Compact DC Modeling of Organic Field-Effect Transistors - IEEE Xplore

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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 61, NO. 2, FEBRUARY 2014

Compact DC Modeling of Organic Field-Effect Transistors: Review and Perspectives Chang-Hyun Kim, Member, IEEE, Yvan Bonnassieux, and Gilles Horowitz

Abstract— In spite of impressive improvements achieved for organic field-effect transistors (OFETs), there is still a lack of theoretical understanding of their behaviors. Furthermore, it is challenging to develop a universal model that would cover a huge variety of materials and device structures available for state-ofthe-art OFETs. Nonetheless, currently there is a strong need for specific OFET compact models when device-to-system integration is an important issue. We briefly describe the most fundamental characters of organic semiconductors and OFETs, which set the bottom line dictating the requirement of an original model different from that of conventional inorganic devices. Along with an introduction to the principles of compact modeling for circuit simulation, a comparative analysis of the reported models is presented with an emphasis on their primary assumptions and applicability aspects. Critical points for advancing OFET compact models are discussed in consideration of the recent understanding of device physics. Index Terms— Circuit simulation, compact modeling, device physics, organic field-effect transistors (OFETs).

I. I NTRODUCTION

O

RGANIC electronics is a rapidly growing technological field that makes use of the semiconducting properties of small molecules and polymers for realizing flexible, lightweight electronics through cost-effective solution-based manufacturing techniques. Organic field-effect transistors (OFETs) are representative devices that form a basic building block for various microelectronic systems [1]. Primary applications of OFETs cover display backplanes, sensors, memories, actuators, radio-frequency identification systems, and so on [2]–[6]. As a result of extensive research efforts on molecular engineering, process development, and device optimization, recent reports show that both p- and n-type OFETs can outperform amorphous silicon (a-Si) thin-film transistors (TFTs) [7]–[9]. With moving demand from the device-level investigation to the circuit-level integration, specific compact models for OFETs gain significant importance. The term ‘compact model’ is generally used for mathematical descriptions that practically reproduce the electrical characteristics; various electronic devices are represented by the corresponding compact models in SPICE-type simulators [10]. Although the behavioral aspect Manuscript received June 18, 2013; revised August 19, 2013; accepted September 2, 2013. Date of publication September 25, 2013; date of current version January 20, 2014. The review of this paper was arranged by Editor M. J. Deen. The authors are with the Laboratoire de Physique des Interfaces et des Couches Minces (LPICM), École Polytechnique, CNRS, 91128 Palaiseau, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TED.2013.2281054

is an obvious emphasis, compact models should capture the essential device physics through correlating model parameters to physical parameters. However, a general lack of theoretical understanding and a wide range of material and structural variations make physics-based compact modeling of OFETs particularly a challenging task. As a consequence, many commercially available SPICE simulators do not currently contain a specific OFET model, and MOSFET or a-Si TFT models have been often adopted to simulate OFET characteristics [11]–[13]. In this article, we present a review on the progress and perspectives of OFET dc compact modeling. The fundamental characters of organic semiconductors and OFETs are described with their implications for device modeling. The principles of compact modeling are concisely discussed to address the necessity for realistic and applicable models. We revisit several representative models with a particular interest in their applicable range and different physical and empirical emphasis. Critical issues related to the existing models are assessed in terms of various key parameters with perspectives on further development toward more physically based models that have a maximum coverage of materials and structures. II. F UNDAMENTALS A. Semiconducting Molecular Solids An organic semiconductor contains carbon atoms linked by alternating single–double bonds in its molecular structure [14]. These π-conjugation units result from weakly bound π-electrons that are delocalized throughout the conjugation blocks. The highest-occupied molecular orbital (HOMO) and lowest-unoccupied molecular orbital (LUMO) of this type of material are separated by a small energetic gap reflecting weak π bonds, which dictates the semiconducting properties of organic molecules. Fig. 1(a) is the chemical structure of pentacene, a prototypical p-type organic semiconductor. Small-molecular semiconductors can be densely packed in a highly periodic manner to form macroscopic single crystals. Fig. 1(b) shows the molecular arrangement in a pentacene crystal. Even though the chemical bonds that build a molecule are covalent bonds, the molecule–molecule interactions are based on weak van der Waals force. This is at variance with crystalline inorganic semiconductors entirely constructed by strong covalent bonds. This most fundamental bonding picture of organic solids implies both new possibilities and ultimate limitations as compared with conventional Si-based electronics. Most importantly, weak intermolecular forces require low chemical or

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KIM et al.: COMPACT DC MODELING OF OFETs

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Fig. 2. Different OFET geometries determined by the order of stacks and relative positions of contact electrodes. G: gate. S: source. D: drain.

