Model for Predicting Film Thickness Depending on the Values of the Atomic Layer Deposition Process Parameters

Abstract

The ALD method is promising for producing thin films used in microelectronics. Optimization of ALD process parameters allows for shorter film production times and reduced reagent consumption. A model for predicting film thickness depending on process parameters was developed. Software was developed to determine the model’s kinetic constants based on experimental data. The algorithm for finding the constants was tested using simulated experimental data. With a film thickness measurement error of ±2.6 nm, ten experimental points may be sufficient to find the optimal process parameters. The fundamental feasibility of using chemical-kinetic models with a large number of constants for practical application is demonstrated.

1. Introduction

Atomic layer deposition (ALD) is a promising technology widely used in microelectronics. One of the important applications of ALD is the production of high-k dielectrics for the manufacture of metal oxide semiconductors (CMOS) and field effect transistors (FETs). Also, the ALD method is used to obtain antireflective coating, , surface passivation coatings − and chemical barrier coatings. In parallel with the ALD method, there is a method called molecular layer deposition (MLD), which uses the same principles as ALD, but to produce organic films from organic reagents. Films obtained on the basis of MLD are used, for example, for the fabrication of desalination membranes, the production of organic light-emitting diodes, , etc. The wide range of applications has generated active interest in research in the field of ALD. −

Despite the many advantages of ALD, such as high conformability of the resulting films and precision control of film thickness, the disadvantage of the method is the relatively low rate of film growth. − Initially, ALD was used to obtain inorganic films using inorganic precursors. − However, one of the drawbacks of such precursors is their low volatility, which results in slow film growth rates. Organometallic precursors, with higher volatility and, consequently, faster film growth rates, have emerged as an alternative to inorganic precursors. − Another solution to the problem of low film growth rate is the optimization of the ALD process parameters. At the same time, optimization of process parameters requires the presence of theoretical developments that allow predicting the film thickness depending on the process parameters.

Theoretical developments in the field of ALD have been actively carried out since the 2000s, − and the proposed theories are based on the idea of adsorption of a reactant (oxidizer precursor or coreagent precursor) on the surface of the resulting film and the reaction of precursor vapors with the adsorbed reactant. In the case of water as a reactant, adsorption leads to the formation of hydroxyl groups, which play a key role in film growth, as they fix the precursor on the surface. − The film thickness dependences on the precursor/reactant step time predicted by these theories have the shape of Langmuir-like curves, but in practice, the shape of the curves may differ from those theoretically predicted. The theory we proposed previously is based on taking into account the coadsorption of both reagents involved in the film growth process. The proposed theory is based on chemical-kinetic equations, on the basis of which a model of uniform film growth is presented in the form of a solution of linear differential equations in analytical form. Based on the proposed model, it was shown that the theory is capable of predicting not only Langmuir-like curves, but also more complex dependencies of film thickness on process parameters. In addition, the principles on which the theory is based allow it to be applied to the MLD method. However, the model uses eight unknown chemical-kinetic constants. This fact complicates the practical application of this theory. In this paper, we propose the LAFOR (Langmuir Adsorption First Order Reaction) model, which takes into account the uneven growth of the film during the first ALD cycles. The LAFOR model is a logical continuation of the theory proposed in. Software for visualizing the dependence of film thickness on process parameters (ALD-LAFORM Visual Tool) is developed and its capabilities is demonstrated. Software for finding kinetic constants based on experimental data (LAFORM Software) is developed and this paper presents a test of its algorithm for finding constants.

2. Material and Methods

The ALD-LAFORM Visual Tool and ALD-LAFORM Software code is written in Python using the Numpy, SciPy, Numba, and Matplotlib libraries. ALD-LAFORM Visual Tool and ALD-LAFORM are available as compiled executable files for Windows, , compiled by Pyinstaller.

Python scripts were used to test the algorithm, and the Numpy library was used to generate random values with a normal distribution. To determine the minima and maxima of functions, the Nelder–Mead method was used. For statistics, the gamma distribution and log–normal distribution were used, as well as the Kolmogorov–Smirnov test.

Testing of the algorithm included the generation of quasi-experimental data based on previously known kinetic constants, followed by finding kinetic constants based on the generated quasi-experimental data and comparison of the calculation results using the original and obtained (calculated) kinetic constants.

The fitting of the gamma and log–normal distributions was carried out using the optimization method of the p-value of the Smirnov–Kolmogorov test. For each sample, this method was used to fit the gamma distribution, log–normal and normal distributions.

To test the performance of the algorithm, data were simulated based on eqs –, and then the obtained data were used to find suitable constants. The values of the kinetic constants used in all calculations to test the algorithm are given in Table in the “Initial” column. The chosen constants are random and are within the acceptable ranges and far from the limits of the constraints described in Appendix A. The values of other physical parameters were chosen so that they were within reasonable limits of real physical quantities: precursor vapor pressure (P _p ) = 1 Pa, reactant vapor pressure (P _w ) = 30 Pa, film molar mass (M) = 0.15 kg/mol, film density (ρ) = 4000 kg/m³, stoichiometric coefficients γ _p and γ _w are 1 and 2, respectively.

1. Values of Calculated Kinetic Constants.

constant	initial	10 points	13 points	16 points	8 points (WP)	Unit
a p	0.12	0.203	0.204	0.157	0.119	Pa–1s–1
a w	0.92	0.952	0.951	0.951	0.920	Pa–1s–1
d p	8.2 × 10–2	9.99 × 10–15	1.01 × 10–15	4.57 × 10–2	8.30 × 10–2	s–1
d w	2.70 × 10–2	1.44 × 10–15	6.52 × 10–10	2.23 × 10–5	2.70 × 10–2	s–1
k p (0)	0.92	0.933	0.933	0.933	0.920	Pa–1s–1
k w (0)	1.2	1.68	1.67	1.38	1.19	Pa–1s–1
C p max	9.12 × 10–7	4.93 × 10–7	4.95 × 10–7	6.83 × 10–7	9.21 × 10–7	mol/m2
C w max	8.34 × 10–6	7.53 × 10–6	7.53 × 10–6	7.53 × 10–6	8.34 × 10–6	mol/m2
RMSD	-	1.1 × 10–18	1.1 × 10–17	8.5 × 10–16	3.2 × 10–16	m

To test the ability of the algorithm to find the values of kinetic constants based on experimental data, experimental film thicknesses were simulated by adding to the exact value of the film thickness a random variable with a normal distribution with zero mathematical expectation and standard deviation (σ) values of 0.1 nm, 0.25 nm, 0.5 nm, 0.75 and 1 nm (see eq B.1). To estimate the error in determining the film thickness using the found constants, the values of the maximum relative difference (MaxRD) and the mean relative difference (MeanRD) were used. To assess the feasibility of finding optimal process parameters using the constants found by the algorithm, the optimal growth rate relative error (OGRRE) and optimal cycle time relative error (OCTRE) values were used. Definitions for MaxRD, MeanRD, OGRRE, and OCTRE are presented in Appendix B.

