Threshold voltage of advanced MOSFETs: Physical criteria and experimental extraction methods

The threshold voltage (VTH) is a fundamental parameter for any MOSFET, characterizing the transition from weak to strong inversion, i.e., from the exponential to linear charge control regimes. Thus accurate and reliable determination of the VTH value is very important for CMOS design and modeling, and becomes increasingly important as the gate oxide thickness as well as power supply voltage scale down. Traditionally, the threshold voltage of a MOSFET has been defined as gate voltage needed to provide the critical surface potential equal to twice the Fermi potential in the semiconductor body. However, this VTH criterion is inadequate in the modern MOSFETs featuring ultra-thin or high-k gate dielectrics [1], and it is meaningless in the case of advanced silicon-on-insulator (SOI) MOSFETs with undoped silicon body [2]. Thus a variety of different threshold criteria have been proposed for these devices [1], [2]. As a result, the VTH value turns out dependent on the particular VTH criterion [2]. Presently there are many VTH determination methods, which differ by the underlying VTH criteria, extraction procedures and sensitivity to parasitic effects. Thus it is highly desirable to have a clear physical understanding of these methods. In this work, we review the existing VTH criteria and demonstrate that, in case of nano-scale MOSFETs with ultra-thin gate dielectrics or/and undoped SOI thin body, featuring gradual transition between weak and strong inversion, the most physically adequate VTH criteria are: (i) the maximum of the second derivative of the inversion charge with respect to the gate voltage (d2Qinv/dVg2), corresponding to the inflection point in the rate of the inversion charge growth, and (ii) the equality between the drift and diffusion drain current (ID) components, Idiff=Idrift. The experimental methods for the VTH extraction based on these criteria are examined. Particular attention is given to the recently proposed method, using the transconductance-to current ratio (gm/ID), in which VTH is evaluated from the gate voltage of the minimum of the d(gm/ID)/dVg versus gate voltage function [3]. Using analytical modeling, it is shown that the VTH criterion of the d(gm/ID)/dVg method is the maximum of d2Qinv/dVg2, just as of the popular transconductance change method, in which VTH is defined from the position of the maximum of the transconductance derivative (dgm/dVg≡d2Id /dVg2) [4]. However, though dgm/dVg and d(gm/ID)/dVg methods rely on the same VTH criterion, they feature different sensitivity to the mobility variation with gate voltage and drain voltage value. With analytical modeling and experimental data for advanced SOI FinFETs and ultra-thin-body SOI MOSFETs, it is demonstrated that the d(gm/ID)/dVg method is much less sensitive to the Vg-dependent mobility and drain voltage value, and thus is more reliable than the dgm /dVg method. Furthermore, it is found that at the point of the minimum of d(gm/Id)/dVg, at vanishingly small drain voltage and ideal MOSFET operation, the value of gm /Id is equal to 2/3 of its maximum value. This can be used as alternative quick method for the VTH extraction corresponding to the criterion of the maximum of d2Qinv/dVg2. At the same time, at the condition Idiff=Idrift, the ratio of gm /Id to its maximum value is found to be 1/2. The difference between the two VTH extractions in the ideal MOSFET is estimated to be ~1.2 kT/q. However, such VTH extractions are possible only when the gm /Id (Vg) curve has a clearly defined plateau in the range of the maximum, whereas it is frequently distorted, for example, due to gate-induced drain leakage current, short-channel effects, or Vg-dependent mobility. Thus the d(gm /Id)/dVg method seems to be more preferable.