Gummel Symmetry Test on charge based drain current expression using modified first-order hyperbolic velocity-field expression

• Published in: Solid-state electronics 2017-03, Vol.129, p.188-195
• Main Authors: Singh, Kirmender, Bhattacharyya, A.B.
• Format: Article
• Language: English
• Subjects: Drain current; Gummel Symmetry Test (GST); Modified first-order hyperbolic [formula omitted]-E model; Strong inversion; Velocity overshoot; Weak inversion
• ISSN: 0038-1101; 1879-2405
• DOI: 10.1016/j.sse.2016.11.006

•Charge based drain current expression using modified first-order velocity-field model.•Drain current and its higher-order derivatives are shown to pass gummel symmetry test (GST).•GST is shown to pass till sixth-order derivative in strong inversion region.•In weak inversion GST is shown to pass till fifth-order derivative. Gummel Symmetry Test (GST) has been a benchmark industry standard for MOSFET models and is considered as one of important tests by the modeling community. BSIM4 MOSFET model fails to pass GST as the drain current equation is not symmetrical because drain and source potentials are not referenced to bulk. BSIM6 MOSFET model overcomes this limitation by taking all terminal biases with reference to bulk and using proper velocity saturation (v-E) model. The drain current equation in BSIM6 is charge based and continuous in all regions of operation. It, however, adopts a complicated method to compute source and drain charges. In this work we propose to use conventional charge based method formulated by Enz for obtaining simpler analytical drain current expression that passes GST. For this purpose we adopt two steps: (i) In the first step we use a modified first-order hyperbolic v-E model with adjustable coefficients which is integrable, simple and accurate, and (ii) In the second we use a multiplying factor in the modified first-order hyperbolic v-E expression to obtain correct monotonic asymptotic behavior around the origin of lateral electric field. This factor is of empirical form, which is a function of drain voltage (vd) and source voltage (vs). After considering both the above steps we obtain drain current expression whose accuracy is similar to that obtained from second-order hyperbolic v-E model. In modified first-order hyperbolic v-E expression if vd and vs is replaced by smoothing functions for the effective drain voltage (vdeff) and effective source voltage (vseff), it will as well take care of discontinuity between linear to saturation regions of operation. The condition of symmetry is shown to be satisfied by drain current and its higher order derivatives, as both of them are odd functions and their even order derivatives smoothly pass through the origin. In strong inversion region and technology node of 22nm the GST is shown to pass till sixth-order derivative and for weak inversion it is shown till fifth-order derivative. In the expression of drain current major short channel phenomena like vertical field mobility reduction, vel