In recent years, organic field effect transistors (OFET) have triggered intensive research for their potential applications in large-area, low-cost and flexible integrated circuits. Perhaps, in the future, it will be feasible to ‘print’ disposable organic circuits in a similar fashion as printing on paper. Discrete OFETs are usually used for research, however, in commercial applications, OFETs generally need to be integrated. Compact models are important in device simulators to predict and optimize the performance of organic integrated circuits [1–3].

Several dynamic current models have been developed and verified by experimental data [1, 3–5]. Torricelli et al [1] derived an analytic I–V model for OFETs based on a density-dependent mobility model. Marinov et al [3] proposed a compact model for the quasistatic charge and capacitances in OFET, which was implemented for the simulation of organic circuits. Chang et al [4] proposed a theoretical description of the charge distribution and the contact resistance in coplanar OFETs and derived an analytical formulation for the charge distribution inside the organic layer in coplanar OFETs. Tejada et al [5] proposed a compact model for the current–voltage characteristics at the contact region that unified different trends found in experiment data. Basile et al [6] presented a numerical model of charge transport in organic semiconductors, the model was also applied to OFETs fabricated on SiO₂ substrates. Soon after, Basile and Fraboni [7] extended their model [6] to calculate the I–V curves of single crystal rubrene OFETs, where band-like transport occurred [8].

The mobility model is a foundation for organic device modelling. Although there are many different kinds of mobility models in the literature, only a few of them have been widely accepted and applied. The exponential mobility model first proposed by Pai [9] is the most popular one (the exponential mobility model). Pai [9] has ever proposed that the mobility of amorphous semiconductor is an exponential function μ(F_x) of the root of electric field F_xwith two temperature-dependent parameters μ₀ and γ determined by fitting experimental data, where x represents the direction perpendicular to the current flow. Dunlap et al [10] analytically derived the expression of mobility for organic semiconductors, which confirmed the exponential model of Pai [9] and obtained mobility in quantitative agreement with what is observed experimentally. Subsequently, Blom et al [11] and Novikov et al [12] proposed different temperature-dependent expressions for parameters in the exponential model.

The second widely accepted mobility model is derived by Vissenberg and Matters (the VM model) [13], which is based on the variable-range hopping transport theory and exponential density of state (DOS), and expressed as a power function μ(n) of density n. Although the VM model can fit the density-dependent carrier mobility data measured by Tanase et al [14] at high density, Torricelli et al [15] pointed out that the VM model can not fit I–V data of organic diodes unless introducing electric field dependent effective temperature. Torricelli et al developed analytic I–V formulae for OFETs based on a modified VM model [16, 17], but they did not consider electric field-dependent effective temperature, so the applicability of resulting formulae would be limited.

The third popular mobility model is proposed by Pasveer et al 18 (the Pasveer model). Pasveer et al fitted the numerical results of the master equation by the factorizing parametrization scheme. The mobility was expressed as a function μ(F_x, n) of both electric field F_x and density n of carriers. Although the Pasveer model can fit well I–V data of some organic diodes [18], the application of the Pasveer model and the exponential model to single-crystal rubrene organic diodes has been presented in our previous literature [19]. The results [19, 20] showed that the exponential mobility model could well describe I–V data of single-crystal rubrene, moreover, the extracted parameters showed correct temperature dependence and the extracted barriers were non-symmetric. However, the model of Pasveer et al [8] failed, and some parameters showed inconsistent temperature dependence which should be constants in the theoretical framework of Pasveer et al [19]. The Pasveer model was conceived for disordered organic semiconductors, hence its application to a highly ordered system such as a single crystal is not meaningful.

In the last popular mobility model [2–5, 21, 22], the mobility is expressed as a power function of the gate-source (the power-function mobility model)Although this is a semi-empirical one, it is very popular for the resulting I–V formulae is the simplest for OFETswhere C_i is the gate insulator capacitance per unit area. However, equation (2) is not realistic, the more realistic version should be expressed as a power function of electric fieldEquation (4) is more realistic than equation (2) as equation (2) is merely a simple approximation of equation (4). The I–V formulae for OFETs based on equation (3) were unavailable in literature, and the temperature dependence of two parameters extracted from equations (1) and (3) should be different, but this is important for the explanation of physical meanings and device modelling.

There are other mobility models in the literature, just not as popular as those mentioned above. For example, the ‘mobility edge’ (ME) model was first used to depict organic materials by Horowitz and Delannoy [23] and advocated by Salleo et al [24]. The model proposed by Oelerich et al [25] is based on the ‘transport energy concept’ and variable-range hopping transport theory. These models are complicated and it is impossible to derive analytic I–V formulae for both organic diodes and OFETs, so we would not consider them in this work.