3. Theory and Algorithms

3.1. ALD-LAFOR Model Equations

In, a chemical-kinetic model of ALD was introduced, where for each step of the cycle, systems of linear differential equations were presented:

Reactant step

$$\{\begin{array}{l}{Ċ}_w=A_w-D_w{C}_p-{\alpha }_w{C}_w\\ {Ċ}_p=-{\alpha }_p{C}_p\\ \mathrm{\Delta }{f}_i[w]=k_w{\int }_0^{{\tau }_w}{C}_{p}dt=\frac{k_w}{{\alpha }_p}{C}_p(t_i)(1-e^{-{\alpha }_p{\tau }_w})\end{array}$$ 1

Postreactant purging step

$$\{\begin{array}{l}{Ċ}_w=-d_w{C}_w\\ {Ċ}_p=-d_p{C}_p\\ \mathrm{\Delta }{f}_i[d]=0\end{array}$$ 2

Precursor step

$$\{\begin{array}{l}{Ċ}_w=-{\omega }_w{C}_w\\ {Ċ}_p=A_p-D_p{C}_w-{\omega }_p{C}_p\\ \mathrm{\Delta }{f}_i[p]=k_p{\int }_0^{{\tau }_p}{C}_{w}dt=\frac{k_p}{{\omega }_w}{C}_w^0(1-e^{-{\omega }_w{\tau }_p})\end{array}$$ 3

Postprecursor purging step

$$\{\begin{array}{l}{Ċ}_w=-d_w{C}_w\\ {Ċ}_p=-d_p{C}_p\\ \mathrm{\Delta }{f}_i[f]=0\end{array}$$ 4

where:

$${\alpha }_p={\gamma }_p{k}_w+d_p;A_w=a_w{P}_w{C}_w^{\max };{\alpha }_w=a_w{P}_w+d_w;D_w=a_w{P}_w\frac{C_w^{\max }}{C_p^{\max }},{\omega }_w=d_w+{\gamma }_w{k}_p;{\omega }_p=a_p{P}_p+d_p;A_p=a_p{P}_p{C}_p^{\max };D_p=a_p{P}_p\frac{C_p^{\max }}{C_w^{\max }}$$

a _p , a _w precursor and reactant adsorption rate constants respectively, d _p , d _w precursor and reactant desorption rate constants respectively, C _p , C _w maximum precursor and reactant surface concentrations respectively, C _p , C _w precursor and reactant surface concentrations respectively, C _p (t _i )initial concentration of precursor at reactant step (initial precursor concentration at the i-th cycle), C _w initial concentration of reactant at precursor step, P _p , P _w precursor and reactant partial pressures respectively, k _p reaction constant of the adsorbed reactant with precursor vapor, k _w reaction constant of the adsorbed precursor with reactant vapor. Δf _i [w], Δf _i [d], Δf _i [p], Δf _i [f]change in the amount of oxide in the reactant step, the purge step after the reactant, the precursor step and the purge step after the precursor, respectively, τ _p , τ _w , τ _f , τ _d precursor, reactant, postprecursor, postreactant step time respectively, γ _p , γ _w the proportionality coefficients for the precursor and reactant respectively for the formation of 1 mol of the film compound.

The change in the amount of film substance per cycle is the sum of Δf _i [w], Δf _i [d], Δf _i [p], Δf _i [f], where Δf _i [d] = 0 and Δf _i [f] = 0, therefore

$$\mathrm{\Delta }{f}_i=\mathrm{\Delta }{f}_i[w]+\mathrm{\Delta }{f}_i[p]$$ 5

A solution to these equations was obtained for the case of uniform film growth for such values that (α _w ≠ α _p ) ∪ (ω _p ≠ ω _w )­

$$\mathrm{\Delta }{f}_i=L\frac{\mathrm{\Phi }+\mathrm{\Psi }(1-W)\mathrm{{\rm Y}}}{1-\mathrm{\Omega }(T-W)\mathrm{{\rm Y}}-\mathrm{\Pi }T}(1-T)+\mathrm{\Gamma }(\frac{\mathrm{\Psi }}{\mathrm{\Omega }}(1-W)+(T-W)\frac{\mathrm{\Phi }+\mathrm{\Psi }(1-W)\mathrm{{\rm Y}}}{1-\mathrm{\Omega }(T-W)\mathrm{{\rm Y}}-\mathrm{\Pi }T})\mathrm{{\rm Y}}$$ 6

where: W = exp­(–α _w τ _w); T = exp­(–α _p τ _w ); Υ = (1 – HXZW)⁻¹; Φ = QY(1 – S); Ψ = FRHY(Z – S); Ω = GRHY(Z – S); Γ = GUH(1 – Z); Π = SOY; S = exp­(–ω _p τ _p ); Z = exp­(–ω _wτ _p ); H = exp­(–d _w τ _d ); O = exp­(–d _p τ _d ); X = exp­(–d _w τ _f ); Y = exp­(–d _p τ _f ); $F=\frac{A_w}{{\alpha }_w}$ ; $G=\frac{D_w}{{\alpha }_p-{\alpha }_w}$ ; $Q=\frac{A_p}{{\omega }_p}$ ; $R=\frac{D_p}{{\omega }_w-{\omega }_p}$ ; $L=\frac{k_w}{{\alpha }_p}$ ; $U=\frac{k_p}{{\omega }_w}$ .

In this case, the film thickness in this approximation is presented in the form of an equation

$$d=N{\rho }^{-1}M·(\mathrm{\Delta }{f}_i)$$ 7

where d is the film thickness, N is the number of cycles, ρ is the density of the film substance, M is the molar mass of the film substance.

In general, the solution of the equations for each step depends on the initial concentrations obtained by solving the equations for the previous step, so that series can be obtained for the concentrations of the precursor and reactant at each step

$$\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{e}({\alpha }_w\ne {\alpha }_p)\cup ({\omega }_p\ne {\omega }_w):$$

$$\{\begin{array}{l}{C}_w(t_{i+1})=\mu +\nu C_w(t_i)+\sigma C_p(t_i)\\ C_p(t_{i+1})=\phi +\xi C_p(t_i)+\chi (\psi +\theta C_w(t_i)+\kappa C_p(t_i))\\ \mathrm{\Delta }{f}_i[w]=\lambda C_p(t_i)\\ \mathrm{\Delta }{f}_i[p]=\gamma (\psi +\theta C_w(t_i)+\kappa C_p(t_i))\end{array}$$ 8

where: λ = L(1 – T); ψ = FH(1 – W); θ = HW; κ = GH(T – W); γ = U(1 – Z); μ = FHXZ(1 – W); ν = HXZW; σ = GHXZ(T – W); φ = QY(1 – S); ξ = OYST; χ = RY(Z – S). C _w (t _i )initial reactant concentration at the i-th cycle. C _p (t _i+1), C _w (t _i+1)initial precursor and reactant concentration respectively at the (i +1)-th cycle.

$$\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{e}({\alpha }_w={\alpha }_p)\cup ({\omega }_p\ne {\omega }_w):$$

$$\{\begin{array}{l}{C}_w(t_{i+1})={\mu }_{\alpha }+{\nu }_{\alpha }{C}_w(t_i)+{\sigma }_{\alpha }{C}_p(t_i)\\ C_p(t_{i+1})=\phi +{\xi }_{\alpha }{C}_p(t_i)+\chi ({\psi }_{\alpha }+{\theta }_{\alpha }{C}_w(t_i)+{\kappa }_{\alpha }{C}_p(t_i))\\ \mathrm{\Delta }{f}_i[w]={\lambda }_{\alpha }{C}_p(t_i)\\ \mathrm{\Delta }{f}_i[p]=\gamma ({\psi }_{\alpha }+{\theta }_{\alpha }{C}_w(t_i)+{\kappa }_{\alpha }{C}_p(t_i))\end{array}$$ 9

where: λ_α = L _α (1 –W _α); ψ_α = F _α H(1 – W _α); θ _α = HW; κ _α = –D _w HW _α τ _w ; μ_α = F _α HXZ(1 – W _α ); ν _α = HXZW _α ; σ_α = –D _w HXZW _α τ _w ; ξ_α = OYSW _α ; W _α = exp­(–ατ _w ); Υ _α = (1 – HXZW _α )^–1; Ψ _α = F_αRHY(Z – S); Ω _α = D_wRHY(Z – S); Γ _α = D_wUH(1 – Z); θ_α = F_αUH(1 – Z); F_α = A_w /α; L_α = k_w /α; α = γ_pk_w +d_p = a_wP_w +d_w.