Fig. 1. (a) Chemical structure of pentacene. (b) Pentacene molecular arrangement in single crystal phase. (c) Typical AFM image of vacuumevaporated pentacene thin film showing polycrystalline configuration (scan size: 2 × 2 μm2 ).

thermal budget for material processing, enabling dispersion of molecules in organic solvents or evaporation of molecular powders at relatively low temperature. However, weak electronic coupling between molecules results in less efficient charge transport and imposes severe limitations to the electrical performance of organic-based optoelectronic devices.

doping level of ∼1014 cm−3 [28], while a solution-processed poly(3-hexylthiophene) film showed ∼1016 cm−3 [29]. Low density of unintentional doping has a critical implication for understanding and modeling organic devices. Even a small built-in potential can fully deplete a thin organic layer so that there is no effective contribution of thermally generated charge carriers from the organic material itself [28], [30]. Rather, the device current is predominantly carried by injected charge carriers. This aspect considerably simplifies the electrostatics upon modeling the charge distribution in an organic electronic device. Also, the role of charge-injecting contacts becomes significant [31], [32].

B. Structural and Energetic Disorder Organic single crystals obtained by controlled growth are advantageous for theoretical studies of genuine electronic properties of a given material [15]. More generally, thermally evaporated small-molecules or solution-cast polymers develop complex process-dependent microstructures including amorphous, polycrystalline, and partially crystallized phases [16]–[19]. The atomic-force microscope (AFM) image of a vacuum-evaporated pentacene film in Fig. 1(c) shows the polycrystalline structure composed of a number of domains with different shapes and sizes. A long-range crystal-packing order of Fig. 1(b) is not expected in such a polycrystalline film, mainly due to noticeable discontinuity at grain boundaries. Structural disorder also introduces perturbation in the energetic landscape, and gives rise to the creation of charge-trapping localized states [20]–[22]. The existence of impurities and gradual formation of degradation-related species are other major origins of the structural and energetic disorder in organic semiconductor films [23], [24]. As a result, modeling charge transport in organic semiconductors is further complicated, and the performance of organic devices tends to be seriously affected by charge-localization effects. C. Doping Except for applications in which a deliberate tuning of electrical properties is necessary [25], organic semiconductors are used without controlled doping. Nonetheless, especially when films are processed without a purification step, small amounts of impurities due to chemical residues or contamination can unintentionally dope the molecular films. In addition, adsorbed oxygen molecules serve as unintentional dopants, which is monitored by an initial increase in conductivity of p-type organic semiconductors exposed to the ambient air [26], [27]. Unintentional dopants are expected to generate only moderate number of carriers; an evaporated pentacene film showed a

D. OFET Geometries Fig. 2. shows possible OFET structures according to the primary position of the gate electrode and the relative position of the source/drain electrodes with respect to the semiconducting layer. In addition, it is possible to group two of the four geometries as staggered and the others as coplanar OFETs; coplanar OFETs have the conducting channel on the same plane with the source/drain electrodes, whereas the channel is separated from the source/drain electrodes by the semiconductor in staggered structures [33]. It should be noted that the source/drain electrodes form direct metal/semiconductor junctions with the organic film in OFET structures. This is another major difference with MOSFETs or a-Si TFTs in which the source/drain regions contain locally doped high-carrier-density zones [34]. Due to a high defect-state density, the metal Fermi-level tends to be pinned within the HOMO–LUMO energy gap, resulting in a substantial charge-injection barrier at most realistic metal/organic interfaces [35]. This injection problem is directly reflected on the transistor characteristics as a contact resistance (Rc ), which is a key element in OFET modeling because Rc can be comparable with or even higher than the connected channel resistance (Rch ) especially in short-channel devices [32]. III. P RINCIPLES OF C OMPACT M ODELING To address the principles of compact modeling approach, we revisit the simplified MOSFET-based current-voltage (I –V ) equations usually adopted for experimental OFET analysis. In the transistor structure shown in Fig. 3, we assume p-type FET behavior; (VGS − VT ) < 0 V for accumulating holes in the channel and VDS < 0 V for normal biasing. Here, VGS is the gate-to-source voltage, VDS is the drain-tosource voltage, and VT is the threshold voltage of the OFET. When the magnitude of applied VDS is low, VDS is uniformly

