A commonly used tool to investigate the complex behavior of devices and how defects affect their operation is numerical simulation. The kind of simulation ranges from circuit level simulations, where very abstract degradation models are used to capture the essence of the effects on device and circuit properties caused by defects, to ab-initio simulations investigating the atomistic behavior of specific defect structures. Inbetween are physical device simulations, which are commonly used in conjunction with measurements to calibrate physical models and to find defect parameters.
Ab-initio simulations can be used to study the properties of materials or molecules. In microelectronics they have been traditionally used to identify the electronic structure of materials, material interfaces and specific defect configurations. This is done by approximately solving the Schrödinger equation for a number of atoms of the material, together with appropriate boundary conditions, e.g. to represent bulk material. This allows, among other things, to estimate energy levels of defects, barrier energies or formation mechanisms of defects. The most commonly used method for defect studies is the density functional theory (DFT) method, which is, compared to other ab initio methods, computationally inexpensive and allows to simulate thousands of atoms on current hardware [77].
Defect models, e.g. the SRH, NMP and hydrogen release models, as discussed in the previous section, are typically used in TCAD (technology computer-aided design) simulations which aim at simulating the degradation of components on the device level. For this, a device model taking geometry, material properties, dopand densities, and mobility models into account is used in conjunction with a defect model as outlined in Section 3.1 to simulate the altering of the device. Such simulations are often performed in a Monte Carlo [78] fashion by placing a large number of defects with randomly drawn parameters in the device. This, however, requires prior knowledge about the distributions of defect parameters and their density. To obtain this information TCAD simulations can be calibrated to explain dedicated measurements, but also DFT studies on the suspected defects can be used to estimate defect parameters.
The simulations typically start with the calculation of an initial equilibrium state of the system. Afterwards, the evolution of defects in the device is calculated based on the applied bias conditions and temperatures for each following time step. Depending on the density of defects in the devices, the calculations may have to be performed self-consistently. For high densities of defects the charges captured by the defects can significantly alter the device electrostatics. If this is the case, the charge of the defects has to be considered in the Poisson equation as otherwise the results will not be accurate. If possible, however, calculations are performed in a non self-consistent manner as this decreases the convergence time of the solver.
To estimate the impact of the defects on the channel without knowing the exact distribution of the random dopands in the device, the charge sheet approximation can be used. It allows to approximate the threshold voltage shift at low defect concentrations and for little deviations of the results from the ones a defect free device. In this approximation, a charge trapped at a defect is considered to be distributed homogeneously over the entire gate area in a thin sheet parallel to the insulator/semiconductor interface. The resulting threshold voltage shift can be simply calculated from the capacitance between the charge sheet and the gate:
with the vacuum permittivity 
, the relative permittivity of the oxide 
, the gate area 
, the gate oxide thickness 
, the defects distance from the interface 
, and the elementary charge 
. It should be noted that this approach generally leads to an underestimation of the average impact of the defects. For a more accurate estimation of the average impact of a single defect on the threshold voltage shift, the
distribution function of step heights has to be determined [79, 39].
Most simulators used for the development of integrated circuits are based on a simulation program with integrated circuit emphasis (SPICE) [80] written by L. Nagel in 1973. SPICE models describe components within the circuit and thus are physically more abstract than device simulation. They are based on relatively simple analytical models, often with a number of empirical parameters. Even though it is generally accepted that BTI is due to the interaction of single defects with the device, to date compact BTI models do not directly calculate the impact of such defects for this. The impact the defects have, e.g. on the threshold voltage and mobility, is calculated based on model parameters calibrated to device simulation or measurements on a specific technology. However, these models cannot describe many facets of BTI which might be important for the development of circuits considering scaled technologies. The most fundamental problem is that calculating the vast number of defects per device becomes inefficient when it has to be done for a number of transistors. For this more compact descriptions of the phenomena are required. One approach which is based on the observations from single defects is the defect-centric perspective, which will be discussed next.
