Phenomenological Single-Particle
Modeling of Reactive Transport
in Semiconductor Processing

Chapter 7 Conclusion and Outlook

This thesis has thoroughly addressed the issue of phenomenological models for reactive single-particle transport in topography simulation. At a first glance, these models might appear to be straightforward, thus not warranting an in-depth investigation. However, as discussed in detail throughout this work, the correct application of single-particle modeling requires very careful consideration and deep knowledge of the involved processing technique to obtain both a reproduction of experimental surfaces and useful chemical insights.

The fundamental constituents of modern topography simulation were introduced: The LS method, and reactive transport models, as well as their interface through the surface advection velocity field. The fundamental phenomenological model at the core of reactive single-particle transport, first-order reversible Langmuir kinetics, was reviewed and discussed with respect to the underlying chemical and physical assumptions. However, a full description of the Langmuir kinetics is incomplete without a discussion of the calculation method for the local reactant fluxes. Therefore, an overview of the four main approaches to calculate the local fluxes was given, namely: Constant flux, bottom-up visibility calculation, top-down pseudo-particle tracking, and 1D models.

From the category of 1D models, Knudsen diffusive transport was chosen to merit a more in-depth examination due to its comparatively simple implementation, ability to provide direct physical insights, and, importantly, abundance of historical and contemporary misconceptions. To address this issue, Knudsen diffusivity was reformulated in a new and more modern fashion using an analogy to radiative heat transfer. First, the well-established expression of Knudsen diffusivity in a long cylinder is recovered. Then, the issue at the heart of the misconceptions was addressed: How to handle geometries with lateral cross-sections different from a cylinder. Particularly, the divergence of the mathematical formulation of Knudsen diffusivity for an infinitely wide trench is discussed. In conclusion, it is shown that 3D geometries are not completely equivalent to their 2D cross-sections. Although commonly used mappings, such as the hydraulic diameter approximation, attempt to simplify the involved geometries into an EAR, the underlying 3D complexity often resists such coarse simplifications.

With a firmer understanding of Knudsen diffusive transport processes, this 1D model was then applied to thermal ALP. Since these processing techniques exploit the self-limiting nature of the surface reactions, the first-order irreversible Langmuir kinetics had to be integrated with the diffusive transport model, showing the profound relationship between reactant transport and surface reactions. Additionally, the novel integration of this reactive transport model with the LS method through the introduction of an artificial time unit was shown.

After manual calibration of parameters to reported experimental data of ALD of Al₂O₃, the temperature dependence of the phenomenological parameters is determined. From this analysis, the importance of the reversibility of the H₂O reaction, particularly at low temperatures, is demonstrated. Additionally, an activation energy comparable with recent direct experimental studies is extracted. By reproducing multiple reactors in the TMA-limited regime, remarkable parameter stability is achieved. This is evidence that the phenomenological parameters are strongly linked to the reactor parameters, most importantly the substrate temperature. Thus, the parameters can be interpreted as a proxy of the reactor setup. Finally, a qualitative analysis of both ALD and ALE of HfO₂ is performed. Due to its efficient integration into a commercial simulator, a path for future investigations of device performance is highlighted.

Attention is then placed on the process of low-bias SF₆ plasma etching of Si. Even though this process is employed due to its near-isotropic etch characteristics, the final surfaces are known to be different from those obtained from ideal isotropic etching. To accurately reproduce these surfaces using topography simulation, the available flux calculation approaches were evaluated. The top-down pseudo-particle tracking approach using a constant effective sticking coefficient was shown to be sufficient to reproduce the experimental surfaces. Moreover, by reproducing multiple geometries reported in the literature using the same surface model and flux calculation approach, the phenomenological model parameters again appear to serve as a proxy of the reactor configuration.

Through a series of computational experiments, a novel empirical relationship was then constructed between an experimentally-accessible quantity of interest and the model parameters. Thus, information about the phenomenological model and, consequently, the surface chemistry and the state of the reactor, can be directly extracted from experimental topography measurements. This information can be used not only for accelerating calibration procedures but also to optimize the reactor setup by interpreting the parameters as, for example, surrogate variables of the fluorine radical density.