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Fig. 3. Illustration for deriving I –V characteristics of OFETs based on the gradual channel approximation. The condition (VGS − VT ) < VDS < 0 V is assumed for normal p-type linear-regime operation.

distributed along the channel as shown by V (x) in Fig. 3. A conducting channel is formed by capacitive effect, which means that the induced hole charge density per unit area Q(x) is proportional to the local effective voltage across the gate capacitance Q(x) = −Ci [(VGS − VT ) − V (x)]

(1)

where Ci is the insulator capacitance per unit area. The linearregime drain-to-source current Ilin is estimated by drift current of the gate-induced holes dV Ilin = Q(x)μ (2) W dx which can be regarded as Ohm’s law with the current density and conductivity multiplied by the y-thickness of the current path [36]. Here, W is the channel width and μ is the hole mobility. We separate the variables and estimate the integrals as follows: VDS L Ilin dx = Q(x)μd V (3) 0 W 0 where the channel length L appears. By inserting (1) into (3), we arrive at 2 VDS W . (4) Ilin = − μCi (VGS − VT )VDS − L 2 When |VDS | increases to meet VDS = (VGS − VT ), the channel is pinched off at drain [Q(L) = 0], and we assume that there is no more increase in current by VDS . The saturation-regime drain-to-source current Isat is obtained by replacing VDS by (VGS − VT ) in (4) W μCi (VGS − VT )2 . (5) 2L The first principle of compact modeling is to include realistic device behaviors. Equations (4) and (5) are widely used as a characterization tool because they allow easy access to μ and VT . However, realistic OFETs do not exhibit ideal behaviors shown √ in Fig. 4(a) and (b); perfectly linear dependence of Ilin and |Isat | on VGS , and absence of subthreshold conduction. Related assumptions under (4) and (5) are constant Isat = −

Fig. 4. Calculated I –V characteristics by (4) and (5). (a) Linear-regime transfer characteristic. (b) Saturation-regime transfer characteristic. (c) Output characteristics. Parameters are summarized on the right-hand side of (a).

μ, absence of contact resistance, and complete subthreshold switchoff. For capturing realistic device characteristics, compact models should go beyond these simplified conditions and embrace various effects with relevant physical bases. The second principle concerns applicability. Note that (4) and (5) are discrete models valid only in limited voltage ranges. For a circuit simulation to reach an effective convergence, devices need to be described by equations that have a maximum applicable range [10]. As will be shown in Section IV, for OFET compact models, this objective is generally fulfilled by modeling the smooth linear-to-saturation transition and belowto-above threshold transition. These principles underline the necessity of developing compact models for circuit simulation. More rigorous requirements of compact models, which indicate the criteria for assessment, are discussed in [37]. IV. C OMPARISON OF D IFFERENT M ODELS From the literature, we selected representative OFET compact models that are consistent with the major principles defined above. We will concisely present the stepwise development of the model equations and discuss distinctive features. All models will be written for p-type operation with intentional modifications on notations for clarity and ease of comparison. Useful information about the sign conversion rule for p- and ntype transistors and the relationship between device and circuit nodal convention can be found in [38]. A. Model by Marinov et al. The development of this model is described in [37] and [39]. In particular, this model provides a practical below-to-above threshold transition and linear-to-saturation transition by a small number of parameters. The model also accounts for the


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exponential dependence of the subthreshold current on gate voltage. Instead of using conventional VGS and VDS , the authors use independent electrode voltage as VG , V D , and VS . This requires the expression of hole charge density Q Mar (x) as follows: (6) Q Mar (x) = −Ci [(VG − VT ) − V (x)] where V (0) = VS and V (L) = V D . The mobility is assumed to follow a power-law dependence featuring the characteristic exponent γ μ0 (7) μMar (x) = γ {−[(VG − VT ) − V (x)]}γ Vaa where μ0 is the unit conversion factor that we will set as 1 cm2 /V·s, and Vaa is the mobility enhancement voltage. The choice of a power-law mobility is related to the charge-transport through localized states that have exponential density-of-states (DOS). Both the mobility prefactor and characteristic exponent reflect the energetic distribution of DOS and charge-transport mechanism. Extensive physical correlation is found in [40], in which the analysis of various materials also shows realistic range of those parameters. gen The generic TFT model IMar starts from

