A Mathematical Model for the Threshold Voltage of a Partially and Fully Depleted MOS/SOI Structure with a Gaussian Distribution in the Film C. Ravariu * , A. Rusu * , D. Dobrescu * , L. Dobrescu * , F. Ravariu ** , C. Codreanu ** , M. Avram ** * POLITEHNICA University of Bucharest — Faculty of Electronics and Telecommunications, 313 Splaiul Independentei, 77206, Bucharest, Romania. Phone: +40-1-4104740; Fax: +40-1-4104740; e-mail: cristir@mcma.pub.ro **National Institute for Research and Development in Microtechnologies (IMT Bucharest) Str.Erou Iancu Nicolae 32B,72996 Bucharest, Romania ABSTRACT The analytical models for the electric field and potential distributions are necessary to establish the inversion or accumulation conditions at the front and back interfaces for a lot of SOI devices. The paper refers to a one-dimensional analysis, both for partially and fully depleted devices on films with non-uniform doping. The goal of this paper is to obtain an accurate model of the electric field and potential distribution in the SOI capacitors with gaussian dependence of the doping profile in the film. In the fully depleted film case, the model takes into account the depletion of the silicon substrate. The model has been used for the threshold voltage computing, but they also represent a reference point in the development of news models for SOI-devices fabricated on gaussian profile films. The results were compared with PISCES numerical simulations and were in a good agreement. Keywords: SOI films, Gaussian profile, Threshold voltage. 1 INTRODUCTION In the latest years, the interest for SOI technologies increased at a rapid rate [1]. Sometimes, an additional doping of the film can be achieved simultaneously with the SOI layers fabrication, or separately, by a diffusion process. After this, a gaussian profile of concentration in the Si-film appears. Then a modelling of these structures becomes necessary. Lim and Fossum reported a unitary model, which accounts of front and back gates coupling both for uniform concentration in the film and for deep impurity implant in the film, approximated by a step profile [2]. For uniform impurity concentration in the film, a model that considers the depletion in the substrate in some particulars regimes were reported by Ravariu et all. [3]. The aim of this paper is to derive the expression of the electric field and potential in a Metal-Insulator-Silicon- Insulator-Substrate capacitor for a gaussian profile in fully- or partially- depleted films. This additional doping will influence the threshold voltage in order to avoid inversion at back/front interface. The work hypotheses are: - the neglect of the gate-body work function differences, interface charges and positive fixed charges in insulators. - a one-dimensional analysis, considering the depletion approximation for the silicon film and eventual substrate. - a uniform doping concentration in the substrate. - p-type film and substrate. - positives bias of the front gate and null potential (reference potential) on back gate. The main notations are: N A1 (x) - variable doping concentration in p-type film (after a diffusion process), N AS1 - surface doping concentration in p-type film, N A2 - doping in p-type substrate (constant), x S1,S2 - film, respectively substrate thickness; x d1,d2 - space charge region thickness in film, respectively in substrate; x ox1,ox2 - thickness of front oxide, respectively of buried oxide; ε Si - dielectric permittivity of silicon, ε ox - dielectric permittivity of oxide, V G - front gate voltage; φ F1 - Fermi potential at the surface of the film. After a diffusion process in the SOI film, the impurity concentration looks like a gaussian function: √ √ ↵  - ? = Dt x exp N ) x ( N AS A 4 2 1 1 (1) where D is diffusion coefficient, t - diffusion time, x - the distance into the film. 2 THE MATHEMATHICAL MODEL FOR THE THRESHOLD VOLTAGE 2.1 Partially Depleted Films A slice from a MOS/SOI partially depleted structure is shown in figure 1. If the front gate will be positive biased, than a space charge region appears in the film, which will advance from 0 to x 2 with the V G increasing. The electric field and potential distribution are derived by