Abstract
The problem of finding an exact analytical closed-form solution of some families of transcendental equations, which describe the equilibrium critical thickness of misfit dislocation generation in epitaxial thin films, is studied in some detail by the Special Trans Functions Theory (STFT). A novel STFT mathematical approach with an analytical closed-form solution is presented. Structure of the STFT exact solutions, numerical results and graphical simulations confirm the validity of the basic principle of the STFT. The proposed STFT analytical approach shows qualitative improvement in theoretical sense (a novel gradient coefficient genesis), and, in accuracy when compared to the conventional analytical and numerical methods.
Highlights
The critical thickness of any strained layer is defined as the minimum thickness required to form misfit dislocations at the strained interface
Special Trans Functions Theory (STFT) ensures reaching extreme precisions in numerical results, which is reflected in this paper as well, where we show the highest precision in defining the equilibrium critical thickness achieved so far
Let us note that the problem of finding an exact analytical closed-form solution to the transcendental Eq (8) using the STFT, is presented in more detail in Refs. 21 and 26
Summary
The critical thickness of any strained layer is defined as the minimum thickness required to form misfit dislocations at the strained interface. A number of papers present new methods for determining the value of the equilibrium critical thickness.[1,2,3,4,5,6,7] Some of them apply an iterative solution based on Newton-Raphson method for solving Eq (6).[1,7] most of the existing methods employ Lambert W function with an iterative procedure to find numerical solutions of the equations describing the equilibrium critical thickness of misfit dislocation generation in epitaxial thin films.[1,7] Braun et al.[7] provided a method for solving the Lambert W function, based on Halley’s iteration. Some transcendental equations describing equilibrium critical thickness of misfit dislocation generation in epitaxial thin films are presented in some detail and analytical closed-form solutions are derived for them, applying STFT. Let us note that the problem of finding an exact analytical closed-form solution to the transcendental Eq (8) using the STFT, is presented in more detail in Refs. Let us note that the problem of finding an exact analytical closed-form solution to the transcendental Eq (8) using the STFT, is presented in more detail in Refs. 21 and 26
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
