20 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 56, NO. 1, JANUARY 2009 A Charge-Based OTFT Model for Circuit Simulation Fabrizio Torricelli, Zsolt M. Kovács-Vajna, Senior Member, IEEE, and Luigi Colalongo, Member, IEEE Abstract—In this paper, a mathematical model for the dc/ dynamic current of organic thin-film transistors is proposed. The model is based on the variable-range hopping transport theory, i.e., thermally activated tunneling of carriers between localized states, and the mathematical expression of the current is formu- lated by means of the channel accumulation charge. It accurately accounts for below-threshold, linear, and saturation operating conditions via a single formulation, and does not require the explicit definition of the threshold and saturation voltages. Basing on the charge control approach, the dc model is straightforwardly generalized to dynamic conditions; the resulting mathematical expressions are simple and suitable for CAD applications. Index Terms—Charge control approach, circuit simulation, compact modeling, organic thin-film transistors (OTFTs), variable-range hopping (VRH). I. INTRODUCTION O RGANIC thin-film transistors (OTFTs) have recently gained considerable interest due to their potential appli- cations in large-area, low-performance, and low-cost integrated circuits. Among them are driving devices for active-matrix flat- panel displays based on organic light-emitting diodes, low- end smart cards, radio-frequency identification tags, sensors, etc. Organic semiconductors for low-cost integrated circuits are typically deposited from solution, resulting in amorphous or polycrystalline thin films. Field-effect transistors based on such materials present several appealing features: the techniques for depositing films allow large areas to be coated, the films can be vacuum-deposited at moderate temperatures, and many polymers and oligomers are soluble and may be processed by spin coating. Furthermore, owing to their intrinsic structural flexibility, in the case of all-polymer systems, OTFTs allow for the production of flexible integrated circuits [1]. The ef- ficient design of complex integrated circuits based on OTFTs requires anyway preliminary optimization and modeling, and the availability of accurate analytical models (SPICE like) is particularly attractive. With respect to crystalline-silicon MOSFETs, the development of an analytical model for OTFTs is complicated by the peculiar nature of the material: due to the low conductivity of organic semiconductors, these devices compare more closely to amorphous TFTs rather than to con- ventional crystalline-silicon MOSFETs. Furthermore, OTFTs are primarily operated as accumulation field-effect transistors, as opposed to the usual inversion mode of silicon MOSFETs; hence, they are normally conducting at zero gate voltage, and Manuscript received March 18, 2008; revised September 24, 2008. Current version published December 19, 2008. The review of this paper was arranged by Editor J. Kanicki. The authors are with the Department of Electronics for Automation, Univer- sity of Brescia, 25123 Brescia, Italy (e-mail: fabrizio.torricelli@ing.unibs.it; zsolt.kovacs@ing.unibs.it; colalong@ing.unibs.it). Digital Object Identifier 10.1109/TED.2008.2007717 the field-effect mobility is dependent on the gate voltage [3]. In recent years, several mathematical models of the OTFT dc current–voltage characteristics were developed [4]–[8]. These models are based on the classical MOS transistor ones [4]–[6], slightly modified by introducing fitting empirical parameters [4], [6] of difficult and uncertain definition. Other approaches based on multiple trapping and release (MTR) theory [4], [5] were proposed but, as reported in [9], the MTR mechanism shows some inconsistencies. In this paper, we present a dc model for OTFTs where the charge transport mechanism is based on the variable-range hopping (VRH) theory, i.e., ther- mally activated tunneling of carriers between localized states (electrons in conduction states) around the Fermi level in the tail of an exponential distribution. This approach has successfully been used to calculate the field-effect mobility of the OTFTs operating in the linear region [7], [10]. The model implicitly accounts for the linear and saturation operating regions via a single formulation, and the relationship between the surface and the channel potential is correctly accounted for. This model does not require the explicit definition of the threshold and saturation voltages as input parameters, which are difficult to be evaluated. Furthermore, without simplifying assumptions, the dc current is formulated as a function of the accumulation charge below the channel, leading to a simple mathematical expression. Both for its easy formulation and for the explicit dependence on the accumulation charge, as pointed out in [11], this approach is very suitable to the extension to the dynamic regime (ac and transient). The generalization to time-dependent conditions is analytically worked out basing on the charge- oriented approach [12] and on the quasi-static approximation. Such approach is particularly suited for OTFTs since, due to their intrinsic low mobility, they are used in low-frequency applications. The resulting expressions characterize the device in all regions of operation without any discontinuities in the capacitance–voltage characteristics, both in the ac and in the transient regime. The paper is organized as follows. In Section II, the basic concepts and the mathematical derivation of the dc model are presented. In Section II-B, the dc model is compared to exper- imental results. In Section III, the derivation of the dynamic model is presented, and in Section IV, conclusions are drawn. In the following, for the sake of simplicity, the model will be presented for n-channel OTFTs only; similar considerations hold for p-channel transistors as well. II. DC MODEL In this section, the analytical expression of the dc current of amorphous OTFTs is derived. The cross section of a typical OTFT is shown in Fig. 1. This layer structure is well suited for 0018-9383/$25.00 © 2009 IEEE