$$\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{e}({\alpha }_w={\alpha }_p)\cup ({\omega }_p={\omega }_w):$$

$$\{\begin{array}{l}{C}_w(t_{i+1})={\mu }_{\alpha \omega }+{\nu }_{\alpha \omega }{C}_w(t_i)+{\sigma }_{\alpha \omega }{C}_p(t_i)\\ C_p(t_{i+1})={\phi }_{\omega }+{\xi }_{\alpha \omega }{C}_p(t_i)+{\chi }_{\omega }({\psi }_{\alpha }+{\theta }_{\alpha }{C}_w(t_i)+{\kappa }_{\alpha }{C}_p(t_i))\\ \mathrm{\Delta }{f}_i[w]={\lambda }_{\alpha }{C}_p(t_i)\\ \mathrm{\Delta }{f}_i[p]={\gamma }_{\omega }({\psi }_{\alpha }+{\theta }_{\alpha }{C}_w(t_i)+{\kappa }_{\alpha }{C}_p(t_i))\end{array}$$ 10

where: γ _ω = U _ω (1 – Z _ω); μ_αω = F _α HXZ _ω (1 – W _α ); ν_αω = HXZ _ω W _α ; σ_αω = –D _w HXZ _ω W _α τ _w ; φ _ω = Q _ω Y(1 – Z _ω ); ξ_αω = OYZ _ω W _α ; χ_ω = –D _p YZ _ω τ _p ; Υ _αω = (1 – HXZ_ωW_α )^–1; Φ _αω = Q_ωY(1 – Z_ω ); Ψ _αω = F_αD_pHYZ_ωτ_p ; Ωαω = D_wD_pHYZ_ωτ_p ; Γ _αω = D_wU_ωH­(1 – Z_ω ); θ _αω = F_αU_ωH(1 – Z_ω ); Π _ω = OYZ_ω ; Z _ω = exp­(–ωτ_p ); Q_ω = A_p /ω; U_ω = k_p /ω; ω = d_w + γ_wk_p = a_pP_p + d_p .

The case for which (α _w ≠ α _p ) ∪ (ω _p = ω _w ) is not presented, since the solution in this case can be obtained by mutual replacement of variables and constants with indices p ↔ w and f ↔ d for the solution in the case for which (α _w = α _p ) ∪ (ω _p ≠ ω _w ).

Thus, the values of the initial concentrations C _p (t _i ) and C _w (t _i ) for the i-th ALD cycle can be sequentially calculated with known initial concentrations C _p (t ₁) and C _w (t ₁) of the first ALD cycle. If we consider the general case and take into account that the substrate material differs from the film material, then a significant complication of the model is required. To implement this model, it is necessary to take into account adsorption on the wafer surface, as well as taking into account incomplete coverage of the wafer with film during the first ALD cycles. Such a complication of the model requires a significant increase in unknown kinetic constants, which needs to be determined somehow. At the same time, there is absolutely no guarantee that this complication provides a significant improvement in the quality of film thickness prediction. Given the above, the simplest way would be to assume initial concentrations C _p (t ₁) and C _w (t ₁) equal to zero, then eqs – allow us to determine the film thickness (d _theor) using the following equation

$$d_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}}=\frac{M·}{\rho }\sum _{i=1}^N(\mathrm{\Delta }{f}_i[w]+\mathrm{\Delta }{f}_i[p])$$ 11

In it was noted that reaction rate constants may have a complex nonlinear dependence on partial pressures and, accordingly, are functions of partial pressures

$$\{\begin{array}{l}{k}_p=f_1(P_p)\\ k_w=f_2(P_w)\end{array}$$ 12

The functions f ₁(0) = 0 and f ₂(0) = 0 can be represented as a Taylor series, given that f ₁(0) = 0 and f ₂(0) = 0

$$\{\begin{array}{l}{k}_p=f_1(P_p)=f_1^{\prime }(P_p)+\frac{f_1^{″}}{2}{(P_p)}^2+\frac{f_1^{‴}}{6}{(P_p)}^3+...\\ k_w=f_2(P_w)=f_2^{\prime }(P_w)+\frac{f_2^{″}}{2}{(P_w)}^2+\frac{f_2^{‴}}{6}{(P_w)}^3+...\end{array}$$ 13

This series can also be expressed in the following form

$$\{\begin{array}{l}{k}_p=k_p^{(0)}{P}_p+k_p^{(1)}{(P_p)}^2+k_p^{(2)}{(P_p)}^3+...\\ k_w=k_w^{(0)}{P}_w+k_w^{(1)}{(P_w)}^2+k_w^{(2)}{(P_w)}^3+...\end{array}$$ 14

On the one hand, the Taylor series allows one to approximate the dependence over a wide range with any accuracy, depending on the number of terms in the expansion of the function. On the other hand, such series requires finding a large number of coefficients k _i ^(j), which introduces significant complications for the practical application of the model. In this regard, the dependence of reaction constants on the partial pressures of reagents in ALD-LAFORM Software is presented in the form of a linear approximation

$$\{\begin{array}{l}{k}_p\approx k_p^{(0)}{P}_p\\ k_w\approx k_w^{(0)}{P}_w\end{array}$$ 15

3.2. ALD-LAFORM Software Algorithm

RMSD in eq A.1 is a complex function of eight constants, and the task of the NLSM is to find the global minimum of this function. ALD-LAFORM Software implements the Lamarckian genetic algorithm for finding the global minimum of RMSD. Each constant is represented in logarithmic form as an individual’s genes (g ₁ ÷ g ₈)­

$$\{\begin{array}{l}{g}_1=\log a_p,g_2=\log a_w,g_3=\log d_p,g_4=\log d_w\\ g_5=\log k_p^{(0)},g_6=\log k_w^{(0)}\\ g_7=-\left|\ln \frac{C_p^{\max }}{C_p^{\mathrm{U}\mathrm{L}}}\right|,g_8=-\left|\ln \frac{C_w^{\max }}{C_w^{\mathrm{U}\mathrm{L}}}\right|\end{array}$$ 16

The parameters C _p ^UL and C _w ^UL in eq correspond to the maximum limit values C _p ^max and C _w ^max respectively, the values of these parameters are specified by the user of the ALD-LAFORM Software. Three options are provided for entering these values: the value itself, the indication of the van der Waals radius through which the value C _p ^UL or C _w ^UL is calculated according to eq A.12, the indication of the element by the van der Waals radius of which it is necessary to calculate C _p ^UL or C _w ^UL.

In the first generation, a population is initially formed from individuals with a random set of genes. The genes of each individual are used as starting points for optimization by the Nelder–Mead algorithm (NMA) with a maximum number of steps limited to 200. The result of optimization of each set of genes is used to form a population with optimized genes, and the RMSD obtained during optimization is used as the fitness function value of the individual. From the population obtained in this way, ancestors are selected to form the next generation. When forming subsequent generations, a population is first formed, obtained from the genes of the ancestors by the action of mutation and crossing-over operators with the addition of a subpopulation consisting of individuals with a random set of genes. Next comes the optimization of the genes of individuals by the NMA with a maximum number of steps limited to 200 and the formation of a population with optimized genes. Next, the selection of ancestors for the next generation occurs, and this is repeated until the limit on the number of generations is reached (the maximum number of generations is set by the user). After reaching the maximum number of generations, the fittest individual genes can be optimized by the NMA with a user-specified limit on the optimization steps. The described algorithm is shown schematically in Figure .

1.

Diagram of the algorithm implemented in the ALD-LAFORM software.

4. Results

4.1. Test of the Algorithm Using Exact Theoretical Film Thickness Values

The most common practice in ALD is to analyze the dependence of film thickness on precursor step time (or reactant step time) with fixed values of other process parameters. Testing of the algorithm showed that this type of data is poorly suited for determining the values of kinetic constants; the constants found differ significantly from the initially specified values. However, increasing the number of points does not lead to significant improvements and even with 16 points some constants differ from the initial ones. As an example, Table presents the data on the found values of constants for different numbers of points (columns “10 points”–“16 points”. From Table it can be seen that even at 13 points the values of the constants are far from those initially specified, and at 16 points the values of the precursor desorption constant do not coincide with those initially specified. In general, selecting points by varying only one process parameter is ineffective and, on the one hand, requires a large set of experimental data, and, on the other hand, does not guarantee that all values of kinetic constants will be correctly determined. The approach in which the process parameters τ _p , τ _w , τ _f and τ _d are varied over a wide range of values shows a significantly better result. Moreover, even with only 8 points, it is possible to obtain constant values close to those initially specified (see column “8 points (WP)” in Table ). Therefore, testing with simulated experimental film thicknesses was carried out using data with varied parameters τ _p , τ _w , τ _f and τ _d .

