Dissertation

D I S S E R T A T I O N

Phenomenological Modeling of
Reactive Single-Particle Transport
in Semiconductor Processing

ausgeführt zum Zwecke der Erlangung des akademischen Grades
eines Doktors der technischen Wissenschaften

unter der Betreuung von

Associate Prof. Dipl.-Ing. Dr.techn. Josef Weinbub
O.Univ.Prof. Dipl.-Ing. Dr.techn. Dr.h.c. Siegfried Selberherr

eingereicht an der Technischen Universität Wien
Fakultät für Elektrotechnik und Informationstechnik
von

Luiz Felipe Aguinsky, MSc.
Matrikelnummer: 11834250

Wien, im Dezember 2022

Abstract

The continuing evolution of micro- and nanotechnology puts ever-increasing pressure on the involved manufacturing processes. In the context of this expanding complexity, the experimental and empirical knowledge of process developers must be complemented with physically-sound modeling and simulation. Historically, the focus of simulations of fabrication processes has been on manufacturing steps which directly impact the electrical characteristics of the final devices. In more recent decades, however, increased attention has been placed on directly investigating the manufactured device structure. The computational modeling of evolving surfaces during their fabrication processes is the purview of topography simulation.

Topography simulation is composed of two main elements: A method for describing the advecting surfaces, and reactive transport models which determine the surface advection velocity fields. The focus of this thesis is on the latter, building upon previous work which has already established the level-set (LS) method to treat the former. Reactive transport can be directly modeled through a combination of reactor-scale simulations, which determine the physical and chemical properties of the reactant species, and first-principle simulations describing the interaction of such species with the surface. However, these simulations are computationally very costly and complex. Also, there is still substantial debate about the intricacies of the chemical phenomena in many manufacturing processes, thus, this type of modeling might not be possible.

Instead, this thesis presents phenomenological models for reactive transport based on first-order reversible Langmuir kinetics using a single effective particle. A particle can either represent a specific chemical species or it can be an aggregate proxy of multiple and often unknown reactants. Although several processes require the consideration of multiple particles, the spirit of parsimony required for phenomenological modeling motivates this thesis to explore the profound complexity already present in reactive single-particle transport.

By reducing the physical and chemical complexity to a restricted number of parameters, not only can experimental surfaces be reproduced, but also insights into the surface chemistry can be gained. To achieve this, this thesis presents an overview of the existing approaches for reactant flux distribution calculations. A particular focus is given on viewing the venerable but often misunderstood Knudsen diffusive transport through a novel lens, which is the major methodological contribution of this work. Then, these reactive transport models are applied to specific problems.

The first novel contribution of this thesis with respect to applications is the novel integration of a Knudsen diffusion-based model with the LS method for thermal atomic layer processing (ALP) in high aspect ratio (AR) structures. This integration permits a thorough analysis of the model parameters and the qualitative investigation of a platform for three-dimensional (3D) integration of novel memories. In another contribution, existing flux calculation approaches are evaluated for the process of low-bias sulfur hexafluoride (SF₆) plasma etching of silicon (Si). This enables new interpretations and analyses, most notably the extraction of an empirical relationship between experimentally-accessible measurements and surface-chemical properties. In a final novel application, the capabilities of topography simulation are showcased in the optimization of Si microcavity resonators through a custom robust automatic calibration procedure.

In conclusion, it is shown that the final topography of a processed device carries the fingerprint of the surface chemistry occurring during the manufacturing process. This fundamental result is the enabler of direct modeling of experimentally processed surfaces as well as inverse modeling: The extraction of chemical information from experimental surfaces. In summary, topography simulation using phenomenological single-particle reactive transport modeling is a powerful tool which is able to complement reactor-scale and first-principle calculations.

Kurzfassung

Die fortschreitende Entwicklung der Mikro- und Nanotechnologie erfordert einen immer höheren Druck auf die betroffenen Herstellungsprozesse. Im Kontext dieser wachsenden Komplexität muss das experimentelle und empirische Wissen der Prozessentwickler_innen durch physikalisch fundierte Modellierung und Simulation ergänzt werden. Historisch lag der Fokus der Simulationen von Herstellungsprozessen auf jenen Schritten, die sich direkt auf die elektrischen Eigenschaften der Halbleiter-Bauelemente auswirken. In den letzten Jahrzehnten wurde jedoch verstärktes Interesse auf die direkte Untersuchung der Struktur der hergestellten Bauelemente gelegt. Die computergestützte Modellierung der sich verändernden Oberflächen während des Herstellungsprozesses ist der Aufgabenbereich der Topographiesimulation.