Fig. 5. Calculated I –V characteristics by the model by Marinov et al. (10) and (12). (a) Transfer characteristic at V D = −1 V. (b) Transfer characteristic at V D = −30 V. (c) Output characteristics. Parameters are summarized on the right-hand side of (a).

gen

IMar dV = Q Mar (x)μMar (x) W dx and the integrals to be calculated are VD L gen IMar dx = Q Mar (x)μMar (x)d V. W 0 VS

(8)

(9)

captured. The authors additionally suggested a replacement of VS in (10) or (12) by the contact voltage drop Vc to include contact resistance effect [37], [39], [42]. However, the term Vc itself depends on the current, and as a result, a fully analytical expression for current is not achieved when γ > 0.

By inserting (6) and (7) into (9), we get W μ0 Ci L Vaaγ {−[(VG − VT ) − V D ]}γ +2 − {−[(VG − VT ) − VS ]}γ +2 . × γ +2 (10) gen

IMar =

The linear-to-saturation transition and below-to-above threshold transition are simultaneously enabled by introducing the effective gate overdrive function (VG − VT ) − V f (VG , V ) = VSS ln 1 + exp − (11) VSS where VSS determines the slope of exponential subthreshold com can be written as current. The final compact TFT model IMar W μ0 = γ Ci L Vaa [ f (VG , V D )]γ +2 − [ f (VG , VS )]γ +2 ×(1+λ|V D − VS |) × γ +2 (12) com IMar

where λ models an imperfect saturation in the output curves. Fig. 5 shows typical I –V characteristics drawn by (10) and (12). The model well describes the upward bending of transfer curves on a linear scale by gate-voltage enhancement of mobility, which is often encountered in experimental results [41]. Furthermore, essential realistic transition behaviors are

B. Model by Estrada et al. This model is based on the so-called unified model and parameter extraction method (UMEM) approach [43], [44]. This model also puts a strong emphasis on the use of a powerlaw relationship between mobility and gate overdrive voltage. A major characteristic is an independent definition of off-state, subthreshold, above-threshold equations that are merged into a single expression by external transition functions. There is no need to build characteristic integrals for this model, and we can directly start from (4). By neglecting 2 /2 term (low |V | assumption), and by including a the VDS DS contact voltage drop in (4), we write the Rc -containing linear R as regime current Ilin R =− Ilin

W R μCi (VGS − VT )(VDS − Ilin Rc ) . L

(13)

R into This can be arranged for Ilin R = Ilin

−W L μCi (VGS − VT )VDS 1−

W L μCi Rc (VGS

− VT )

The power-law mobility is assumed as μ0 μEst = γ [−(VGS − VT )]γ . Vaa

.

(14)

(15)

Note that μEst only supports the gate-voltage dependence, whereas μMar in (7) basically allows for spatial variation.

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Further analysis of this mobility model based on hopping transport through exponential DOS is put forward in [45]–[47]. Effective drain voltage VDSe is defined as m − m1 VDS (16) VDSe = VDS × 1 + αs (VGS − VT ) where m controls the abruptness of linear-to-saturation transition, and αs models the deviation of the saturation drain voltage from the ideal (VGS − VT ) point. The above-threshold above is obtained by replacing μ by (15), V current IEst DS by (16), and by adding a λ term in (14) above IEst

=

W μ0 γ +1 γ C i [−(VGS − VT )] L Vaa μ0 γ +1 1+ W γ C i Rc [−(VGS − VT )] L Vaa m − m1 VDS ×VDS 1 + αs (VGS − VT )

× (1 − λVDS ). (17)