Finally, the capabilities of topography simulation are showcased by exploring the optimization of the fabrication process of silicon microcavity resonators. In order to move away from manual parameter calibration, a more robust automatic calibration procedure was introduced, including a custom feature detection algorithm. With that procedure, the entire phenomenological parameter set was calibrated using only the final state of the surface. Having achieved a calibrated simulation, the etch times and photoresist opening diameter of the involved two-step etching process were computationally investigated to optimize parameters indicative of device performance.
An increase in the first step etch time, ideally by doubling it, appears to positively impact the resonator quality. Simultaneously, the second etch time can be significantly reduced, leading to a reduction in overall etch time and, consequently, a reduction in cost and complexity.

In summary, it can be concluded that the final topography of a processed device possesses substantial information about the involved surface-chemical phenomena. Therefore, both direct modeling of experimental topographies and inverse modeling to gather information about the surface chemistry are possible. This strongly supports the applicability of first-order reversible Langmuir kinetics to a wide range of semiconductor processing applications. Through the judicious use of approximations, by paying careful attention to the calibration procedure, and by choosing the correct flux calculation approach, reactor-scale or first-principle simulations can either be complemented, or even completely bypassed in a few cases, using phenomenological modeling of reactive single-particle transport in topography simulation.

For future research, a natural extension of this work is the combination of the most physically rich flux calculation approach, the top-down pseudo-particle tracking, with first-order reversible Langmuir kinetics without the constant effective sticking coefficient approximation. For self-limiting reactions, a similar pulse time integration approach to that employed for thermal ALP can be employed to handle variable sticking coefficients. Nevertheless, for processes in the steady-state, a different self-consistent methodology for the calculation of the sticking coefficients will be necessary. This will require substantial computational resources, but the possible gains in accuracy and insight are very attractive. Additionally, as already hinted in the examination of aspect ratio dependent RIE, an increased number of phenomenological particle species will be necessary for certain processes.

In addition, the realization that the final surfaces carry the fingerprint of the surface chemical processes imposes additional pressure on the calibration method. The here-presented automated calibration procedure shows the tremendous promise of this approach, however, it had to be manually tailored to the application. To move beyond these custom solutions, recent advances in computer vision and artificial intelligence could be leveraged to create a truly general automatic calibration procedure and earnestly close the loop between experiment and simulation.

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List of Own Publications

Journal Articles

[1] L. F. Aguinsky, F. Rodrigues, T. Reiter, X. Klemenschits, L. Filipovic, A. Hössinger, J. Weinbub, "Modeling non-ideal conformality during atomic layer deposition in high aspect ratio structures", submitted to Solid-State Electronics, under review, 2022. arXiv: 2210.00749
[2] L. F. Aguinsky, F. Rodrigues, G. Wachter, M. Trupke, U. Schmid, A. Hössinger, J. Weinbub, "Phenomenological modeling of sulfur hexafluoride plasma etching of silicon", Solid-State Electronics, vol. 191, (invited), 108262-1–108262-8, 2022, doi: 10.1016/j.sse.2022.108262.
[3] L. F. Aguinsky, G. Wachter, P. Manstetten, F. Rodrigues, M. Trupke, U. Schmid, A. Hössinger, J. Weinbub, "Modeling and analysis of sulfur hexafluoride plasma etching for silicon microcavity resonators", Journal of Micromechanics and Microengineering, vol. 31, 125003-1–125003-9, 2021, doi: 10.1088/1361-6439/ac2bad.
[4] A. Toifl, F. Rodrigues, L. F. Aguinsky, A. Hössinger, J. Weinbub, "Continuum level-set model for anisotropic wet etching of patterned sapphire substrates", Semiconductor Science and Technology, vol. 36, 045016-1 – 045016-12, 2021, doi: 10.1088/1361-6641/abe49b.

Book Contributions

[5] L. F. Aguinsky, G. Wachter, F. Rodrigues, A. Scharinger, A. Toifl, M. Trupke, U. Schmid, A. Hössinger, J. Weinbub, "Feature-scale modeling of low-bias SF₆ plasma etching of Si", in Joint International EUROSOI Workshop and International Conference on Ultimate Integration on Silicon (EUROSOI-ULIS). IEEE, pp. 1–4, 2021, doi: 10.1109/EuroSOI-ULIS53016.2021.9560685.
[6] F. Rodrigues, L. F. Aguinsky, A. Toifl, A. Scharinger, A. Hössinger, J. Weinbub, "Surface reaction and topography modeling of fluorocarbon plasma etching", in International Conference on Simulation of Semiconductor Processes and Devices (SISPAD). IEEE, pp. 229–232, 2021, doi:
10.1109/SISPAD54002.2021.9592583.