4.2. Test of the Algorithm Using Simulated Experimental Film Thickness Values

For the first ten points from Table (rows “Point” from 1 to 10), 1000 sets of simulated experimental data were generated for each of the values of σ = 0.1, 0.25, 0.5, 0.75, 1.0 (see eq B.1). Based on the obtained data sets, the values of kinetic constants were found; statistical data for the found values of kinetic constants for σ = 0.1 nm, 0.25 nm, 0.5 nm are presented in Table . For σ = 0.75 and 1 nm, there are outliers in the statistics of the constant’s values distribution, due to which the average values differ greatly from the average values obtained for σ = 0.1 nm, 0.25 and 0.5 nm. As an example, Figure shows the distribution of the values of the constants a _p for σ = 0.5 and 1 nm. Figure b shows the presence of calculated constants with values of the order of 10¹⁴÷10¹⁵ Pa^–1s^–1 (∼5% of all calculated values) for σ = 1 nm. Moreover, despite the fact that the average value of a _p for σ = 0.5 nm is close to the initial value, the distribution of the calculated values varies over a wide range on the one hand and has a complex distribution that differs from the normal one (see Figure a).

2. Values of Process Parameters and Corresponding Calculated Theoretical Thicknesses.

point	d init, nm	τ p , s	τ w , s	τ f , s	τ d , s	N	τALD
1	25.05	0.05	0.1	1	3	2000	2 h:18 m
2	47.66	0.1	0.1	1	3	2000	2 h:20 m
3	138.45	5	0.1	1	3	1000	2 h:32 m
4	22.32	0.1	0.05	1	3	1000	1 h:09 m
5	122.07	1	1	1	3	1000	1 h:40 m
6	61.22	5	10	1	3	400	2 h:07 m
7	63.15	10	5	1	3	400	2 h:07 m
8	24.24	0.1	5	1	3	1000	2 h:32 m
9	113.80	1	0.1	10	3	1000	3 h:55 m
10	17.64	0.1	1	1	15	1000	4 h:45 m
11	114.76	2	2	1	3	800	1 h:47 m
12	104.16	3	3	1	3	700	1 h:57 m
13	76.53	5	5	1	3	500	1 h:57 m
14	77.79	7	7	1	3	500	2 h:30 m
15	121.50	1	1	5	3	1000	2 h:47 m
16	101.63	1	1	1	10	1000	3 h:37 m

3. Values of Kinetic Constants Found for Different Values of σ.

const	σ = 0.1 nm	σ = 0.25 nm	σ = 0.5 nm	unit
a p	0.105	0.113	0.139	Pa–1s–1
a w	0.918	0.918	0.940	Pa–1s–1
d p	10.0 × 10–2	9.64 × 10–2	9.89 × 10–2	s–1
d w	2.71 × 10–2	2.71 × 10–2	2.71 × 10–2	s–1
k p (0)	0.921	0.922	0.925	Pa–1s–1
k w (0)	1.05	1.10	1.13	Pa–1s–1
C p max	5.84 × 10–6	7.11 × 10–6	7.36 × 10–6	mol/m2
C w max	8.34 × 10–6	8.33 × 10–6	8.30 × 10–6	mol/m2

2.

Precursor adsorption constant values distribution for different values of σ: (a) Distribution for σ = 0.5 nm. (b) Distribution for σ = 1 nm.

The distribution of MaxRD values for σ = 0.1 nm, 0.25 nm, 0.5 nm, 0.75 and 1 nm is shown in Figure a–e, from which it can be seen that with an increase in the value of σ, the range of the distribution of MaxRD values increases. For values of σ = 0.75 and 1 nm, the MaxRD distributions have a wide range of values and thus differ qualitatively from the MaxRD distributions for smaller values of σ. Moreover, if for σ = 0.75 nm 95% of the MaxRD values are less than ∼25%, then for σ = 1 nm there are only 90% of such MaxRD values.

3.

Distribution of the MaxRD obtained for different values of σ: (a) Distribution for σ = 0.1 nm. (b) Distribution for σ = 0.25 nm. (c) Distribution for σ = 0.5 nm. (d) Distribution for σ = 0.75 nm. (e) Distribution for σ = 1 nm (10 points). (f) Distribution for σ = 1 nm (16 points).

Based on the 16 points (rows “Point” 1 to 16 in Table ), 1000 data sets for σ = 1 nm were generated, and MaxRD values were obtained based on the calculated constants. Figure f shows the distribution of MaxRD for the calculated constants. For the obtained constants, the average value of MaxRD is 7%, while the values of MaxRD do not exceed 25%. Table shows the statistical distribution parameters of MaxRD values for the corresponding σ values approximated by the log–normal distribution. Figure shows the result of approximating the distribution of MaxRD values for σ = 0.25 nm and σ = 0.5 nm as examples of the worst and best approximations. The result of the approximation is 99th percentiles (P99), presented in Table , shows that as the σ value increases from 0.1 to 0.25 nm, there is a slight increase in the P99 value, and as the σ value increases from 0.5 to 0.75 nm, the P99 value becomes two times larger. Moreover, from Table it can be seen that the distributions for σ = 0.5 nm and σ = 1 nm (16 points) have similar characteristics and close P99 values.

4. Statistics for MaxRD under Log-normal Distribution.

σ, nm	mean, %	standard deviation, %	P95, %	P99, %
0.1	4.5 (4.4)	2.2 (2.1)	8.2 (7.8)	9.8
0.25	5.8 (5.7)	2.4 (2.3)	9.8 (9.3)	11
0.5	7.3 (7.4)	3.3 (3.9)	13 (13)	17
0.75	9.9 (10)	5.8 (7.4)	21 (22)	31
1 (10 points)	13 (13)	8.9 (9.2)	29 (34)	46
1 (16 points)	7.0 (7.0)	3.4 (3.3)	13 (12)	16

^aValues in parentheses are calculated directly from the sample.

4.

Histogram of the MaxRD distribution and the corresponding log–normal distribution (lognorm): (a) Distribution for σ = 0.25 nm. (b) Distribution for σ = 0.5 nm.

The statistics of the distribution of MeanRD values are presented in Table , and Figure presents histograms of distributions with the approximation of the log–normal distribution for σ = 1 nm (10 points) and σ = 1 nm (16 points) as examples of the worst and best approximations. In contrast to the MaxRD distribution statistics, in this case there is no significant increase in P99 with an increase in σ from 0.5 to 0.75 nm, which is due to the underestimation of P99 for σ = 0.75 nm by the obtained gamma distribution. In this case, the 95th percentile values (P95), calculated directly from the sample with an increase in σ from 0.25 to 0.5 nm, increase less than 2-fold (1.73 times) and with an increase in σ from 0.5 to 0.75 nm, the P95 value increases more than 2-fold (2.37 times).

5. Statistics for MeanRD under Log-normal Distribution.

σ, nm	mean, %	standard deviation, %	P95, %	P99, %
0.1	0.45 (0.44)	0.24 (0.22)	0.86 (0.79)	1.1
0.25	0.59 (0.58)	0.32 (0.31)	1.2 (1.1)	1.6
0.5	0.86 (0.90)	0.59 (0.76)	2.0 (1.9)	3.0
0.75	1.3 (1.4)	1.2 (1.6)	3.3 (4.5)	5.9
1 (10 points)	1.8 (2.0)	1.9 (2.5)	4.8 (6.5)	9.2
1 (16 points)	0.88 (0.89)	0.51 (0.58)	1.8 (1.8)	2.6

^aValues in parentheses are calculated directly from the sample.

5.

Histogram of the MeanRD distribution and the corresponding log–normal distribution (lognorm): (a) Distribution for σ = 1 nm (10 points). (b) Distribution for σ = 1 nm (16 points).

