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.

Topics
Thermodynamic states and processes, Theoretical computer science, Software engineering, Epitaxy, Thin films, Functional analysis, Integral equations, Partial differential equations, Complex functions, Chemical elements

The critical thickness of any strained layer is defined as the minimum thickness required to form misfit dislocations at the strained interface. 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. 1–12. The existence of a critical thickness for dislocation generation in thin films was proposed by Frank and Van der Merwe.2 In 1974, Matthews and Blakeslee3 derived an equation for the critical thickness, based on the force equilibrium criterion. A theoretical approach used to calculate the critical film thickness of dislocation generation in epitaxial thin films based on another criterion, namely the stability criterion of dislocation, was introduced by Freund4 in 1987. The two models present significant mathematical similarities.

Let us describe the problem discussed in this article using the equation for the equilibrium critical thickness obtained by Freund1,4 in the following form:

(1)

where b is the magnitude of the Burgers vectors, ν the Poisson ratio of the epilayer material, ε0 the misfit strain, r0 the cut-off radius for the dislocation core, and α the angle between the glide plane and the interface.

After structural modification, Eq. (1) takes a form of a new transcendental equation:

(2)

On the other hand, equilibrium critical thickness equation (Eq. (1)) can theoretically be modeled as the Matthews and Blakeslee equation of the form:

(3)

where A, B and C represent material constants with the following formulae:

(4)

After structural modification by an abbreviation of the form:

(5)

Eq. (3) takes a form of a well-known transcendental equation:

(6)

where:

(7)

A number of papers present new methods for determining the value of the equilibrium critical thickness.1–7 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. Let us note that solutions obtained in Refs. 1 and 7 are not obtained in an analytical closed form. In other words, presented mathematical methods in Refs. 1 and 7 are nothing more than advanced iteration methods.

Mathematical form of the relationship between the equilibrium critical thickness and material parameters can be reduced to a transcendental equation (6). 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,13–23 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 theory15 and Hopfield neuron analysis,18 as well as solar cell analysis20,25 etc. Note that, also, a broad class of nonlinear functional equations of the transcendental type which appear in Refs. 5, 6, and 8–12 can be solved by direct applications of the STFT.

In this article, 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.

The biggest impact of this work is in the usage of a new STF theory approach in formulae genesis for the equilibrium critical thickness of misfit dislocation generation in epitaxial thin films. All investigations and analysis of the equilibrium critical thickness in this paper have been consistently accomplished with the usage of a new STF theory. STFT ensures reaching extreme precisions in numerical results (arbitrary number of accurate digits in the numerical structure of the transcendental numbers), which is reflected in this paper as well, where we show the highest precision in defining the equilibrium critical thickness achieved so far. That, of course, implies that the relevant constants (as, b etc.) have been used with greater number of exact digits than in conventional approaches.

Driven by the thorough analysis of the results obtained, we believe that STFT (supported by Mathematica software), as a theoretical approach, represents a novel theoretical standard in analysis of the equilibrium critical thickness of misfit dislocation generation in epitaxial thin films.

In this section, as above mentioned, the problem of finding an exact analytical closed-form solution to the transcendental equation (6) using the Special Trans Functions Theory,13–27 will be discussed in detail.

For convenience, we restrict ourselves to the one dimensional transcendental equation given in section I. Let us note that Eq. (6) has two real solutions: z> (for z > 1) and z< (for z < 1),19,21,26 which are presented in Fig. 1. Consequently, the subject of the theoretical analysis presented here is the transcendental equation of the form:

(8)
FIG. 1.
FIG. 1. Graphical presentation of two solutions to Eq. (6).
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Graphical presentation of two solutions to Eq. (6).

FIG. 1.
FIG. 1. Graphical presentation of two solutions to Eq. (6).
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Graphical presentation of two solutions to Eq. (6).

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Authors’ Let us note that the analytical solution to the transcendental equation (8), for Z < 1, has been reported previously in several articles.13,14,19 Namely, for Z < 1, the equation (8) has an analytical closed-form solution of the form:

(9)

where trans< (β) is a new special trans function defined as:

(10)

with [x] denoting the greatest integer ≤x. For practical analysis and numerical calculation, expression (10), for Z<, takes the following form:

(11)

where 〈z<〉P< denotes numerical values of the transcendental number z< given with

(12)

accurate digits, where the error function G≺ is defined as:

(13)

More explicitly, for a fixed variable M, 〈z<〉P< takes the form:

(14)

Expression (14) gives the number of accurate digits in the numerical structure of the transcendental number z<.

