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Emulation and Simulation of
Microelectronic Fabrication Processes

2.2 Numerical Material Representations

The choice of numerical methods used to represent the materials in a simulation has great implications on the modelling capabilities of a simulator. The material representation must be chosen depending on the requirements of the models to be carried out. Therefore, in order to choose the correct representation for a specific task, it is important to develop a deep understanding of the required capabilities, as well as the properties of each material representation.

As discussed in Section 1.2, the goal of this work is to allow for the description of entire electronic circuits in a computationally efficient manner applicable for design technology co-optimisation (DTCO). Therefore, fast simulation times and the ability to represent large structures for the developed simulator is critical. In the following, different modelling approaches and their relative advantages and disadvantages are presented.

2.2.1 Atomistic Modelling

The most physically rigorous way to represent a material is to consider the structure of the atoms and molecules forming the substance. Computationally, atomistic models can be simulated using molecular dynamics [31] or Monte Carlo (MC) methods [32]. Using these methods, each fabrication process is simulated considering every single atom or molecule impinging on the atoms of the substrate and modelling each chemical reaction and forces between these atoms. Even surface roughness, which describes the properties of a surface at the smallest scale, can thus be modelled. Therefore, this approach results in the most rigorous physical description possible, but as every single atom has to be considered, it is computationally very costly and it is unfeasible to model large structures, even when large-scale computational resources are available. Hence, the number of atoms which can be modelled using these approaches is limited [33], so these methods cannot be employed to simulate the manufacture of entire microelectronic devices. They are rather applied to understand specific regions of a device, such as a certain material - gas molecule interactions on the interface [34]. These models have also been used to extract certain process parameters which are then applied in a kinetic Monte Carlo or continuum model, which work at larger time scales [35].

2.2.2 Continuum Approach

Due to the computational limitations of atomistic modelling, a continuum approach is commonly employed to model semiconductor manufacturing processes. In this approach, the surface of a material is considered continuous with no abrupt changes or steps [36], as would be the case when considering individual atoms. Therefore, a modelled material is considered a single solid body in the region \(\mathcal {M} \in \mathbb {R}^D\), where \(D\) is the number of spatial dimensions of the simulation domain. The interface of the material is thus described by the bounding surface \(\mathcal {S}\) of \(\mathcal {M}\). Any change to the interface of a material is modelled by moving the surface \(\mathcal {S}\) and thus changing \(\mathcal {M}\).

During semiconductor manufacturing, several materials are combined in a single structure, where each material has specific properties which are commonly assumed to be constant in the continuum model. Therefore, each material \(\mathcal {M}_i\) describes a region of space with a specific set of physical properties. The interface between two separate materials is described by the two bounding surfaces, so the physical properties change abruptly across the material interface. Hence, smooth transitions in material composition cannot be modelled straight-forwardly using this approach. However, if there is no long range smooth transition of physical properties and the modelled structure sizes are greater than the lattice constant of the modelled materials, it is sufficient to model the bounding surfaces of all materials to achieve an appropriate description of all materials. Additionally, the continuum approach allows for the resolution of the modelled interfaces to be adjusted to the requirements of a specific simulation, compared to the fixed resolution of the atomistic approaches given by the size of atoms.

Due to the improved computational efficiency and greater flexibility of the continuum approach, it is the principal method employed in commercial and academic process TCAD tools [37] and was chosen for the modelling carried out in this work. Several numerical material representations using this approach are discussed in the following sections.