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Emulation and Simulation of
Microelectronic Fabrication Processes

3 Surface Rate Calculation

In order to move a surface, its evolution must be captured by a velocity field \(\vec {V}(\vec {x}_s)\) defined at every point on the surface \(\vec {x}_s\), as described in Section 2.4. In the case of the sparse-field level set method, this means that there must be a velocity field \(\vec {V}(\vec {a})\) defining the surface movement for every active point \(\vec {a}\) describing the interface. As all topographical changes of the surface resulting from a manufacturing process are captured in this velocity field, it is crucial that it is as accurate and as robust as possible. Analogously to physical growth or etch rates, these surface velocities are often referred to as surface rates, even when the physical behaviour is not considered directly. In order to capture these surface rates appropriately, process models are used to simulate how a fabrication process will affect the surface and thus give accurate approximations for the surface velocities. Two main approaches to modelling fabrication processes to find the surface rates are discussed in the following.

Firstly, empirical modelling, or process emulation, describes extracting geometric parameters, such as etch depth or isotropy from experimental data and using them to formulate simple models for the surface rates. When using the iterative advection described in Section 2.4.2, a velocity field can only be applied for a single advection step and then must be recalculated since the level set (LS) surface has changed and thus the velocity field must be adjusted as well. However, geometric process parameters may also be expressed as a geometric advection distribution, as described in Section 2.4.3, and the entire process could be emulated in a single step.

The second approach is chemical modelling and encompasses a rigorous description of all relevant physical processes leading to a change in topography by considering the underlying physical principles, including particle transport, surface kinetics and surface chemistry. Since chemical modelling requires the explicit simulation of process time, only iterative advection may be used to advance a LS surface, while geometric advection is not suited for this type of modelling.

Due to its simplicity, empirical modelling is usually computationally more efficient, while chemical modelling describes the underlying physics of a process and can thus be used for physical analysis or to simulate processes for which no experimental data is available.

3.1 Empirical Surface Rate Modelling

In this approach to finding the surface velocities \(\vec {V}(\vec {x}_s)\) capturing the topographical changes, the result of a fabrication step must already be known and geometric parameters of the final geometry extracted. Once these parameters, such as etch depth, deposition distance, isotropy, or directionality are known, surface velocities are generated algebraically. Since no physical behaviour is modelled, but only geometric considerations are used to mimic the final surface, this approach is also referred to as process emulation [116]. A simple example is isotropic deposition of a material, which is modelled by setting the velocity field in the normal direction of the surface \(V(\vec {x}_s)\) to the thickness of the deposited material. Applying iterative advection for a unit of time results in the final surface showing the expected deposition behaviour. If the surface is moved over large distances, this velocity field must be applied several times in order not to violate the Courant-Friedrichs-Lewy (CFL) condition.

However, for certain types of processes, it is possible to express the velocity field \(\vec {V}(\vec {x}_s)\) as a geometric advection distribution. Then, for isotropic deposition, the initial surface is simply shifted outwards by a constant distance, resulting in a film of constant thickness, as if grown perfectly isotropically. There is no general limit on the complexity of a process modelled using geometric advection, although the algebraic descriptions required to capture the geometric properties of complex fabrication process may sometimes be described more naturally using a velocity field and applying it using iterative advection. Due to the simplicity and straight-forward calculations of the final surface, geometric advection is highly efficient and commonly employed for the generation of large structures in order to characterise entire microelectronic circuits[117, 118, 119]. Since time is not modelled explicitly, but just the initial interface and the final surface, the geometric advection algorithm discussed in Section 2.4.3 is the most appropriate method for simulating this type of model.