The distribution of OGRRE and OCTRE values for σ = 0.5 nm and σ = 1 nm (16 points) is shown in Figure , which show that for σ = 0.5 nm, the maximum obtained OGRRE and OCTRE values do not exceed 2.5%, and 95% of the OGRRE and OCTRE values do not exceed 1%. Figure c,d show that for σ = 1 nm, the maximum obtained OGRRE and OCTRE values also do not exceed 2.5%, and 95% of the OGRRE and OCTRE values also do not exceed 1%. It is worth noting that the statistics of the distribution of OGRRE and OCTRE values differs from the statistics of the distribution of MaxRD and MeanRD values and is described by a gamma distribution.

6.

Distribution of OCTRE and OGRRE, obtained for different values of σ: (a) Distribution of OCTRE for σ = 0.5 nm. (b) Distribution of OGRRE for σ = 0.5 nm. (c) Distribution of OCTRE for σ = 1 nm (16 points). (d) Distribution of OGRRE for σ = 1 nm (16 points).

The statistics of the OGRRE and OCTRE values distributions are presented in Tables and , and Figure shows the histograms of the OGRRE value distributions for σ = 0.1 nm and σ = 1 nm (10 points) as examples of the worst and best approximations. It is worth noting that the OGRRE and OCTRE distributions for the same σ values are almost identical. It is also worth noting that the highest P99 value for OGRRE and OCTRE does not exceed 3%. Moreover, with an increase in σ from 0.5 to 0.75 nm, no significant increase in P99 is observed for the MaxRD and MeanRD distributions, which is associated with an overestimated P99 for σ = 0.5 nm, while the characteristics of the distributions for σ = 0.1 nm and σ = 1 nm (16 points) are also similar. Increasing σ from 0.5 to 0.75 nm does not lead to a significant increase in P99, which is associated with an overestimation of P99 for σ = 0.5 nm. It is worth noting that, as for the MeanRD and MaxRD distributions, the distribution characteristics for σ = 0.1 nm and σ = 1 nm (16 points) are also similar.

6. Statistics for OGRRE under Gamma Distribution.

σ, nm	mean, %	standard deviation, %	P95, %	P99, %
0.1	0.10 (9.4 × 10–2)	0.10 (8.2 × 10–2)	0.30 (0.25)	0.46
0.25	0.16 (0.15)	0.13 (0.13)	0.42 (0.39)	0.62
0.5	0.26 (0.26)	0.24 (0.24)	0.74 (0.67)	1.1
0.75	0.46 (0.47)	0.43 (0.45)	1.3 (1.4)	2.0
1 (10 points)	0.64 (0.64)	0.58 (0.63)	1.8 (1.8)	2.7
1 (16 points)	0.32 (0.32)	0.28 (0.28)	0.88 (0.84)	1.3

^aValues in parentheses are calculated directly from the sample.

7. Statistics for OCTRE under Gamma Distribution.

σ, nm	mean, %	standard deviation, %	P95, %	P99, %
0.1	0.10 (9.4 × 10–2)	9.8 × 10–2 (8.2 × 10–2)	0.30 (0.25)	0.45
0.25	0.16 (0.15)	0.13 (0.13)	0.42 (0.38)	0.61
0.5	0.26 (0.26)	0.24 (0.24)	0.74 (0.68)	1.1
0.75	0.47 (0.47)	0.45 (0.45)	1.4 (1.4)	2.1
1 (10 points)	0.67 (0.67)	0.63 (0.61)	1.9 (2.0)	2.9
1 (16 points)	0.32 (0.32)	0.28 (0.27)	0.88 (0.82)	1.3

^aValues in parentheses are calculated directly from the sample.

7.

Histogram of the OGRRE distribution and the corresponding gamma distribution (gamma): (a) Distribution for σ = 0.1 nm. (b) Distribution for σ = 1 nm (10 points).

For N = 1000, the film thickness value d _theor = 132.6 nm, knowing the OGRRE value, one can obtain the difference between the film thicknesses (Δd) of the optimal growth rate point calculated for the initial constants and the optimal growth rate point obtained for the calculated constants. Considering that in eq B.1 a random variable with a normal distribution was used, 99% of the d _e values lie within d _theor ± 2.6 σ, thus the error of film thickness measurement (EFTM) can be determined as 2.6 σ for the 99% confidence interval. Table presents a comparison of EFTM and Δd, which shows that the values of these parameters are of the same order of magnitude.

8. Comparison of the Error in Determining the Film Thickness with the Error in Determining the Optimal Film Growth Point.

σ, nm	P99 OGRRE, %	EFTM (2.6 σ), nm	Δd, nm
0.1	0.46	0.26	0.61
0.25	0.62	0.64	0.82
0.5	1.1	1.3	1.5
0.75	2	1.9	2.7
1 (10 points)	2.7	2.6	3.6
1 (16 points)	1.3	2.6	1.7

5. Results and Discussion

For fixed values of kinetic constants (a _p , a _w , d _p , d _w , k _p , k _w , C _p ^max and C _w ^max), the film thickness d _theor (eq ) is a function of 7 process parameters (τ _p , τ _w , τ _f , τ _d , P _p , P _w and N). In the most general case, eq describes a 7-dimensional surface in space (τ _p , τ _w , τ _f , τ _d , P _p , P _w , N, d _theor), visualization of this surface by two-dimensional cross-section gives 7 variants of cross-sections, and visualization by three-dimensional cross-section gives 21 variants of cross-sections. The ALD-LAFORM Visual Tool implements visualization of two-dimensional cross-sections by planes (τ _p , d _theor), (τ _w , d _theor) and (N, d _theor) and three-dimensional cross-section by a three- dimensional plane (τ _p , τ _w , d _theor) as the most important among all variants. An example of film growth inhibition by a reactant was shown in. Table presents the values of constants and parameters corresponding to the case of film growth inhibition by a precursor, and Figure shows an example of visualization of the film thickness dependence on process parameters for these values of constants and parameters, made using the ALD-LAFORM Visual Tool. Figure a,b shows the dependence of the film thickness on τ _p , τ _w , the peculiarity of which is the presence of a “surface ridge”. Figure c,d show sections of the surface shown in Figure a,b. The appearance of the film thickness dependence with the presence of a maximum in Figure c was shown and explained in. Figure d shows that increasing the value of τ _p changes the appearance of the dependence such that at τ _p = 100 ms the curve has a “classical” appearance with a plateau, and at τ _p = 5 s a practically linear “CVD-like” dependence of the film thickness on τ _w is observed. This “CVD-like” dependence is explained by the dominance of precursor adsorption on the film surface: due to the low reaction rate of the adsorbed precursor with reactant vapors, the surface remains predominantly covered with the precursor. In this case, the main contribution to film growth is made by the reactant step, and the rate of film growth is almost proportional to the product k _w C _p (t _i )­τ _w . Figure e,f shows the nonlinear dependence of film thickness on the number of steps at a small number of ALD steps. The decrease in the thickness growth rate per ALD cycle at the initial stage is explained by the saturation of adsorption sites with the precursor after ∼20 ALD steps, which reduces the growth rate. Overall, this visualization example shows that the presented model can explain complex dependencies not only on the film thickness from the precursor and reactant steps, but also on the number of ALD cycles.

9. Values of Constants and Process Parameters for an Example of Inhibition of Film Growth by a Precursor.

const	value	const	value
γ p	1	γ w	2
a p	0.1 Pa–1s–1	a w	0.1 Pa–1s–1
d p	1 × 10–4 s–1	d w	1 × 10–5 s–1
k p (0)	0.5 Pa–1s–1	k w (0)	1 × 10–3 Pa–1s–1
C p max	1 × 10–6 mol/m2	C w max	1 × 10–5 mol/m2
P p	2 Pa	P w	50 Pa
M	0.1 kg/mol	ρ	4000 kg/m3
τ f	5 s	τ d	5 s

8.

Visualization of film thickness dependences on process parameters for the case of film growth inhibition by a precursor, performed using the ALD-LAFORM Visual Tool (for a–d N = 500): (a) Visualization of film thickness as a 3D function of precursor step time and reactant step time, with fixed purge step times. (b) Visualization of film thickness as a function of precursor step time and reactant step time (with fixed purge step times) using a color gradient. (c) Visualization of film thickness as a function of precursor step time and several fixed values of reactant step times and fixed purge step times. (d) Visualization of film thickness as a function of reactant step time and several fixed values of precursor step times and fixed purge step times. (e) Visualization of the dependence of film thickness on the number of ALD steps at fixed step times. (f) Visualization of the dependence of film growth per cycle (average film thickness per cycle) on the number of ALD steps at fixed step times.

