1  Introduction

The ongoing trend in the semiconductor industry towards ever smaller devices has lead to ever faster digital circuits at decreasing cost. However, scaling devices down to the nano-meter regime increases the probability of functional failure due to single point defects or parametric fluctuations, such as fluctuations in the doping. Single point defects are studied in reliability, whereas parametric fluctuations are often referred to as variability. The widespread and readily available drift diffusion (DD) model and its first-order quantum corrected versions have often been used to assess both reliability  [1] and variability in nano-scale devices, often despite better knowledge or lack of alternatives  [2]. The challange in selecting the appropriate transport model lies in the exact definition of the problem to be assessed and the quantities of interest. In reliability as well as in variability, one is interested in changes in device parameters, such as the absolute threshold voltage shift in MOSFETs, due to charges in the semiconductor and the insulator of the device. Variability, however often needs to be accounted for when investigating reliability. This is the case in sub-100nm node devices, where fluctuations of the doping cannot be ignored anymore  [34]. For example, in nano-scale FinFets the discrete nature of the doping needs to be accounted for since on average there are only very few dopant atoms in the channel of the device. Additionally, in order to avoid unphysically overestimation of carrier concentrations in nano-scale devices, first-order quantum correction models  [567] or solutions of the Schrödinger-Poisson equation are required. Identifying and capturing essential quantum mechanical effects in a charge transport model, which needs to be computationally efficient and accurate at the same time is challanging.

Considering device degradation, two effects have received wider attention of reliability researchers and engineers alike: the bias temperature instability (BTI) and hot-carrier degradation (HCD). BTI is mainly attributed to oxide defects sensitive to temperature and the electric field. HCD, on the other hand, is attributed to highly energetic carriers, which impinge on the oxide-semiconductor interface in MOSFETs, thereby generating interface defects. Most studies investigating reliability on nano-scale devices are either strongly based on measurement data or on oversimplified device or degradation models. The selection of an appropriate transport model to be coupled with a reliability model is highly dependent on the device design and the degradation effect of interest. For example, when investigating BTI for a nano-scale FinFet  [8], carrier confinement and variability of the doping are crucial, whereas, e.g. on a planar 500nm n-channel MOSFET a classic DD simulation is often sufficient. However, if one is interested in understanding HCD, the drift diffusion transport model has proven to be insufficient in any device, since HCD requires exact modelling of the energy distribution of the charge carriers.

Due to the delicate relation of transport-, reliability-models and variability Chapter 2 gives an overview of charge transport models and solution methods based on the Schrödinger equation and Boltzmann’s Transport equation (BTE). In this chapter various transport models, the basic assumptions during their derivation as well as solution methods is be discussed. Since modern hot-carrier degradation models require a solution of the BTE, the spherical harmonics expansion (SHE) method is used to solve the BTE in a time efficient manner. Chapter 3 describes the spherical harmonics expansion of the Boltzmann transport equation, the derivation of the respective equations for the expansion coefficients and its numeric implementation. In Chapter 4 first-order quantum correction methods, especially the density gradient (DG) model  [9], are briefly discussed. The density gradient model is required in order to efficiently assess variability due to random discrete dopants. This is the focus of Chapter 5, which is mainly concerned with the ΔVth variability of single discrete oxide traps in the presence of random discrete dopants. Additionally, it will be shown that a first-order quantum corrected drift diffusion transport model is insufficient to capture the distribution of ΔVth steps caused by single point defects in planar nano-scale MOSFETs. Chapter 6 describes the typical setups for simulation and measurement of the bias temperature instability. Additionally, the latest model for BTI and the appropriate selection of a transport model for BTI are discussed. The chapter ends with an investigation of BTI in p-channel MOSFETs, using an integrated poly-heater technology, at high temperatures. The differences of the current BTI model to the standard SRH model as well as the general deficits of the SRH model will be shown. The last chapter (Chapter 7) is exclusively about HCD. Important characteristics of HCD are discussed, followed by an exhaustive explanation of our current HCD model. Additionally, suitable transport models for the simulation of HCD in MOSFETs are discussed. It is shown that SHE is at present the better BTE solution technique, compared to Monte Carlo approaches, in the context of HCD. Finally the application of the current HCD model using distribution functions obtained by SHE for two different devices and four different stress conditions is be discussed.