|
Thin-film deposition with magnetron sputtering is a substantially complex process. Modeling of these processes require forethought as to what the modeling objectives are. The objectives often dictate what modeling method one should adopt. In broad strokes, there are two distinct methodologies, namely, the stochastic method and the analytical method.
Monte Carlo is the most prominent technique in the category of stochastic modeling. In these models the behavior of a PVD process emerges statistically after following a large number of events that can be (relatively) easily described by mathematical formula. The stochastic nature stems from initial conditions that are spawned through computer-generated “random” events.
Analytical modeling applies mathematical formula to all physical phenomena considered. The behavior of a PVD process is computed and presented deterministically.
When a magnetron sputtering source is employed, a PVD process involves a number of physical phenomena. Some of the important phenomena are: generation of gas-discharge plasma, sheath formation (the potential drop in front of the target surface), sputtering of the target material, transport of depositing particles in space, collisions of the particle with other gas species in transport, blockage of the particle by obstacles (uniformity masks, collimators, chimneys etc.), condensation of the particles onto the substrate, movement and binding of the deposited particles with the growing film.
In high-power impulse magnetron sputtering, or HIPIMS, other physical phenomena become prominent, including ionization of the material species through collisions, acceleration of ionized species towards the substrate (under the influence of a potential gradient in front of the substrate), absorption of the momentum (carried by the ionic species) by the growing film, energy-induced surface mobility, re-sputtering of the deposited material etc.
There are many common misconceptions about both the stochastic modeling and the analytical modeling. From a literature survey*, which has been a multi-year project, we can offer the following take-away.
1. Monte Carlo models are hybrid models
In what are labeled “Monte Carlo” models, typically only one or a few physical processes are handled by the Monte Carlo computing codes. The rest physical processes are treated by applying known mathematical (analytical) formulae. Such hybridization is both reasonable and necessary – to treat all physical processes stochastically is computation-time prohibitive.
2. Monte Carlo models do not include all important phenomena
In published literature of Monte Carlo studies, researchers elect what physical phenomena to include and others to ignore. This approach is universal in computer simulation. Because of the computation-time constraint, Monte Carlo models often leave out some consequential physics.
3. No Complex-motion tooling has been modeled through Monte Carlo
In existing literature, Monte Carlo method was applied to deposition processes that were static and simple in geometry except few single-axis rotation cases that were simulated with a superposition of crude stepping. When a substrate is moved in a complex tooling, such as dual-axis rotation or rotation + translation, the changes in flux pattern require breaking down of the process into thousands or more steps, each of which must be computed individually. Given that a Monte Carlo simulation of a static geometry typically takes hours of computing time, modeling of complex-motion tooled systems is cost and time prohibitive.
4. Uniformity masks are rarely modeled through Monte Carlo
To optimize a correction mask (shadow mask), the travels of depositing species must be modeled with a fine spatial resolution in order for small adjustment in a mask to show effects. Trial-and-error iterations are essential. Again, the cost and computing time constraints make Monte Carlo method impractical for the tasks.
5. Monte Carlo can mimic thin-film growth microscopically
An area of success for Monte Carlo is simulation of thin-film growth microscopically. Such a study can take into account the kinetic energy carried by the incident species, mobility of the deposited atoms under the influence of the kinetic energy and the thermal energy (substrate heating). Columna structure or other microscopic features of the grown films can be well simulated; their linkage with the growth condition can be found or validated. These studies can be particularly helpful in understanding HIPIMS in which bombardment of substrate surface by energetic ions plays an important role in determining the physical properties of resultant materials. (A caveat: in these studies the flux of incident species, in front of the substrates, are usually modeled with an analytical formula from certain assumptions and simplification.)
6. Analytical models are proven successful
There exist, in published literature, a wealth of examples of sputter deposition being modeled with analytical approaches. In these models all the physical processes are simulated using (continuous) mathematical formulae that are known to describe the processes to certain accuracy. Angular distributions of the sputtered atoms from targets, for instance, are often described by a polynomial of the cosine, which is verifiable through experiments.
In the simplest implementation, an analytical model adopts the “line of sight” approach to describe the transport of depositing species from the target to a substrate; the condensation is modeled with the “hit and stick” assumption. The line-of-sight approach ignores collisions in the gas phase; the hit-and-stick assumes that the condensation is 100%. Despite the simplifications, these models have shown remarkable accuracy and utility in engineering magnetron-sputtering based PVD processes.
7. Analytical models can include collisions and non-unity sticking
In more sophisticated analytical models, the gas-phase collisions are brought into consideration through the introduction of “collision probability” and “average deflection angle”, parameters that allow diffusive transport (due to gas-phase collisions) to be treated phenomenologically in a statistic-average fashion. These models are capable of accounting for and predict thickness distributions in a broad range of chamber pressure. In fact, compared with published Monto Carlo studies of thickness distributions at various chamber pressure the analytical models yield results that are indistinguishable from the former.
Analytical models can also simulate non-sticking events for depositing species impinging on a substrate surface. The primary cause for non-sticking is inefficient transfer of momentum: when a particle is incident obliquely, the surface may be incapable of capturing it by absorbing the momentum parallel to the surface. Through a simple modification in the condensation term, analytical models can well encapsulate the angle dependent non-sticking. Such modeling has proven effective in further improving the accuracy by capturing the minor yet noticeable thickness roll-off that typically occurs at edges of large substrates.
8. Analytical modeling is indispensable for complex processes
Presently, analytical approach is the only practical method for modeling sputter deposition that involves complex motion of substrates, vapor-obstruction elements (shadow masks, chimneys, partitioning walls, vapor collimators etc.), large-scale re-emittance of condensed particles (from heated surfaces), complex-geometry targets etc. Until computing speed takes a leap in orders of magnitude, potentially though quantum computing, analytical modeling will remain indispensable.
Because of the rapidity and the spatial resolution of the analytical approach, routines have been created to automatically retrieve unknown angular distributions of sputtered particles by means of lease-square fitting of experimental data, and to automatically optimize uniformity masks.
| 