It is worth noting that the LAFOR model is not universal and implies a number of assumptions that were described in. These assumptions include, in particular, the absence of adsorption of reaction byproducts and the Langmuir adsorption model. Table provides examples of reactions that are not included in the LAFOR model and may affect the quality of the model’s prediction. In the case of reactants such as H₂O and NH₃, dissociation of the reactant on the film surface may occur, which leads to the fact that the dependence of the adsorption and desorption rates to not fit into the framework of the Langmuir adsorption model. In the case of organic precursors, in the presence of decomposition reactions on the surface of the film, for example, there may be several different ligands-in addition to the precursor ligands, there may be ligands formed as a result of decomposition reactions. At a very low rate of decomposition, this reaction can be neglected. At a very high rate of decomposition reaction (if we are not talking about the formation of a nonvolatile product, for example, a metal on the surface of a film from its precursor), the reaction rate constant k _w corresponds to the rate constant of the reaction of the reactant vapor with the decomposition product of the precursor, and not with the precursor itself. If the rate of decomposition is such that the precursor does not have time to completely decompose during the ALD cycle steps, then two reactions must be taken into account: the reaction of the precursor with reactant vapors and the reaction of the precursor decomposition product with reactant vapors. The presence of two such reactions can lead to the presence of more complex dependencies of film thickness on process parameters, which the LAFOR model cannot describe. In the case of interaction of an organic reactant with oxygen or ozone, water may be formed as a byproduct of the reactions, which can participate in the film formation reaction, which can also affect the quality of the LAFOR model prediction. Plasma-enhanced ALD (PEALD), which employs plasma (e.g., oxygen-based) as a reactant, deserves special mention. In the case of PEALD, film etching may occur (depending on plasma characteristics)a process not accounted for by the model.

10. Reactions Capable of Influencing the Reduction in the Quality of the LAFOR Model Prediction.

	precursor
reactant	inorganic	organic
O2 or O3	-	H2O formation, precursor decomposition
H2O, NH3, etc	Reactant surface dissociation	Reactant surface dissociation, precursor decomposition

When considering the adsorption process within the framework of the Langmuir adsorption model, it is worth noting the relationship between kinetic constants and the fractional occupancy of the adsorption sites (θ _A ) and the equilibrium constant (K _eq) and the Langmuir adsorption equation

$${\theta }_A=\frac{K_{\mathrm{e}\mathrm{q}}·P_A}{1+K_{\mathrm{e}\mathrm{q}}·P_A}$$ 17

where P _A is partial pressure of adsorbed reagent.

Equilibrium constant K _eq,w of the reactant is related to the adsorption and desorption constants by the equation

$$K_{e\mathrm{q},w}=\frac{a_w}{d_w}$$ 18

Accordingly, the Langmuir adsorption equation for the reactant can be represented as follows

$${\theta }_{A,w}=\frac{{a_{w}P}_A}{d_w+{a_{w}P}_A}$$ 19

It is worth noting that the reactant fractional occupancy of the adsorption sites θ _A,w is achieved at equilibrium, which requires infinite time (t = ∞). Thus, the physical meaning of the reactant fractional occupancy of the adsorption sites θ _A,w can be conditionally expressed by the following equation

$${\theta }_{A,w}=\frac{C_w(t=\infty )}{C_w^{\max }}$$ 20

where C _w (t = ∞) is the maximum concentration of the reactant that can be achieved in the adsorption process under given conditions (hereinafter referred to as the “equilibrium concentration of the reactant”; clarification: C _w ^max corresponds to the concentration at which all adsorption centers are occupied by the reactant).

All steps of the ALD cycle have a limited time, so it is more convenient to introduce the following functions to characterize the reactant adsorption

$$\{\begin{array}{l}{\theta }_w(t)=\frac{C_w(t)}{C_w^{\max }}\\ {\eta }_w(t)=\frac{{\theta }_w(t)}{{\theta }_{A,w}}=\frac{d_w+{a_{w}P}_w}{{a_{w}P}_w{C}_w^{\max }}{C}_w(t)=\frac{C_w(t)}{F}\end{array}$$ 21

where F was introduced in eqs –.

The function of reactant fractional occupancy of the adsorption sites θ _w (t) can be used to optimize process parameters along with OGRRE and OCTRE, and can also be used to construct more complex functions. Similarly, the function θ _p (t) for the precursor can be defined in the following form

$${\theta }_p(t)=\frac{C_p(t)}{C_p^{\max }}$$ 22

In the presence of side reactions of precursor decomposition on the film surface, selection of process parameters to minimize the value of θ _p (t) can potentially minimize the presence of impurities in the film associated with precursor decomposition.

The function η _w (t) can take values from 0 to 1 and shows how close the reactant concentration is to the equilibrium one. Moreover, from eqs and () it is evident that the parameter F introduced in eqs – corresponds to the equilibrium concentration of the reactant (using the same reasoning it is easy to show that the equilibrium concentration of the precursor corresponds to the parameter Q, also introduced in eqs –. It follows from the solutions to eqs – that the function η_w (t) decreases or remains constant during the purge and precursor steps, whereas it increases (or equals zero) during the reactant step; accordingly, this function attains its minimum value at the beginning of the reactant step and its maximum value at the end of the reactant step. This function takes a value of unity only when the reactant step time is infinite. However, the time during which the function value approaches unity will depend on the value of the product a _w P _w . Thus, we can introduce a time period parameter τ _{w,(1‑δ)·100}, corresponding to the reactant step time at which the function η _w (t) takes the value 1-δ

$${\eta }_w({\tau }_{w,(1-\delta )·100})=1-\delta$$ 23

where δ is a specified small value, such as 0.01 or 0.001. Thus, τ _w,99 corresponds to the reactant step time at which the reactant concentration at the end of the reactant step is 99% of the equilibrium concentration. It is worth noting that the value of τ _w,99 will depend on the purge and precursor step times. Moreover, in the ALD process optimization problem, finding the process parameters that yield the maximum value of θ _w (t) and the minimum value of τ _w,99 can be used to improve the quality of the resulting films and minimize the time it takes to obtain them.

As noted in, the adsorption, desorption, and reaction constants in eqs – are constant at a fixed temperature but take on different values at different temperatures. The availability of two or more data sets for different temperatures makes it possible to determine the values of the constants at those temperatures. In the absence of side reactions, the temperature dependence of the constants can be approximated by an Arrhenius-type equation

$$k(T)=A·e^{-E_{\mathrm{a}}/\mathrm{R}\mathrm{T}}$$ 24

where: k­(T) is the rate constant (corresponds to the constants a _p , a _w , d _p , d _w , k _p , k _w ), A is the pre-exponential factor or Arrhenius factor or frequency factor, E _a is the molar activation energy for the reaction, R is the universal gas constant.

However, this approach has clear limitations. The first limitation is that this equation implies a simple, single–stage reaction pathway, which may not correspond to the actual reaction mechanism; furthermore, the reaction mechanism may change with temperature. The second limitation relates to the accuracy of determining the constant values (see Appendix A). Another approach involves linear and power-law approximations of the dependence of constant values on temperature. If constant values are available for two different temperatures, a linear approximation of the temperature dependence can be performed, enabling the estimation of values within that temperature range. Having data for three temperatures allows for the use of not only linear but also quadratic approximation, as well as a combined approach where linear approximation is applied to some constants and quadratic approximation to others.