Note that a modified asymptotic expression is proposed in [44] for improved output conductance fitting; however, here we choose the basic expression (1 − λVDS ) that allows a comparison of extracted λ values by different models on an equal base. The subthreshold current is defined as a simple exponential function ln 10 sub (VGS − V0 ) = I0 exp − (18) IEst S where S is the subthreshold swing, V0 is the onset voltage, com is and I0 is the off current. The final compact model IEst constructed by using the following transition functions: 1 com above IEst = IEst × 1 − tanh[B(VGS − V B )] 2 1 sub × 1 + tanh[B(VGS − V B )] + I0 (19) +IEst 2 where the transition voltage V B and the transition parameter B are introduced for accurately modeling the below-to-above threshold transition. The calculated transfer curve in Fig. 6(a) shows how the different regimes are connected by the transition method in (19). This model enables a fine tuning of transition behaviors as indicated in Fig. 6(b) and (c), which requires a larger number of parameters compared with the model by Marinov et al. It is also inferred that this model has a significant flexibility toward various possible expansions or modifications. Central parameters such as μ, Rc , S are kept explicit and independent, which implies that they can be later replaced by a new model defined as a function of physical parameters and bias conditions. C. Model by Li et al. This model described in [48] considers the potential barrier between grains in a polycrystalline organic film. An analytical expression for gate-voltage-dependent mobility is empirically derived from a numerical estimation of the simultaneous change in the barrier height, free carrier density, and trapped carrier density.

Fig. 6. Calculated I –V characteristics by the model by Estrada et al. (19). (a) Transfer characteristic at VDS = −1 V shown with the independent current elements. Parameters are summarized on the right-hand side. (b) and (c) Model-calculated curves showing the effect of transition parameters B and m. The other parameters are kept the same as those in (a).

The model is derived starting again from (4), where we do 2 /2 term, and reorganize (4) into not drop the VDS

W μCi (VGS − VT )2 − (VGS − VT − VDS )2 . (20) 2L The characteristic mobility function is

s (21) μLi = μig exp VGS − κ Ilin = −

where μig is the so-called intragrain mobility, and s and κ are empirical parameters. It is expected that approximate correlation of s and κ to DOS parameters can be found by simultaneously examining the effect of variation of physical and empirical parameters on the mobility value. The final com is obtained by replacing μ by (21), V compact model ILi DS by (16), VT by Von in (20), and by adding a λ term

W s com Ci ILi = − μig exp 2L VGS − κ ⎧ ⎨ × (VGS − Von )2 ⎩ ⎫ m − m1 2 ⎬ VDS − VGS −Von −VDS 1+ ⎭ αs (VGS − Von ) ×(1 − λVDS ).

(22)

We note that the effective drain voltage in the original paper [48] has a different form but is equivalent to (16). The authors fixed m as 10 and did not identify αs as a fitting parameter (αs = 1 assumed). Nonetheless, here we suggest the formula


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Fig. 7. Calculated I –V characteristics by the model by Li et al. (22). (a) Transfer characteristics at VDS = −1 V and VDS = −30 V with the model parameters summarized in the lower panel. (b) Output characteristics calculated by the same model and parameters with a variation of the αs value.

(22) with m and αs as free parameters for improved fitting capability. Fig. 7 shows representative characteristics calculated by (22). A main feature of this model is that there is no distinction between subthreshold and above-threshold regime. The authors used Von to cover the whole range of transfer characteristics. The near-Von regime (conventional subthreshold regime) does not have a strictly exponential dependence on VGS . As a result of this merged description, Von should be defined within conventional deep-subthreshold or off-state regimes [Fig. 7(a)]. In turn, the saturation voltage αs (VGS − Von ) in (22) is overestimated and there is a need to compensate this by inserting αs < 1 as indicated in Fig. 7(b). D. Application to Experimental Data To examine the fitting capability and compare extracted parameters, we applied the three models to the same set of experimental data. Note that there can be significant interdependency between extracted μ and Rc values because the I –V characteristics of a single device can only provide the series sum of Rch and Rc [49]. Transmission-line method on multiple channel lengths [32] or direct potential mapping along the source-drain path [50] are desirable experimental techniques for separately and simultaneously extracting the voltage-dependent Rch and Rc . Here, to focus on the mobility behavior given by different models, we choose a low-Rc sample and assume that Rc is zero. Fig. 8 shows the fitting results with the extracted parameters. In terms of adjustment precision, all models satisfactorily reproduced the measured data. It is reasonable to infer that, when the data do not exhibit particularly nonideal behaviors, there is a large room for the choice of a model. Such nontrivial but often observed characteristics may include diode-like curvature in output curves, pronounced off-state leakage, strong VDS -dependence of the subthreshold current, and so on [13], [51]–[55]. This comparison result indicates that underlying physical assumption gains more importance to check the reliability of a model, and the fitting capability is to be