Conference Contributions

[7] L. F. Aguinsky, "Advanced flux models in Victory Process: Low-bias SF₆ etching and thermal atomic layer processing", in Silvaco Users Global Event (SURGE), (invited), 2022.
[8] L. F. Aguinsky, G. Wachter, A. Scharinger, F. Rodrigues, A. Toifl, M. Trupke, U. Schmid, A. Hössinger, J. Weinbub, "Feature-scale modeling of isotropic SF₆ plasma etching of Si", in Book of Abstracts of the Joint International EUROSOI Workshop and International Conference on Ultimate Integration on Silicon (EUROSOI-ULIS), pp. 54–55, 2021.
[9] F. Rodrigues, L. F. Aguinsky, A. Toifl, A. Hössinger, J. Weinbub, "Feature scale modeling of fluorocarbon plasma etching for via structures including faceting phenomena", in Book of Abstracts of the International Workshop on Computational Nanotechnology (IWCN), pp. 101–102, 2021.
[10] L. F. Aguinsky, P. Manstetten, A. Hössinger, S. Selberherr, J. Weinbub, "An extended Knudsen diffusion model for aspect ratio dependent atomic layer etching", in Book of Abstracts of the International Conference on Atomic Layer Deposition (ALD) Featuring the International Workshop on Atomic Layer Etching (ALE), p. 109, 2019.
[11] L. F. Aguinsky, P. Manstetten, A. Hössinger, S. Selberherr, J. Weinbub, "Three-dimensional TCAD for atomic layer processing", in Book of Abstracts of the Workshop on High Performance TCAD (WHPTCAD), p. 5, 2019.
[12] P. Manstetten, L. F. Aguinsky, S. Selberherr, J. Weinbub, "High-performance ray tracing for nonimaging applications", in Book of Abstracts of the Workshop on High Performance TCAD (WHPTCAD), p. 20, 2019.
[13] P. Manstetten, G. Diamantopoulos, L. Gnam, L. F. Aguinsky, M. Quell, A. Toifl, A. Scharinger, A. Hössinger, M. Ballicchia, M. Nedjalkov, J. Weinbub, "High performance TCAD: From simulating fabrication processes to Wigner quantum transport", in Book of Abstracts of the Workshop on High Performance TCAD (WHPTCAD), p. 13, 2019.
[14] L. F. Aguinsky, P. Manstetten, A. Hössinger, S. Selberherr, J. Weinbub, "A mathematical extension to Knudsen diffusion including direct flux and accurate geometric description", in Book of Abstracts of the International Workshop on Computational Nanotechnology (IWCN), pp. 109-110, 2019.
[15] G. Diamantopoulos, P. Manstetten, L. Gnam, V. Simonka, L. F. Aguinsky, M. Quell, A. Toifl, A. Hössinger, J. Weinbub, "Recent advances in high performance process TCAD", in Book of Abstracts of the SIAM Conference on Computational Science and Engineering (CSE), p. 335.

Curriculum Vitae

Education

03/2018 – present

Doctoral candidate in Electrical Engineering
Institute for Microelectronics, TU Wien
Vienna, Austria

10/2014 – 01/2018

Master’s degree in Simulation Sciences
Thesis titled "Modeling Radiative Heat Transfer in Solar Thermochemical Particle Receivers"
RWTH Aachen University and Deutsches Zentrum für Luft- und Raumfahrt (DLR)
Aachen and Cologne, Germany

01/2012–06/2012

Exchange student
University of Montreal
Montreal, Canada

03/2009 – 08/2013

Bachelor’s degree in Physics
Thesis titled "Capillary Electrophoresis in Ultra-High Electric Fields"
Federal University of Rio Grande do Sul (UFRGS)
Porto Alegre, RS, Brazil

Research Projects

2022

FWF Erwin Schrödinger Postdoctoral Fellowship
"Exploring Novel Materials for Next-Generation Optically Stimulated Memristors"
Integrated Systems Laboratory, ETH Zurich
Zurich, Switzerland

Awards and Scholarships

2021

Solid-State Electronics Best Poster Award
EUROSOI-ULIS 2021 Conference

2012

Science without Borders CNPq Scholarship
University of Montreal, Canada.

Teaching

2021

Bachelor Thesis Supervisor
Thesis titled "Rasterization for Flux Calculations in Topography Simulations"
TU Wien

2019–2022

Co-lecturer
Numerical Simulation and Scientific Computing I
TU Wien

2011

Teaching Assistant
Computational Methods of Physics B
UFRGS

Phenomenological Single-Particle Modeling of Reactive Transport in Semiconductor Processing