As we all know, although there are basic I–V formulae of OFETs available in the literature, it is too difficult to derive analytic I–V formulae based on the general mobility model μ(F_x, n) for a complicated two-fold integral. Inspired by the above findings, we reformulated the basic I–V formulae as a simple and convenient form for numerical calculations of any mobility model μ(F_x, n), and which could easily deduce analytic I–V formulae for simple popular mobility model μ(F_x, n). Based on the reformulated I–V formulae, two analytic I–V formulae based on the exponential and power-function mobility models had been derived, and they were applied to three OFET devices with different materials. Moreover, the temperature dependence of model parameters was analyzed, and simple relationships proposed by Blom et al [11] were confirmed for both models.

In the following, for the sake of simplicity, the I–V formula will be deduced for n-type OFETs only. The integral expression of current and voltage of OFET is as follows [16, 17]where W is the channel width, L is the channel length, V_S and V_D are the source and the drain voltages, V_ch is the channel potential (or the pseudo Fermi energy), φ is the electrostatic potential, φ_s is the surface potential at the insulator-semiconductor interface. q is the charge of an electron, n is the density of electrons, and μ is the mobility of electrons.

μ is generally a function μ(T, F_x, n) of temperature T, electric field F_x and density n of carriers. However, for general expressions of μ, equation (5) is difficult to evaluate, so it is necessary to reformulate this expression as a convenient form that can be easily substituted into arbitrary mobility function μ(T, F_x, n). If assuming the gradient channel approximation, ${F}_{x}\gg {F}_{y},$ the Poisson equation ${{\rm{\nabla }}}^{2}\varphi =-\left(\partial {F}_{x}/\partial x+\partial {F}_{y}/\partial y\right)=\rho /\varepsilon $ can be simplified as ${\partial }^{2}\varphi /\partial {x}^{2}=\rho /\varepsilon =qn/\varepsilon .$ With the aid of the electric field ${F}_{x}=-\partial \varphi /\partial x,$ the Poisson equation can be changed as [16, 17]Assuming the Boltzmann statistics for electronsthe electric field can be integrated asAnd applying the Gauss’ law to the semiconductor-insulator interface, one can obtain the surface potential φ_s [16, 17]where C_i is gate insulator capacitance per unit area, V_G and V_FB are the gate voltage and flat-band voltage respectively. Substituting (9) in (8), the equation for solving surface potential is obtained The iterative solution of (11) may be avoided as φ_s is very well approximated by the simplified equation [16, 17] in the triode region, i.e. when the channel is completely formed, and by in the saturation region, i.e. when the channel is pinched off.

By using equations (7) and (8), the expression of the drain current (5) can be rewritten asBecause the integrand for φ would diverge as φ approaching zero, equation (16) is also inconvenient both for numerical calculations and analytic derivations. We thus propose alternative variable ξ instead of the variable φ.Substitution of equation (17) into equations (7) and (8) yields Combining equations (17–19), the current equation (16) can be changed to following form In Equations (20, 21), ξ should be determined by substituting equation (10) into equation (17)

In this section, analytic I–V expressions for two mobility models are derived by using those reformulated equations (20–22). The first mobility model is the power-function mobility model [2–5, 21, 22] in equation (4). Firstly, at the low bias voltage, ${V}_{{\rm{D}}}\lt {V}_{{\rm{G}}}-{V}_{\mathrm{FB}},$ by using equations (4) and (19), equation (20) is changed to the following formBy using equations (12, 22), equation (23) can be changed as follows Secondly, at the high bias voltage, ${V}_{{\rm{D}}}\geqslant {V}_{{\rm{G}}}-{V}_{\mathrm{FB}},$ by using equations (14) and (19), equation (21) is changed as followsBy using equations (12, 22), equation (26) can be changed as follows If comparing equation (25) with (3) proposed in references [2–5, 21, 22], it can be seen that their forms are very different. First, equation (25) is more reasonable than equation (3), and it is more valuable for practical applications. Because the material or device parameters extracted by the two equations may be different. The parameters extracted by equation (25) should be more reasonable than that by equation (3), for equation (4) is more realistic than equation (2). Second, the I–V properties of OFETs predicted by equation (3) and real material or device parameters may be incorrect, which is a severe limitation to device modelling. Moreover, the temperature variations of parameters of equations (25) and (3) should be different. Our following results show that temperature variations of parameters extracted from equation (25) are very simple as that in the exponential model with some theoretical foundation. Whereas, temperature variations of parameters extracted from equation (3) may be obfuscated and more complicated.