Die Topographiesimulation besteht aus zwei Hauptelementen: Eine Methode zur Beschreibung der sich bewegenden Oberflächen und reaktive Transportmodelle, welche die Geschwindigkeitsfelder der Oberflächenadvektion bestimmen. Der Schwerpunkt dieser Arbeit liegt auf letzterem, aufbauend auf früheren Arbeiten, die bereits die "Level-Set"-Methode zur Behandlung des ersteren etabliert haben. Der reaktive Transport kann durch eine Kombination aus Simulationen im Reaktormaßstab, welche die physikalischen und chemischen Eigenschaften der reaktiven Spezies bestimmen, und "First-Principles"-Simulationen, welche die Interaktion dieser Spezies mit der Oberfläche beschreiben, direkt modelliert werden. Allerdings sind diese Simulationen sehr rechenintensiv und komplex. Außerdem gibt es immer noch umfangreiche Diskussionen über die Besonderheiten der chemischen Phänomene in vielen Herstellungsprozessen, sodass diese Art der Modellierung unter Umständen nicht möglich ist.

Stattdessen werden in dieser Dissertation phänomenologische Modelle für den reaktiven Transport auf Basis der reversiblen Langmuir-Kinetik erster Ordnung unter Verwendung eines einzigen effektiven Partikels vorgestellt. Ein Partikel kann entweder eine spezifische, chemische Spezies repräsentieren oder es kann ein aggregierter Ersatz für mehrere und oft unbekannte Reaktanten sein. Obwohl mehrere Prozesse die Berücksichtigung mehrerer Partikel erfordern, motiviert der für die phänomenologische Modellierung erforderliche Sinn für Parsimonie diese Dissertation dazu, die tiefe Komplexität zu erforschen, die bereits im reaktiven Einzelpartikeltransport vorhanden ist.

Indem die physikalische und chemische Komplexität auf eine begrenzte Anzahl von Parametern reduziert wird, können nicht nur experimentelle Oberflächen reproduziert werden, sondern auch Einblicke in die Oberflächenchemie gewährt werden.
Um dies zu erreichen, wird in dieser Dissertation ein Überblick über die bestehenden Methoden zur Berechnung der Verteilung von Reaktantenflüssen gegeben. Ein besonderer Schwerpunkt liegt auf der Betrachtung des altbekannten, aber oft missverstandenen Knudsen-Diffusionstransports durch eine neue Linse, was der wichtigste methodische Beitrag dieser Arbeit ist. Dann werden diese reaktiven Transportmodelle auf spezifische Probleme angewandt.

Der erste neuartige Beitrag dieser Dissertation in Bezug auf Anwendungen ist die neuartige Integration eines auf Knudsen-Diffusion basierenden Modells mit der "Level-Set"-Methode für die thermische Atomlagenbearbeitung in Strukturen mit hohem Aspektverhältnis. Diese Integration erlaubt eine gründliche Analyse der Modellparameter und die qualitative Untersuchung einer Plattform für die dreidimensionale Integration von neuartigen Speichern. In einem weiteren Beitrag werden bestehende Methoden zur Flusskalkulation für den Prozess des Schwefelhexafluorid-Plasmaätzens von Silizium mit geringer Bias bewertet. Dies ermöglicht neue Interpretationen und Analysen, insbesondere die Extraktion einer empirischen Beziehung zwischen experimentell zugänglichen Messungen und oberflächenchemischen Eigenschaften. In einer finalen, neuartigen Anwendung werden die Möglichkeiten der Topographiesimulation bei der Optimierung von Si-Mikrokavitätenresonatoren durch ein eigenes robustes automatisches Kalibrierungsverfahren demonstriert.