The mentioned outline in Ref. 26 is also applicable for derivation of the analytical closed-form solution to equation (8) for Z > 1 (trans>(β)), but for the appropriate equation for identification (EQID) in a form of an integral equation. Note that in the aforementioned outline of the special trans functions (trans<(β) and trans>(β)) derivation, some of its most elementary properties are discussed. It is not difficult to define necessary and sufficient conditions that the partial differential equation and the integral equation have to satisfy in order to be equation candidates for the EQID of the transcendental equation (8). Namely, the transcendental equation (8) for Z > 1, can be reformulated as follows:

(15)

where:

(16)

with z< defined in Eqs. (10) and (14). Eq. (15) has the analytical closed-form solution:

(17)

where trans > β , a is a new special trans function defined as:

(18)

and F > b , u given as:

(19)

where:

(20)

and, similarly as before, u / a denotes the greatest integer ≤u/a. Since R ′ b , u = − b e − a b R ( b , u − a ) ⁠, one obtains:

(21)

where:

(22)

with a defined as follows: a ∈ R + and a > 1.

The practically applicable formula (16) takes the form:

(23)

where 〈Z>〉P> is the value of the transcendental number Z> given with P> accurate digits. Analogously to the previous case: P > = 1 − log G > ⁠, where the error function G> is defined as G > = Z > − B ⋅ e Z > ⁠. More explicitly, formula (23) takes the form:

(24)

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.

In this article, transcendental equations for the critical thickness analysis are presented in some detail and analytical closed-form solutions are obtained for them, using the Special Trans Function Theory. Let us note that, for practical applications, we have to recall that there are two possible values of z: z< and z>. To find the value of z that correctly describes the evolution of the critical thickness, additional considerations are required. We identify z> as the value that correctly describes the evolution of the critical thickness as a function of the lattice misfit ε0, due to the fact that a misfit ε0 → 0 (e.g. β → 0) corresponds to hc → ∞. In other words, an increasing misfit must cause a decreasing critical thickness. Thus, from Eqs. (7) and (18) we have:

(25)

or, more explicitly: h c = A lim x → ∞ ln F − β , x − 1 F − β , x + A lim x → ∞ ln F > b , u − a F > b , u ⁠, where F − β , x ⁠, F > b , u are defined by Eqs. (10) and (19) respectively. For practical analysis and numerical calculation, formula (25), for critical thickness hc with P accurate digits in numerical structure, takes the form:

(26)

In this section some numerical results for the equilibrium critical thickness of misfit dislocation generation in epitaxial thin films, based on the Eq. (26), will be presented. Tables I and II show some numerical results for different values of r0. The graphical simulations are presented in Fig. 2, where functions of the form h c = h c ε o ⁠, as well as the gradient coefficients of the type ∂ h c / ∂ ε o = φ ε o are displayed for different values of r0.

TABLE I.

Numerical results for the equilibrium critical thickness of misfit dislocation generation in epitaxial thin films based on formula (26).

