Solution for the critical thickness models of dislocation generation in epitaxial thin films using the Lambert W function

On the exact analytical solution of some families of equilibrium critical thickness transcendental equations

“…A short overview and description of the currently known approaches for the critical thickness of strained epitaxial films is presented in Ref. 1 and several mathematical models formulated to describe this can be found in Refs. [1][2][3][4][5][6][7][8][9][10][11][12].…”

“…Let us describe the problem discussed in this article using the equation for the equilibrium critical thickness obtained by Freund 1,4 in the following form:…”

“…[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 However, 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. This method gives an advantage over Newton's due to the fact it converges faster.…”

“…Perovich's Special Trans Functions Theory (STFT) has been proved to be a very powerful theory for solving transcendental equations and obtaining exact analytical closed-form solutions without any assumptions, hypotheses nor any additional proofs, [13][14][15][16][17][18][19][20][21][22][23] as opposed to the approaches presented in papers. 1,7 Examples of its application are shown in papers concerning the calculation of the closed-form solutions in some families of transcendental equations, 19 Lambert transcendental equation, 24 engineering materials, 26 theory of neutron slowing down, 13,14 linear transport theory, 17,23,27 Plutonium temperature estimation, 21 temperature estimation, 22 nonlinear circuit theory 15 and Hopfield neuron analysis, 18 as well as solar cell analysis 20,25 etc. Note that, also, a broad class of nonlinear functional equations of the transcendental type which appear in Refs.…”