel length is 12 µm. The oxide thickness was not specified from the experiment, hence a typical thickness of 200 nm was chosen. We first start by fitting the transfer characteristics given a set of parameters (i.e. for the models defined in the previous section). Once a good fit is achieved, the output characteristics is simulated using the same model parameters. Before going straight into the comparison, it is important to first understand the behaviour of the transfer characteristics. This is useful especially in the presence of fixed charges and interface (or  ) 10 nm Contact length (Con l ) 50 nm Contact thickness (Con th ) 100 nm bulk) traps, where each one has an effect on transistor performance, and such effects can be seen from the transfer curves.

Trap-free versus fixed charges and interface/bulk traps
We start by investigating the effect of negative fixed charges at the oxide-organic interface (will be referred to from now on as 'FC'. From figure 2, it can be seen that a positive shift in the threshold voltage is induced as we move from a 'pristine' device towards higher concentrations of fixed charges. This behaviour follows the relation: where ∆V th is the threshold voltage shift, Q int the fixed charge concentration, and C ox the oxide capacitance respectively. This ∆V th is better depicted in the inset of figure 2 by plotting the square root of the drain current ( √ Id), and the threshold voltage V th is simply the intersection at the x-axis when the current is zero. Following the inclusion of fixed interface charges, we can now further introduce interface or bulk traps. Traps are regarded neutral if empty, then either positively or negatively charged if occupied by a hole or an electron (will be referred to as 'ITH' and 'ITE' respectively). A single trap level was chosen above the valence band. The effect of both types of charged traps, modelled at the oxide-organic interface, is depicted in figures 3(a) and (b). On one hand, positively charged traps leads to less current drive (i.e. lower drain current and a downward shift in the transfer curve). This is true, as a higher gate voltage is needed to induce more charges in the channel region. Conversely, negatively charged traps lead to a decrease in the inverse sub-threshold slope and as a result a high sub-threshold current is evaluated, eventhough the transistor is operated at high positive gate voltages (i.e. in the off state). At high concentrations of electron traps, the positive applied gate voltage becomes insufficient to deplete the channel of holes leading to high currents in the off-state. Under negative applied gate voltages no change is observed in the drain current. This is self-evident, as in this voltage regime the channel is filled up with holes as mobile charges and trapped electron concentrations have no effect on the transfer characteristics. Furthermore, the negatively charged traps also lead to ∆V th shifts (inset of figure 3(b)) in the absence of fixed charges. This is not apparent in the main panel of figure 3(b) simply due to the already high presence of fixed charges as a starting point before the interface traps were applied.
Finally, we investigated the effect of positively charged bulk traps (will be referred to as 'BTH') on the transistor performance with a single level above the valence band. As illustrated in figure 3(c), they also have a similar effect to that of the positively charged interface traps. Between bulk and interface traps, the former has a more pronounced effect on the inverse sub-threshold slope. This is a consequence of the bulk traps being located deeper in the band gap. In the supplementary information (stacks.iop.org/JPhysD/53/105102/ mmedia), we provide a charge density map projected on our device for the ITH (figure (S1)), ITE (figure (S2)), and BTH (figure (S3)) cases at different densities. From this analysis we were able to show that fixed or trapped charges have their own distinct effects on the transfer characteristics of an organic transistor, but another crucial conclusion from this analysis is that using either fixed charges alone 2 or as a combination with either interface or bulk traps might be insufficient to get a good description of the transfer curves. Possibly a combination between all three of them is required to achieve a good fit with the experimental results.