ε0 β Z> G hc [nm] ∂hc/∂εo
0.0001  0.00122  8.8905  6.64ߙ⋅ߙ10−22  1997.62  −2.25ߙ⋅ߙ10−2 
0.0002  0.00244  8.1049  6.19ߙ⋅ߙ10−22  910.54  −5.19ߙ⋅ߙ10−3 
0.0003  0.00367  7.6404  6.40ߙ⋅ߙ10−22  572.24  −2.19ߙ⋅ߙ10−3 
0.0004  0.00489  7.3083  2.24ߙ⋅ߙ10−22  410.52  −1.18ߙ⋅ߙ10−3 
0.0005  0.00612  7.0490  4.49ߙ⋅ߙ10−32  316.76  −7.38ߙ⋅ߙ10−4 
0.0006  0.00734  6.8360  1.06ߙ⋅ߙ10−33  255.99  −4.99ߙ⋅ߙ10−4 
0.0007  0.00856  6.6550  1.31ߙ⋅ߙ10−35  213.61  −3.59ߙ⋅ߙ10−4 
0.0008  0.00979  6.4975  1.88ߙ⋅ߙ10−36  182.49  −2.69ߙ⋅ߙ10−4 
0.0009  0.01101  6.3581  1.86ߙ⋅ߙ10−37  158.73  −2.09ߙ⋅ߙ10−4 
0.001  0.01224  6.2328  9.60ߙ⋅ߙ10−40  140.04  −1.66ߙ⋅ߙ10−4 
0.002  0.02448  5.3954  4.73ߙ⋅ߙ10−48  60.61  −3.72ߙ⋅ߙ10−5 
0.003  0.03672  4.8920  2.51ߙ⋅ߙ10−53  36.63  −1.53ߙ⋅ߙ10−5 
0.004  0.04896  4.5267  1.32ߙ⋅ߙ10−58  25.42  −8.15ߙ⋅ߙ10−6 
0.005  0.06120  4.2375  1.63ߙ⋅ߙ10−63  19.04  −4.98ߙ⋅ߙ10−6 
0.006  0.07344  3.9967  3.17ߙ⋅ߙ10−60  14.96  −3.32ߙ⋅ߙ10−6 
0.007  0.08568  3.7892  7.39ߙ⋅ߙ10−57  12.16  −2.36ߙ⋅ߙ10−6 
0.008  0.09792  3.6062  7.43ߙ⋅ߙ10−54  10.12  −1.75ߙ⋅ߙ10−6 
0.009  0.11016  3.4417  3.96ߙ⋅ߙ10−51  8.59  −1.34ߙ⋅ߙ10−6 
0.01  0.12240  3.2918  1.29ߙ⋅ߙ10−48  7.39  −1.06ߙ⋅ߙ10−6 
0.015  0.18360  2.6812  4.93ߙ⋅ߙ10−38  4.01  −4.27ߙ⋅ߙ10−7 
0.02  0.24480  2.1921  5.73ߙ⋅ߙ10−29  2.46  −2.26ߙ⋅ߙ10−7 
0.025  0.30600  1.7353  9.23ߙ⋅ߙ10−20  1.55  −1.47ߙ⋅ߙ10−7 
ε0 β Z> G hc [nm] ∂hc/∂εo
0.0001  0.00122  8.8905  6.64ߙ⋅ߙ10−22  1997.62  −2.25ߙ⋅ߙ10−2 
0.0002  0.00244  8.1049  6.19ߙ⋅ߙ10−22  910.54  −5.19ߙ⋅ߙ10−3 
0.0003  0.00367  7.6404  6.40ߙ⋅ߙ10−22  572.24  −2.19ߙ⋅ߙ10−3 
0.0004  0.00489  7.3083  2.24ߙ⋅ߙ10−22  410.52  −1.18ߙ⋅ߙ10−3 
0.0005  0.00612  7.0490  4.49ߙ⋅ߙ10−32  316.76  −7.38ߙ⋅ߙ10−4 
0.0006  0.00734  6.8360  1.06ߙ⋅ߙ10−33  255.99  −4.99ߙ⋅ߙ10−4 
0.0007  0.00856  6.6550  1.31ߙ⋅ߙ10−35  213.61  −3.59ߙ⋅ߙ10−4 
0.0008  0.00979  6.4975  1.88ߙ⋅ߙ10−36  182.49  −2.69ߙ⋅ߙ10−4 
0.0009  0.01101  6.3581  1.86ߙ⋅ߙ10−37  158.73  −2.09ߙ⋅ߙ10−4 
0.001  0.01224  6.2328  9.60ߙ⋅ߙ10−40  140.04  −1.66ߙ⋅ߙ10−4 
0.002  0.02448  5.3954  4.73ߙ⋅ߙ10−48  60.61  −3.72ߙ⋅ߙ10−5 
0.003  0.03672  4.8920  2.51ߙ⋅ߙ10−53  36.63  −1.53ߙ⋅ߙ10−5 
0.004  0.04896  4.5267  1.32ߙ⋅ߙ10−58  25.42  −8.15ߙ⋅ߙ10−6 
0.005  0.06120  4.2375  1.63ߙ⋅ߙ10−63  19.04  −4.98ߙ⋅ߙ10−6 
0.006  0.07344  3.9967  3.17ߙ⋅ߙ10−60  14.96  −3.32ߙ⋅ߙ10−6 
0.007  0.08568  3.7892  7.39ߙ⋅ߙ10−57  12.16  −2.36ߙ⋅ߙ10−6 
0.008  0.09792  3.6062  7.43ߙ⋅ߙ10−54  10.12  −1.75ߙ⋅ߙ10−6 
0.009  0.11016  3.4417  3.96ߙ⋅ߙ10−51  8.59  −1.34ߙ⋅ߙ10−6 
0.01  0.12240  3.2918  1.29ߙ⋅ߙ10−48  7.39  −1.06ߙ⋅ߙ10−6 
0.015  0.18360  2.6812  4.93ߙ⋅ߙ10−38  4.01  −4.27ߙ⋅ߙ10−7 
0.02  0.24480  2.1921  5.73ߙ⋅ߙ10−29  2.46  −2.26ߙ⋅ߙ10−7 
0.025  0.30600  1.7353  9.23ߙ⋅ߙ10−20  1.55  −1.47ߙ⋅ߙ10−7 
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TABLE II.