We also can derive analytic I–V expression for the exponential model in equation (1). Firstly, at the low bias voltage, ${V}_{{\rm{D}}}\lt {V}_{{\rm{G}}}-{V}_{\mathrm{FB}},$ by using equations (1) and (19), equation (20) is changed as followsBy using equations (12), (22), (29) can be changed as followsIf introducing following new variables,Equation (30) can be evaluated as Secondly, at the high bias voltage, ${V}_{{\rm{D}}}\geqslant {V}_{{\rm{G}}}-{V}_{\mathrm{FB}},$ by using equations (1) and (19), (21) is changed as followsBy the same procedure as equations (30–33), (34) can be evaluated asIf comparing equation (35) with (33), it can be seen that equation (35) can be seen as a special case of equation (33) as u_D = 0.

In order to verify the practicability of analytic I–V formulae derived above, we apply them to three different devices, a pentacene OFET with a 10 μm channel length and a 20 000 μm channel width [1], a poly (2,5-thienylene vinylene) (PTV) OFET with 10 μm channel length and 20 000 μm channel width [1], and a bottom gate pentacene (BGP) based OFET with 100 μm channel length and 700 μm channel width [26]. Device parameters are taken from references [1 and 26], and listed in tables 1–4. The optimized parameters are listed in tables 1 and 2 for the exponential model; tables 3 and 4 for the power model. In addition to some fixed parameters, such as W, L, ϵ, T, V_FB and C_i that vary with different devices, each mobility model has different values of parameters μ₀ and γ. For pentacene and PTV OFETs, the experimental data involves five temperatures. It is inconvenient to list all optimized parameters in tables 1 and 3, so we separately list the parameters for pentacene and PTV OFETs in tables 2 and 4.

The theoretical I–V curves for three OFETs are compared with experimental points in figures 1–3. Symbols represent experimental data, while lines represent the theoretical data of the two mobility models mentioned above. Figure 1 shows the transfer characteristics I_D (V_GS) of a pentacene OFET [1] at different drain voltages and temperatures. The theoretical curves of the two models fit well with the experimental points in figure 1(a), while in figure 1(b), the power model gives better results. In figure 2, the transfer characteristics of a PTV OFET at different drain voltages and temperatures are shown. Similarly, for the exponential model and power model, a very good agreement between the analytical model and the experimental results is obtained. These figures show that the theoretical curves match experimental I–V points satisfactorily except for the pentacene OFET in figure 1(b).

Figure 3 shows the I_D–V_DS curves (V_GS = −10, −20, −30 and −40) of a BGP OFET [26] at 300 K, 325 K, and 350 K. Curves for two mobility models have achieved good matching results for this material, and curves of the exponential model also performed well. These results confirm the accuracy of the analytic I–V expressions derived above.

Because variations μ₀ and γ of the exponential model and the power model are temperature-dependent, the variations of μ₀ and γ with T are analyzed based on the data in tables 2 and 4 and plotted in figures 4 and 5. Figures 4(a) and (b) exhibit the relationship of μ₀ with temperature for the exponential model in pentacene and PTV OFETs, while figures 4(c) and (d) are for the power model. Figure 4 shows that μ₀ would increase with the increasing temperature and shows a similar Arrhenius relationship as initially proposed by Blom et al [11] for the exponential model. This is meaningful for practical applications, especially for the power model. Because μ₀ has been extracted by using equation (25), if one uses equation (3) to extract μ₀, the simple temperature relationship may be covered up and exhibiting a complicated one.

Figures 5(a) and (b) show the relationship of γ with temperature for the exponential model in pentacene and PTV OFETs, while figures 5(c) and (d) are for the power model. Figure 5 shows a similar relationship for both models, that γ decreases with increasing temperature, and γ is a linear function of inverse temperature initially proposed by Blom et al [11] for the exponential model. The simplicity of these relationships is meaningful for practical applications.

In this paper, the fundamental I–V formula of an OFET has been reformulated to overcome the divergence of the integrand. Two analytic models of OFETs based on exponential and power mobility models are proposed, and numerical calculations have been done for three devices made of three kinds of materials: a pentacene OFET, a PTV OFET and BGP OFET. The results show that the matching degree between theoretical I–V curves and the experimental data is satisfactory for both models, thus confirming the accuracy of the reformulated I–V expression. Moreover, the extracted parameters μ₀ for exponential and power mobility models show simple Arrhenius temperature dependence, and γ shows a simple linear relationship with the inverse of temperature. All of these simple relationships contribute to the simplicity of the reformulated I–V expression. These are very useful for practical applications and device simulations.