Abschließend wird gezeigt, dass die Endtopographie eines gefertigten Bauelements den Fingerabdruck der Oberflächenchemie trägt, die während des Herstellungsprozesses auftritt. Dieses fundamentale Ergebnis ermöglicht die direkte Modellierung von experimentell prozessierten Oberflächen ebenso wie die inverse Modellierung: die Extraktion von chemischen Informationen aus experimentellen Oberflächen. Zusammenfassend lässt sich sagen, dass die Topographiesimulation mit Hilfe der phänomenologischen Modellierung des reaktiven Einzelteilchentransports ein leistungsstarkes Instrument ist, das die Berechnungen im Reaktormaßstab und die "First-Principles"-Berechnungen ergänzen.

Resumo

A evolução contínua da micro e nanotecnologia coloca uma pressão cada vez maior sobre os processos de fabricação envolvidos. No contexto dessa complexidade em expansão, o conhecimento experimental e empírico dos desenvolvedores de processos deve ser complementado com modelagem e simulação fisicamente fundamentada. Historicamente, o foco das simulações de processos de fabricação incidiu sobre as etapas de fabricação que impactam diretamente nas características elétricas dos dispositivos finais. Em décadas mais recentes, no entanto, a maior atenção tem sido dada à investigação direta da estrutura dos dispositivos fabricados. A modelagem computacional de superfícies em evolução durante os processos de fabricação é o âmbito da simulação de topografia.

A simulação de topografia é composta de dois elementos principais: um método para descrever as superfícies advectantes e modelos de transporte reativos que determinam os campos de velocidade de advecção da superfície. O foco dessa tese é no último, baseado em trabalhos anteriores que já estabeleceram o método de conjunto de nível para tratar o primeiro. O transporte reativo pode ser modelado diretamente através de uma combinação de simulações em escala de reator, que determinam as propriedades físicas e químicas das espécies químicas reativas, e simulações de primeiros princípios descrevendo a interação de tais espécies com a superfície. No entanto, essas simulações são muito caras e complexas computacionalmente. Além disso, ainda há um debate considerável sobre os meandros dos fenômenos químicos envolvidos em muitos processos de fabricação, portanto, esse tipo de modelagem nem sempre é possível.

Ao invés disso, essa tese apresenta modelos fenomenológicos para o transporte reativo baseados em cinética de Langmuir reversível de primeira ordem, usando uma única partícula efetiva. Uma partícula pode representar uma espécie química particular ou pode ser um substituto agregado de múltiplos reagentes, os quais por vezes são desconhecidos. Embora diversos processos exijam a consideração de múltiplas partículas, o espírito de parcimônia necessário para a modelagem fenomenológica leva essa tese a explorar a profunda complexidade já presente no transporte reativo de uma única partícula.

Reduzindo a complexidade física e química a um número restrito de parâmetros, não apenas as superfícies experimentais podem ser reproduzidas, mas também é possível obter uma compreensão maior da química de superfície. Para conseguir isso, esta tese apresenta uma visão geral das abordagens já existentes para cálculos de distribuição de fluxo de reagentes. Um enfoque particular é dado na análise do venerável,
mas freqüentemente incompreendido, transporte difusivo de Knudsen através de uma nova perspectiva, o que constitui a principal contribuição metodológica deste trabalho. Em seguida, esses modelos de transporte reativos são aplicados a problemas específicos.

A primeira contribuição original dessa tese com relação a aplicações é a integração inovadora de um modelo de difusão de Knudsen com o método de conjunto de nível para processamento térmico de camadas atômicas em estruturas de alta relação de aspecto. Essa integração permite uma análise completa dos parâmetros do modelo e a investigação qualitativa de uma plataforma de integração tridimensional de novas memórias. Em outra contribuição, as abordagens existentes para o cálculo do fluxo são avaliadas para o processo de corrosão de silício com hexafluoreto de enxofre em baixo viés de voltagem. Isso permite interpretações e análises originais, mais notavelmente a extração de uma relação empírica entre as medidas experimentalmente acessíveis e propriedades químicas de superfície. Em uma última contribuição original, as capacidades de simulação topográfica são demonstradas na otimização de microcavidades ressonadoras de silício através de um procedimento de calibração automática robusto e individualizado.