One of the practical applications of ALD in microelectronics is the production of films on high-aspect-ratio substrates. First of all, it is worth noting two extreme cases. In the first case, the sizes of the substrate “roughnesses” are significantly larger than the sizes of the molecules, for example, the diameters of holes or the distances between pillars in the case of surfaces with holes or pillars, respectively. In this case, the LAFOR model can be used as a first approximation to estimate the coating thickness. When the sizes of the substrate “roughnesses” are comparable to the sizes of the molecules, the LAFOR model cannot be applied. To simulate coating growth on high-aspect-ratio substrates, a numerical simulation method can be applied using the Navier–Stokes equations to model gas flows inside the chamber and over the surface of a substrate with complex geometry and using eqs – to simulate film growth on the substrate surface. For such numerical modeling, the constants for eqs – can be obtained experimentally for a normal substrate using the described algorithm for finding the constants of the LAFOR model.

Varying the parameters of the ALD process to obtain experimental data optimizes the process of finding the values of kinetic constants for the LAFOR model. Thus, the model can be validated using a relatively small amount of experimental data. The obtained result shows that 10 experimental points may be sufficient to find optimal process parameters with a film measurement error of about ±2.6 nm. However, to confirm the point corresponding to optimal film growth, at least two more experimental points, with parameters below and above the optimal ones, are required. These three additional points can be used to refine the constant values and optimal growth parameters. The process of refining the optimal growth parameters can be multistep with sequential refinement of the optimal growth parameters and the addition of experimental points of optimal growth for the previous step. The MaxRD and MeanRD parameters can be used to compare the constants of the previous step with the refined constants of the next step. An estimate of the errors in predicted film thicknesses and the accuracy of determining optimal parameters can be made based on the distribution parameters of the MaxRD, MeanRD, OGRRE, and OCTRE values obtained using the presented method. The presented method allows for the experimental assessment of the model’s applicability to a specific “precursor–reagent” system without the use of additional analytical equipment, as it relies solely on film thickness measurement tools.

The obtained results demonstrate the fundamental feasibility of the LAFOR model’s practical application, and furthermore, the fundamental feasibility of complex chemical-kinetic models with a large number of parameters. The proposed method can be used to assess the possibility of practical application of the new proposed models.

Appendix A

A. Nonlinear least-squares method and its limitations

To be able to predict the thickness of the resulting film for given process parameters, it is necessary to know the values of eight kinetic constants. In the absence of any experimental or theoretical data on the values of the kinetic constants, the estimates of the values of the constants can be obtained from the experimental ALD data using the nonlinear least-squares method (NLSM). The essence of NLSM is to find the minimum of the root-mean-square deviation (RMSD) function between experimental and theoretical values of thickness as a function of kinetic constants

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{D}(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t})=\sqrt{\frac{1}{n}\sum _{i=1}^n{(d_{\exp }({\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}}_i)-d_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}}({\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}}_i,\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}))}^2}$$ A.1

where: consta _p , a _w , d _p , d _w , k _p , k _w , C _p , C _w , param-τ _p , τ _f , τ _w , τ _d , P _p , P _w , N, n is the number of experimental film thickness values, d _exp(param _i )experimental value of film thickness for given parameters param _i .

It is worth noting that NLSM has several limitations. From a technical point of view, the values of τ _p , τ _f , τ _w , τ _d , P _p , P _w have a narrow range of possible values. The parameters τ _f and τ _d must ensure the ability to remove precursor and reactant vapors from the chamber to eliminate the possibility of reactions in the gas phase, so these parameters typically have values greater than ∼1 s. The lower limit of the values of τ _p and τ _w cannot be lower than the opening time of the valves that open the precursor and reagent supply lines, i.e. ∼1÷10 ms. (For example, Swagelok’s atomic layer deposition diaphragm valves are specified with closing–opening speeds of less than 5 ms). Generally, the upper limit of the values τ _p , τ _f , τ _w , τ _d , N is limited by the “reasonable” value of the ALD process time (τ _ALD)/(A.2)­

$${\tau }_{\mathrm{A}\mathrm{L}\mathrm{D}}=N({\tau }_p+{\tau }_f+{\tau }_w+{\tau }_d)$$ A.2

If the ALD process time is limited to 24 h and N = 100, then the upper value of the parameters τ _p , τ _f , τ _w , τ _d will be 144 min (∼10⁴ s). In practice, however, the values of these quantities usually do not exceed 1 min (∼10² s). Thus, the values of τ _f and τ _d do not go beyond the range of 1÷10⁴ s, and the values of τ _p and τ _w do not go beyond the range of 10^–3÷10⁴ s. These ranges limit the possibility of determining a _p , a _w , d _p , d _w , k _p , k _w .

In the case of desorption rate constants (d _p and d _w ), the range of values that can be determined for d _p and d _w can be estimated from the value of the desorption degree (the ratio of the concentration after a certain period of time to the initial value), which can be given by the following equation

$${\phi }_i(\tau )=\frac{C_i(\tau )}{C_i(0)}=e^{-d_i\tau }$$ A.3

where: φ _i (τ(τ) desorption degree of the i-th reagent over a period of time τ, d _i desorption rate constant of the i -th reagent, C _i (τ(τ) surface concentration at the end of time period τ of the i -th reagent, C _i (0)initial surface concentration of the i-th reagent. The value φ _i = 1 corresponds to the absence of desorption, and accordingly, the value φ _i ≈ 1 corresponds to the case of quasi-irreversible adsorption.

The parameters H, O, X and Y in eqs – correspond to the desorption degree of the reactant and precursor at the purge steps. The parameters Φ, Ψ and Γ from eq have a direct dependence on the parameters H, O, X and Y. Thus, for values of H, O, X and Y close to ∼1, the relative difference in film thickness depending on the values of the desorption constants and the durations of the purges can be estimated using the expression

$$\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}(d_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}})\sim \mathrm{\Delta }{\phi }_i=|{\phi }_i({\tau }_{\min })-{\phi }_i({\tau }_{\max })|$$ A.4

where: relative difference­(d _theor)relative difference in film thickness, τ _min, τ _maxminimum and maximum values of the purge step.

With a film determination accuracy of ∼1%, the value of Δφ _i should be greater than 0.01. Using the previously obtained limits of the values of τ _f and τ _d , it can be determined that the condition Δφ _i > 0.01 is fulfilled at a value of d _i > 10^–6 s^–1. If the maximum values of τ _f and τ _d are limited to values of ∼10² s, then the condition Δφ _i > 0.01 is fulfilled at a value of d _i > 10^–4 s^–1. Thus, the lower limit for determining desorption rate constants can be estimated within the range of 10^–6÷10^–4 s^–1. The upper limit for the desorption rate constant for conditions Δφ _i > 0.01 and φ _i ≈ 0 is reached at a value of d _i ∼4.6 s^–1. Thus, a rough estimate of the limits of determination of the adsorption rate constants gives values from 10^–6÷10^–4 s^–1 to 4.6 s^–1.

The values range of the adsorption rate constants (a _p and a _w ) can be estimated from the value of the adsorption rates in the cases of low-rate adsorption and high-rate adsorption. In the case of low adsorption rate with limited adsorption time, the amount of adsorbed reagent (C _i ) is small, so the adsorption rate (rate of change of adsorbed reagent) can be expressed by the equation

$$r_i^{\mathrm{a}\mathrm{d}\mathrm{s}}=a_i{P}_i{C}_i^{\max }\approx 0$$ A.5

where: r _i adsorption rate of i-th reagent.

Film formation due to the reaction of the adsorbed precursor occurs at the reactant step, and film formation due to the reaction of the adsorbed reactant with precursor vapors occurs at precursor step. Accordingly, the value of Δf _i [p] shows the contribution of the reaction of the adsorbed reactant to the formation of the film per cycle, and Δf _i [w] shows the contribution of the reaction of the adsorbed precursor. The amount of reactant at the beginning of the precursor step (C _w ⁰ in eq ) cannot exceed the value a _w P _w C _w ^max τ _w , accordingly, using eq , we can obtain the following inequality

$$\mathrm{\Delta }{f}_i[p]=\frac{k_p}{{\omega }_w}{C}_w^0(1-e^{-{\omega }_w{\tau }_p})<\frac{k_p}{{\omega }_w}{C}_w^0<\frac{1}{{\gamma }_w}{C}_w^0<\frac{1}{{\gamma }_w}{a}_w{P}_w{C}_w^{\max }{\tau }_w$$ A.6

Since the value of a _w P _w C _w ^maxτ _w is small due to the low adsorption rate, the obtained value of Δf _i [p] is also small. Using the value Δf _i [p], one can estimate the thickness of the film that is formed due to the reaction of the adsorbed reactant with the precursor vapors

$$d[p]=N{\rho }^{-1}M·(\mathrm{\Delta }{f}_i[p])$$ A.7

where: d[p]thickness of the film that is formed due to the reaction of the adsorbed reactant with the precursor vapors.