Numerical results for the equilibrium critical thickness of misfit dislocation generation in epitaxial thin films based on formula (26) for different values of r0.

hc [nm] ∂hc/∂ε0
ε0 r 0 = b 2 r 0 = b 4 r 0 = b 10 r 0 = b 2 r 0 = b 4 r 0 = b 10
0.0001  910.545  410.525  140.045  −1.03ߙ⋅ߙ10−2  −4.75ߙ⋅ߙ10−3  −1.66ߙ⋅ߙ10−3 
0.0002  410.525  182.492  60.6146  −2.37ߙ⋅ߙ10−3  −1.07ߙ⋅ߙ10−3  −3.72ߙ⋅ߙ10−3 
0.0003  255.998  112.624  36.6394  −9.99ߙ⋅ߙ10−4  −4.50ߙ⋅ߙ10−4  −1.53ߙ⋅ߙ10−4 
0.0004  182.492  79.5936  25.4276  −5.38ߙ⋅ߙ10−4  −2.41ߙ⋅ߙ10−4  −8.15ߙ⋅ߙ10−5 
0.0005  140.045  60.6146  19.0427  −3.33ߙ⋅ߙ10−4  −1.40ߙ⋅ߙ10−4  −4.98ߙ⋅ߙ10−5 
0.0006  112.621  48.4067  14.9670  −2.25ߙ⋅ߙ10−4  −1.00ߙ⋅ߙ10−4  −3.32ߙ⋅ߙ10−5 
0.0007  93.5585  39.9508  12.1630  −1.61ߙ⋅ߙ10−4  −7.14ߙ⋅ߙ10−5  −2.36ߙ⋅ߙ10−5 
0.0008  79.9365  33.7786  10.1285  −1.20ߙ⋅ߙ10−4  −5.33ߙ⋅ߙ10−5  −1.75ߙ⋅ߙ10−5 
0.0009  68.9559  29.0935  8.59257  −9.35ߙ⋅ߙ10−5  −4.11ߙ⋅ߙ10−5  −1.34ߙ⋅ߙ10−5 
0.001  60.6156  25.4276  7.39654  −7.44ߙ⋅ߙ10−5  −3.26ߙ⋅ߙ10−5  −1.06ߙ⋅ߙ10−5 
hc [nm] ∂hc/∂ε0
ε0 r 0 = b 2 r 0 = b 4 r 0 = b 10 r 0 = b 2 r 0 = b 4 r 0 = b 10
0.0001  910.545  410.525  140.045  −1.03ߙ⋅ߙ10−2  −4.75ߙ⋅ߙ10−3  −1.66ߙ⋅ߙ10−3 
0.0002  410.525  182.492  60.6146  −2.37ߙ⋅ߙ10−3  −1.07ߙ⋅ߙ10−3  −3.72ߙ⋅ߙ10−3 
0.0003  255.998  112.624  36.6394  −9.99ߙ⋅ߙ10−4  −4.50ߙ⋅ߙ10−4  −1.53ߙ⋅ߙ10−4 
0.0004  182.492  79.5936  25.4276  −5.38ߙ⋅ߙ10−4  −2.41ߙ⋅ߙ10−4  −8.15ߙ⋅ߙ10−5 
0.0005  140.045  60.6146  19.0427  −3.33ߙ⋅ߙ10−4  −1.40ߙ⋅ߙ10−4  −4.98ߙ⋅ߙ10−5 
0.0006  112.621  48.4067  14.9670  −2.25ߙ⋅ߙ10−4  −1.00ߙ⋅ߙ10−4  −3.32ߙ⋅ߙ10−5 
0.0007  93.5585  39.9508  12.1630  −1.61ߙ⋅ߙ10−4  −7.14ߙ⋅ߙ10−5  −2.36ߙ⋅ߙ10−5 
0.0008  79.9365  33.7786  10.1285  −1.20ߙ⋅ߙ10−4  −5.33ߙ⋅ߙ10−5  −1.75ߙ⋅ߙ10−5 
0.0009  68.9559  29.0935  8.59257  −9.35ߙ⋅ߙ10−5  −4.11ߙ⋅ߙ10−5  −1.34ߙ⋅ߙ10−5 
0.001  60.6156  25.4276  7.39654  −7.44ߙ⋅ߙ10−5  −3.26ߙ⋅ߙ10−5  −1.06ߙ⋅ߙ10−5 
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FIG. 2.
FIG. 2. Function of the form h c = h c ε o for different values of r (r = r0).
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Function of the form h c = h c ε o for different values of r (r = r0).

