In the full system of equations (cf. Equation (2.27) in Chapter 2) the fixed charges expressed as a charge density on the right hand side of Poisson’s equation are modelled as macroscopic densities. All charges, whether they arise from doping, the free carriers or traps have so far been viewed as macroscopic quantities, neglecting their granular nature. This approach is well-applicable as long as the number of charged particles is large enough, which is the case for semiconductor devices with characteristic lengths larger than devices of the 100nm node [3]. The influence of the granular, random nature of the donor and acceptor atoms in sub-100nm node field effect devices has been well investigated by [3, 92, 93] and many others. Often, the study of this kind of granularity is called the study of random discrete dopants (RDD). The granularity itself stems from the device fabrication, during which the dopants are implanted into the silicon substrate resulting in a Poisson distribution of dopants per unit volume. In deca-nanometer devices the natural random, non-smooth distribution of dopants can lead to significant inter-device variability of important parameters such as the threshold voltage Vth. Nevertheless, random discrete dopants are not the only source of variability. In sub-100nm field effect devices, trapped charges also have to be viewed as discrete, since they can electrostatically interact with the dopants in the channel [4], giving rise to the different variability in ΔVth in degraded MOSFETs. Yet another source of variability emerges in small area metal contacts [94], since the granularity of the metal (MGG) becomes apparent: a volume of metal in a semiconductor device consists, due to fabrication, of multiple grains. The work function of the grains is statistically distributed and is usually assumed to follow a normal distribution. In field effect devices, this is important if a metal gate instead of a highly doped poly-silicon gate is used, which is the case for the sub-32nm nodes [94].
In the course of this thesis RDD and random discrete traps have been investigated since both are essential to describe the ‘step heights’ seen in ΔVth recovery traces [4, 95], especially after bias temperature stress. MGG is important to correctly assess the variability of device parameters of sub-100nm metal-gate MOSFETs and has thus been implemented into the simulator MinimosNT for evaluation purposes. It was found that MGG has negligable impact on the threshold voltage shifts caused by discrete oxide traps close to the semiconductor interface. Therefore, MGG is not considered in the remainder of this chapter.