Em conclusão, mostra-se que a topografia final de um dispositivo processado carrega a impressão digital da química de superfície que ocorre durante o processo de fabricação. Esse resultado fundamental é o que permite a modelagem direta de superfícies processadas experimentalmente, bem como a modelagem inversa: a extração de informações químicas de superfícies experimentais. Em resumo, a simulação topográfica usando a modelagem fenomenológica de transporte reativo por uma única partícula é uma ferramenta poderosa que é capaz de complementar os cálculos em escala de reator e de primeiros princípios.

Acknowledgement

The intense journey that is the pursuit of a doctoral degree is, ultimately, a product of the collaboration between the student and the supervisors. I would like to firstly thank my primary supervisor, Prof. Josef Weinbub, for his outstanding level of support for all of my ideas, plans, and ambitions. I always knew that I could count on him to provide sharp, precise, and timely feedback which in turn has enormously expanded my own scientific skills.

Jointly, I owe a debt of gratitude to my secondary supervisor, Prof. Siegfried Selberherr. I greatly admire and appreciate his immense legacy in establishing the Institute for Microelectronics as a leading center for globally-recognized research. The Institute was an excellent place of work, where I could follow my own ideas while still relying on the incredible expertise of the local researchers. The lively discussions fostered by Prof. Selberherr have made me a more well-rounded scientist, his feedback has ingrained in me the necessity of a sharp eye for detail, and his extensive knowledge has guided me from my beginnings as a complete novice in the field of TCAD.

Additional thanks go to Dr. Andreas Hössinger from the industrial partner Silvaco Europe Ltd., who has worked intensely in conjunction with Prof. Weinbub to create and maintain the Christian Doppler Laboratory for High Performance Technology Computer-Aided Design (HPTCAD). His input has prompted research questions which are simultaneously scientifically engaging as well as industrially relevant. I am grateful for the interesting discussions we had in our regular meetings and for his valuable and efficient feedback.

This thesis would not have been possible without the help of my colleagues and friends from the Christian Doppler Laboratory for HPTCAD and also from the Institute for Microelectronics in general. Special thanks go to Frâncio Rodrigues, co-founder of the Brazilian specialty coffee corner, who has been my editor and proofreader of first resort, and who has indulged with me in lengthy musings about the past, present, and future of the global semiconductor industry. Additional thanks go to Christoph Lenz, Paul Manstetten, Xaver Klemenschits, Roberto Orio, Felipe Ribeiro, Alex Toifl, Michael Quell, Tobias Reiter, and Alex Scharinger for the scientific and also not-so-scientific lunch discussions. This work would also not have happened without the practical help from the entire staff of the Institute, and to that I thank Diana Pop and Petra Kampter-Jonas, who help hold everything together, and Prof. Tibor Grasser for his stewardship.

My support network originally started in that far away land of Porto Alegre, Brazil, but over the years it now extends globally. My everlasting thanks for the friendship and support from my fellow expatriates, exiles, and outcasts: Pedro Perfeito, Arthur Loureiro, Augusto Medeiros, Rodrigo Sieben, Natália Amaral-Skreinig, Gustavo Amaral-Skreinig, Genário Oliveira, Matheus Pinheiro, and Wilton Loch. Your support, companionship, and role-playing game sessions were absolutely necessary for surviving these trying times. My friends from other parts of the planet are also always present at heart and are dearly thanked. Alex Zimmermann, Ahish Vishwanatha, and Bill Heymann have all been fundamental for me to overcome so many difficulties, and I treasure your camaraderie.

Last but absolutely not least I give my heartfelt thanks to my amazing family. To my beloved wife Aline, the best part of my life, I am so grateful for you being my partner, supporter, and number one fan in this amazing journey through this narrow bridge that is life. I also thank my brother João Pedro, who has always strived to be with me everywhere I am, even when geographically impossible, to always give me the best advice. My aunt Susana, who always takes the time to check in on me, is deeply thanked. My incredible parents have supported me throughout my life, and I am so grateful for everything they have done through the years. I thank my father Sergio for the love, support, and encouragement for me to always follow my dreams and passions. Finally, my mother Beatriz has always been my great inspiration for all academic pursuits, showing me that the world can be understood and repaired through both science and loving kindness. My love and thanks to you.

2 Review of Reactive Transport Models for Topography Simulation

2.1 The Level-Set Method

2.2 Langmuir Adsorption Kinetics