Fig. 8. Calculated I –V characteristics in comparison with the experimental data on a pentacene OFET. The lower panel in each model summarizes the model parameters. The inset of (a) and (b) shows the device structure and the photograph of the flexible pentacene OFET arrays, respectively. The experimental data are taken from [44], in which the fabrication details are also available.

examined by application to various organic semiconductors and different device configurations. Fig. 9 shows the mobility calculated by each of the three models [(7), (15), and (21)] into which the extracted parameters in Fig. 8 are inserted; Fig. 9(a) allows comparison of the mobility values at the same gate bias points and Fig. 9(b) traces the characteristic shapes of the mobility versus

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Fig. 9. Comparison of the mobility behavior extracted from the I –V data in Fig. 8 by different models. (a) Mobility as a function of gate voltage on a linear scale. (b) Mobility as a function of characteristic effective gate overdrive voltage of each model on a log–log scale.

effective gate overdrive voltage. Upon applying the so-called H -function integral method, the linear-regime transfer curve (VDS = −2 V) was used for the model by Estrada et al. and the saturation–regime transfer curve (VDS = −50 V) was used for the model by Marinov et al. as suggested in [39] and [43] respectively. This results in the difference in VT and γ values by these two models, although they assume similar powerlaw mobility. The characteristic nonpower-law behavior in the model by Li et al. leads to a curve on a log–log scale. V. I SSUES , C HALLENGES , AND P ERSPECTIVES As we reviewed in the previous section, the reported models already seem to have a fairly good fitting capability for describing realistic electrical characteristics of OFETs. However, many critical device phenomena are not fully integrated or often simplified, which may limit the application of a model to OFETs that inherently feature a large performance variation due to the virtually unlimited range of materials and structures. We discuss here the issues and challenges of the current state of OFET models, and outline the perspectives based on the recent progress of device physics. A. Threshold Voltage and Subthreshold Current Modeling Threshold voltage model is not well established and it appears as a constant parameter in many compact models. The definition of VT in OFETs should differ from that in MOSFETs because OFETs basically operate in accumulation mode. The primary consideration for channel creation is the flat-band voltage VFB . Fig. 10 is an illustration of the energy diagram from the gate electrode to the source electrode in an OFET. In an inorganic MOS capacitor, VFB is defined from the difference between the gate Fermi level and the semiconductor bulk Fermi level. In contrast, as shown in Fig. 10(b), VFB in OFETs is defined from the work function difference between gate and source electrode. It is because a small built-in potential is sufficient to fully deplete an unintentionally doped thin organic film, so that VGS variations do not only influence the insulator/semiconductor interface, but also spread throughout the entire semiconductor thickness [28], [31]. As a consequence of full depletion, both the gate insulator and the organic film have the straight potential

Fig. 10. Energy diagrams for conceptual representation of the different regimes of an OFET. A low hole-injection barrier at the source/organic interface is assumed to predict p-type transistor behavior. The gate electrode is assumed to have a lower work function than the source electrode so that VFB < 0 V.

profiles, so that the VGS > VFB regime in Fig. 10(a) can be called dielectric regime. In this idealized picture, VT is equivalent to VFB , because the energetic of VGS < VFB in Fig. 10(c) favors the movement of injected holes toward the semiconductor/insulator interface. However, traps result in additional contribution; effective VT should include the gate voltage needed to fill sufficient traps before an accumulation layer of mobile carriers is established [56]. In addition, it is expected that the contact configuration has a direct impact on VT because the position of the source electrode determines the effective distance through which injected carriers travel to reach the semiconductor/insulator interface (see Fig. 2) [57]. By definition, once the value of VT is determined, any conduction at VGS > VT (p-type) is regarded as subthreshold current. As shown in Sections IV-A and IV-B, the VGS -dependence of subthreshold current is generally modeled by an empirical exponential function. This is based on the MOSFET theory that accounts for the subthreshold conduction as a diffusion current. A description of the diffusion component in OFETs is proposed in [58]. There is a strong need to develop a physical model for the subthreshold current in OFETs, which possibly considers a contribution of the drift current of bulk carriers, hump-like characteristics related to the interface traps, and so on [55], [59], [60]. Such complex behaviors cannot be described by a simple exponential model. B. Mobility Modeling Charge localization effect due to disorder leads to VGS -dependent mobility in OFETs [61]. The power-law mobility similar to (7) or (15) is strongly based on the assumption of exponential DOS; both the multiple trapping and release (MTR) theory and the hopping theory dictate a power-law mobility with the characteristic exponent reflecting the width of DOS [61], [62]. On the other hand, it has been shown that a Gaussian DOS is suitable for describing intrinsic energetic distribution of states in a disordered organic semiconductor [63], [64]. The characteristic carrier-density-dependence of hopping mobility in Gaussian DOS is expected to be directly linked to VGS -dependent mobility in OFETs. Unfortunately, consideration of purely Gaussian DOS does not lead to an analytical expression for mobility. As an alternative approach, an approximate analytical function that mimics the numerical