Validating experimental data
In this sub-section we aim to present our simulation results and validate them with experimental measurements from [74], by fitting both the transfer and output characteristics of a 12 µm channel Pentacene BG-BC OFET. We start by first achieving the best fit of the transfer characteristics. Once this is completed we use the corresponding parameterizations to simulate the output characteristics. Following the analysis in section 3.1, we are presented with three different scenarios to fit the transfer characteristics starting from a pristine condition, namely: (i) The first scenario includes: • Negative interface fixed charges, to get the correct V th and hence the slope of the transfer curve in the on-state. • Positively charged bulk traps, to control the sub-threshold characteristics. • (If required) Negatively charged interface traps, to have a more precise control over the sub-threshold slope.
(ii) The second scenario includes: • Negative interface fixed charges, to get the correct V th and hence the slope of the transfer curve in the on-state. • Positively charged interface traps, to get the appropriate evaluated current across the entire voltage range. • (If required) Positively charged bulk traps, to control the sub-threshold characteristics.
(iii) The third scenario includes: • Negatively charged interface traps, to get the correct V th keeping in mind its added effect on the sub-threshold. • Positively charged bulk traps to adjust the current drive accordingly.
From here onwards we will refer to the three scenarios mentioned above in the same order as SC1, SC2, and SC3 respectively. Simulation parameters used to describe the above scenarios (i.e. fixed an trap concentrations) will be adjusted accordingly to reach the best fit of the transfer characteristics. All other parameters are summarized in table 2. From table 2, all values referring to the µ 0 , and interface or bulk trap energies are taken as input from the analysis carried out in [74].
These three parameters will also be carefully adjusted, if it is needed to do so, for the fitting process. The site density N 0 , σ, and F are kept fixed. All three scenarios were considered carefully. And within each one different concentrations for the fixed charges, interface and bulk traps (as well as their energy level) were investigated. It would be difficult to present all these results as we went through numerous parameter sets to achieve the best fit. Out of the three scenarios, we could not reproduce the transfer characteristics with SC3. Using SC1 or SC2, we managed to reproduce an almost perfect fit but only within a specific gate voltage range (between the off-state and up to −20 V of the applied gate voltage). The SC1 and SC2 fits are shown in figure 4, both on the linear and logarithmic scales. Beyond the −20 V mark both SC1 and SC2 drift away from the experimental measurement. By re-adjusting any (or all) of the three parameters within either SC1 or SC2, we were only able to fit the lower part of the transfer curve. This included different trap concentrations as well as their energy level within the band gap. Moreover, applying a GDOS in describing the trap energy as opposed to a single level definition did not bring us closer to a better fit. The only way we were able to fit the upper part of the transfer curve was actually using a fourth and different scenario which involved only the presence of FC  figure 4. This approach brought us to the conclusion that the applied gate voltage range is split into two distinct regimes, where the lower regime is described by considering all of FC, ITE (from SC1) or ITH (from SC2), and BTH. While the upper regime is appropriately described using only FC. This conclusion can be further consolidated by looking back at the analysis illustrated in figure 3. Each of ITE, ITH, BTH only affect the transfer curves substantially in the sub-threshold regime and at low gate voltage in the on-state of our transistor. Possibly, ITH can somehow to a slightly larger extent affect the transfer curves at higher gate voltages compared to the other two, but even if we further increase ITH either in concentration or energy level, we lose any fitting in the lower regime of the gate voltage. At high gate voltages the effect of all three is greatly diminished, as all trapped interface and/or bulk charges are freed and join the mobile charges in the channel leading to higher evaluated drain currents. The final simulation parameters used to achieve these fits are summarized in table 3. It is important to note that our starting values for both the trap energies and the µ 0 had to be re-calibrated accordingly to reach our final fits. After achieving a fit for the transfer curve, we then move on to simulating the output characteristics and once more draw out a comparison between our simulated results and the experimental measurements. As two scenarios were used for the fitting process (i.e. SC1/SC2 and FC), the same approach is taken for the output characteristics. Between SC1 and SC2, we decided to go forward with the latter to simulate the lower regime of the gate voltage. FC was used to simulate the upper regime respectively. Nevertheless, the reader will find the complete (i.e. covering the entire range) simulated output characteristics for each scenario provided in the supplementary information (see figure (S4)). Final output characteristics using SC2 and FC are depicted in figure 4(c). Quite clearly, with SC2 we successfully reproduced a perfect match for gate voltages of −10 V and −20 V. Using FC to simulate gate voltages of −30 V and −40 V did not yield a perfect fit but we were still able to reproduce the correct behaviour in both the linear and saturation regimes as well as the highest evaluated current respectively. Potential profiles across the channel region for the fitted output characteristics are given in figure (S5) of the supplementary information.

Extracting performance parameters
A crucial part of our investigation, once an agreement is achieved between the simulation and experimental results, we can then extract all important performance parameters of our transistor. These parameters include the V th , and the effective µ of our device. Naturally, based on the results presented in the previous sub-section extracting one value for each is not possible. This is due to the fact that we have two gate voltage regimes described by two different fitting parameterizations. Hence, when coming to extract both V th and µ, this depends on the operating conditions of our device. If it is to be operated at low gate voltages then parameterization SC2 is used, and vise versa FC if the device is operated at high gate voltages. We begin with extracting V th . Figure 5 shows the √ Id versus Vg relation. At the lower regime (i.e. using SC2), V th = −2.8 V, conversely at the higher regime (i.e. using FC) V th = +2.35 V. The latter coincides with the experimental value. But this difference in V th between high and low operating regimes is a consequence of the fixed charges and/or interface (bulk) traps. In the low regime V th highly depends on the interface and bulk traps, therefore a larger negative voltage is required to switch on the device. In the high regime, it is rather is solely controlled by the fixed charges present and it is shifted towards a more positive value.
A similar approach is taken in extracting the effective µ of our device, and this could easily be done by finding the slope of the linear fits from figure 5. Nevertheless, µ is a rather complex performance parameter that depends on several factors including the temperature, charge density, and the applied voltage (or electric field). We wanted to find it's dependence on the latter, hence we extracted the mobility dependence on the drain voltage (µ versus V d @ V g ). These dependencies are illustrated in figure 6, and as it can be seen the mobility decreases (rather slightly) as we transition from the linear to the saturation regime of our device. Using SC2, the mobility is between 1.14 × 10 −2 cm 2 V −1 s −1 -1.26 × 10 −2 cm 2 V −1 s −1 , and using  FC it is between 7 × 10 −4 cm 2 V −1 s −1 -1.2 × 10 −3 cm 2 V −1 s −1 .
Once again with FC we estimated a mobility closer to the experimental value of 4 × 10 −3 cm 2 V −1 s −1 .

Conclusion
We presented a DD model for the investigation and characterization of OFETs. This model included the GDOS taking into consideration the site density and energetic disorder commonly associated with organic materials, as well as the PF mobility for small molecules (the hopping mobility for polymers). The effects of fixed charges, interface traps and bulk traps (i.e. on the transfer characteristics) were investigated individually. Extending this analysis for the comparison with experimental measurements, we concluded that more than one scenario is required to achieve a good fit depending on the device operating conditions. Eventhough fixed charges are present, the effect of both interface and bulk traps were found to be significant at low gate voltages. Conversely, the currentvoltage characteristics are well described with the influence of fixed charges alone at high gate voltages. This was followed by the extraction of transistor performance parameters.