FIG. 2.
FIG. 2. Function of the form h c = h c ε o for different values of r (r = r0).
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Function of the form h c = h c ε o for different values of r (r = r0).

Close modal

From a theoretical point of view, solutions (26) for hc can be found with an arbitrary order of accuracy by taking appropriate values of x and u / a ⁠. It is not difficult to see that the analytical solution (26) gives impressive results which confirm successful application of the STFT. In an exact way, the number of accurate digits in the practical applicable hc is in accordance with the physical requirements of exactness. Accordingly, the final form of solution (26) continues to stay in the domain of an analytical form, regardless of the number of accurate digits in the numerical structure of hc obtained via computer calculations. Note, the subject of the analysis presented in this section is also the numerical efficiency comparison between the famous Lambert W(β) function and the special Perovich’s trans functions. It is not difficult to realize that trans<(β) and trans>(β) have superior accuracy and computational efficiency for all values of β.19,26

We have used infinite precision arithmetics and validated the accuracy and computational complexity of the two formulas with Mathematica ver. 5.2. The reference value of the solution has been computed by the numerical equation solver Find Root and the built-in function Product Log. The Lambert W(β), along with trans<(β) and trans>(β) functions, has been timed by the Mathematica function Timing to compare their execution times. The Timing command includes only CPU time spent in the Mathematica kernel for computing an expression. All computations were carried out on a 2.53 GHz Intel(R) Core(TM) i5 CPU with 3GB RAM, under Windows 7 Ultimate operating system, with Mathematica program (version 5.2 by Wolfram Research, Inc).

We found that trans<(β) has superior time computational efficiency for all values of β. It becomes clear, from the presented numerical values in Ref. 19, that the proposed formula for trans<(β) gives superior results compared to the Lambert one. Note that the Lambert W series for Z > 1 cannot be used for practical numerical computation. Consequently, for Z > 1, our formula (26) becomes a unique existing exact analytical expression for calculation of hc. Thus, it is a novel theoretical result. In addition, since (26) is an analytical closed-form result, we can obtain new gradient coefficients of type ∂hc/∂εo, ∂hc/∂α, which is useful for theoretical analysis of dynamics of the critical thickness.

From previous sections it is obvious that the Special Trans Functions Theory is a consistent general approach to solving transcendental equations in equilibrium critical thickness domain. This means that, in some manner, we can obtain various Special Trans Functions, including trans>(β).

A new formula within the equilibrium critical thickness theory, Eq. (26), is valid in the numerical sense (See Table I). Thus, the obtained analytical solutions, in addition to having a theoretical value, possess potential for practical application. The theoretical accuracy of the STFT13,14,24,26 is unlimited, and, therefore, extreme precision is attainable with this approach (See Table I). Advantage of the Perovich method is evident comparing to conventional analytical and numerical methods.19,20,23–25 Namely, solutions obtained by the STFT are exact analytic solutions in the closed form. In addition, it has to be underlined that computation complexity is far better than in conventional methods.

In this manuscript, it has been proven that the equilibrium critical thickness transcendental equation is solvable by direct application of the STFT. Without a doubt, this is a significant contribution of the STF theory. We claim that the STFT could stand for a standard theory for the analysis of the equilibrium critical thickness. We have found, using Mathematica programme, that the STF theory implies obtaining numerical results with arbitrary number of significant figures as well as an exact analytical closed-form solution to the equilibrium critical thickness. According to the authors’ knowledge, this is the first direct application of the STF theory to the genesis of the equilibrium critical thickness with extreme precision (with arbitrary number of accurate digits in the numerical structure of the equilibrium critical thickness). In other words, formula (26) is novel and original.

This work is supported by Ministry of Science of Montenegro, through the Project MN01/2337/14.

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