models based on a Gaussian DOS can provide a way to build a semiempirical mobility model [65], [66]. Electric field dependence of mobility is characteristic of hopping-based transport in disordered media and is to be interpreted as VDS -dependence in OFETs. So far, this effect is less emphasized in compact models compared with gate-voltage dependence partly due to the fact that the variation of lateral drain field remains small in long channel devices. Therefore, the future development should take the field-dependence into account as the fabrication technology keeps downscaling the channel dimension. C. Contact-Resistance Modeling Contact resistance is often described as a sum of source and drain resistances of equal magnitude [13], [67]. However, it is physically reasonable to infer that the injecting contact contributes more significantly and the contact voltage drop mostly occurs at the source electrode [50]. In the case of staggered OFETs, the bulk film resistance is a main contribution to the parasitic series resistance because charge carriers have to transport across the semiconductor layer to reach the conducting channel. In contrast, Rc in coplanar OFETs is directly influenced by the charge-injection barrier because of the intimate contact between the source electrode and the conducting channel [68]. Because of this strong geometrical implication, modeling Rc in the literature has been generally dedicated to only one type of contact configuration. Nonetheless, a common characteristic is strong VGS -dependence of Rc as often observed in experimental studies. Therefore, modeling contact effects by including VGS -dependence is of significant importance. Several theoretical models were recently proposed with different approaches [32], [67], [69], [70]. VI. C ONCLUSION We have reviewed the development of compact dc modeling of OFETs. In reported models, possibly extensive physical theories have been simplified and packed into an analytically accessible level. It is found that each model has different emphasis on a characteristic mobility behavior and empirical transition methods. Even though a mature-level description of the device physics is still not established, many fundamental aspects have been recently revealed, and corresponding development of physics-based compact models is under continuous progress. Because of the large variation in OFET materials and structures, it is believed that further modeling efforts will be oriented toward the capability of rearrangement and flexibility for capturing main trends in device characteristics. R EFERENCES [1] G. Gelinck, P. Heremans, K. Nomoto, and T. D. Anthopoulos, “Organic transistors in optical displays and microelectronic applications,” Adv. Mater., vol. 22, no. 34, pp. 3778–3798, Sep. 2010. [2] S. Steudel, K. Myny, S. Schols, P. Vicca, S. Smout, A. Tripathi, B. van der Putten, J.-L. van der Steen, M. van Neer, F. Schütze, O. R. Hild, E. van Veenendaal, P. van Lieshout, M. van Mil, J. Genoe, G. Gelinck, and P. Heremans, “Design and realization of a flexible QQVGA AMOLED display with organic TFTs,” Org. Electron., vol. 13, no. 9, pp. 1729–1735, Sep. 2012.

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Yvan Bonnassieux received the Ph.D. degree from École Normale Supérieure de Cachan, Cachan, France, in 1998. He is currently a Professor with École Polytechnique, Palaiseau, France, where he is the Head of the Organic Electronics Research Team.

Chang-Hyun Kim (S’11–M’13) received the Ph.D. degree from École Polytechnique, Palaiseau, France, in 2013. He is currently a Post-Doctoral Research Associate with École Polytechnique.

Gilles Horowitz received the Ph.D. degree from Université Paris Diderot, Paris, France, in 1975. He is a Senior Research Fellow with the Centre National de la Recherche Scientifique (CNRS) and is currently with École Polytechnique, Palaiseau, France.