Thus, we can obtain the inequality

$$\{\begin{array}{l}d[p]=N{\rho }^{-1}M·(\mathrm{\Delta }{f}_i[p])<\frac{N{\tau }_{w}M}{\rho {\gamma }_w}{a}_w{P}_w{C}_w^{\max }<\frac{{\tau }_{\mathrm{A}\mathrm{L}\mathrm{D}}M}{\rho {\gamma }_w}{a}_w{P}_w{C}_w^{\max }\\ a_w{P}_w{C}_w^{\max }>\frac{\rho {\gamma }_w}{{\tau }_{\mathrm{A}\mathrm{L}\mathrm{D}}M}d[p]\end{array}$$ A.8

The film density ρ values are usually in the range of ∼10³÷10⁴ kg/m³, the molar mass M values are usually in the range of ∼0.1÷1 kg/mol, the proportionality coefficient values γ_w are usually in the range of 0.25÷4. In this case, the d[p] value should be higher than the error in determining the film thickness, which usually does not exceed 10^–10 m (1 Å). For the value of the ALD process time τ _ALD ∼10⁵÷10⁴ s and taking into account similar considerations for the case of precursor adsorption, one can obtain the inequalities

$$\{\begin{array}{l}{a}_w{P}_w{C}_w^{\max }>{10}^{-18}÷{10}^{-13}\mathrm{m}\mathrm{o}\mathrm{l}/\mathrm{s}\\ a_p{P}_p{C}_p^{\max }>{10}^{-18}÷{10}^{-13}\mathrm{m}\mathrm{o}\mathrm{l}/\mathrm{s}\end{array}$$ A.9

The values of the maximum concentration of reagents C _w ^max and C _p ^max cannot be arbitrarily large and are limited by the physical sizes of the adsorbed molecules and the distances between adsorption centers (which are limited by the sizes of the atoms of the substance of which the adsorption surface consists). The maximum concentration of adsorbed reactant on the film surface C _w ^max can be expressed through the equation:

$$C_w^{\max }=\frac{n_{\mathrm{w}}}{S_{\mathrm{U}}{N}_{\mathrm{A}}}$$ A.10

where: n _wnumber of reactant molecules, N _AAvogadro’s number, S _Uunit of surface area (1 m² in SI units).

The sizes of the reactant and precursor molecules cannot be smaller than a hydrogen atom, therefore the maximum possible values of C _w ^max and C _p ^max can be estimated by the value that can be obtained for the hexagonal packing arrangement of hydrogen atoms. The packing density for hexagonal production is $\frac{\pi }{2\sqrt{3}}$ (∼0.9069), so the following equations can be written

$$\begin{array}{l}\frac{\pi }{2\sqrt{3}}=\frac{s_{\mathrm{H}}}{S_{\mathrm{U}}}=\frac{n_{\mathrm{H}}\pi {(r_{\mathrm{H}})}^2}{S_{\mathrm{U}}}\\ \frac{n_{\mathrm{H}}}{S_{\mathrm{U}}}=\frac{1}{2\sqrt{3}{(r_{\mathrm{H}})}^2}\end{array}$$ A.11

where: n _Hnumber of hydrogen atoms, r _Hvan der Waals radius of the hydrogen atom.

Thus, we obtain the inequality

$$C_w^{\max }=\frac{n_{\mathrm{w}}}{S_{\mathrm{U}}{N}_{\mathrm{A}}}<\frac{n_{\mathrm{H}}}{S_{\mathrm{U}}{N}_{\mathrm{A}}}=\frac{1}{2\sqrt{3}{(r_{\mathrm{H}})}^2{N}_{\mathrm{A}}}$$ A.12

with the van der Waals radius of the hydrogen atom being ∼10^–10 m (1.0–1.2 Å − ), the values of C _w ^max and C _p ^max cannot exceed 4.8 × 10^–5 mol/m². Using inequalities (A.9), we obtain

$$\{\begin{array}{l}{a}_w{P}_w>{10}^{-13}÷{10}^{-9}{s}^{-1}\\ a_p{P}_p>{10}^{-13}÷{10}^{-9}{s}^{-1}\end{array}$$ A.13

thus, the lower limit of determination of the adsorption rate constants a _p and a _w depends on the partial pressures of the reagents.

In the case of a high adsorption rate (a _w P _w ≈ + ∞), the solution of eq for the dependence of the reactant concentration on time C _w (t) can be represented in the following form

$$\{\begin{array}{l}{C}_w(t)\approx C_w^{\max }((1-e^{-a_w{P}_{w}t})+(\frac{C_w(t_i)}{C_w^{\max }}+\frac{C_p(t_i)}{C_p^{\max }})e^{-a_w{P}_{w}t}-\frac{C_p(t_i)}{C_p^{\max }}{e}^{-{\alpha }_{p}t})\\ e^{-a_w{P}_{w}t}\approx 0\\ a_w{P}_w\gg d_w\end{array}$$ A.14

The first-time derivative of the function C _w (t) takes form

$$\frac{\partial C_w(t)}{\partial t}\approx a_w{P}_w{C}_w^{\max }{e}^{-a_w{P}_{w}t}+(\frac{\partial C_w(t_i)}{\partial t}-a_w{P}_w{C}_w(t_i))e^{-a_w{P}_{w}t}+\frac{C_w^{\max }}{C_p^{\max }}(\frac{\partial C_p(t_i)}{\partial t}-a_w{P}_w{C}_p(t_i))e^{-a_w{P}_{w}t}+C_w^{\max }({\alpha }_p{C}_p(t_i)-\frac{\partial C_p(t_i)}{\partial t})\frac{e^{-{\alpha }_{p}t}}{C_p^{\max }}$$ A.15

For limited reaction rates and limited rates of precursor adsorption and desorption, since a _w P _w ≈ + ∞, the value of the derivative $\frac{\partial C_w(t)}{\partial t}$ can be expressed as follows

$$\frac{\partial C_w(t)}{\partial t}\approx C_w^{\max }({\alpha }_p{C}_p(t_i)-\frac{\partial C_p(t_i)}{\partial t})\frac{e^{-{\alpha }_{p}t}}{C_p^{\max }}$$ A.16

eq A.16 shows that the derivative $\frac{\partial C_w(t)}{\partial t}$ is almost independent of a _w P _w , therefore, to estimate the maximum value of a _w P _w , which can be determined by the NLSM, we take C _p (t _i ) = 0 and $\frac{\partial C_p(t_i)}{\partial t}$ = 0 (the case of no precursor adsorption). In this case, eq A.16 can be represented as follows

$$\{\begin{array}{l}{C}_w(t)\approx C_w^{\max }(1+(\theta (t_i)-1)e^{-a_w{P}_{w}t})\\ \theta (t_i)=\frac{C_w(t_i)}{C_w^{\max }}\\ 0<\theta (t_i)<1\end{array}$$ A.17

eq A.17 allows us to estimate the difference in concentrations at the maximum and minimum reactant step times

$$|C_w({\tau }_{\max })-C_w({\tau }_{\min })|\approx |C_w^{\max }({|\theta (t_i)|}_{{\tau }_{\max }}-1)e^{-a_w{P}_w{\tau }_{\max }}-C_w^{\max }({|\theta (t_i)|}_{{\tau }_{\min }}-1)e^{-a_w{P}_w{\tau }_{\min }}|$$ A.18

where: τ _min, τ _maxmaximum and minimum values of reactant step times, |θ(t _i )|_τmin, |θ(t _i )|_τmaxvalues of θ(t _i ) at maximum and minimum reactant step times.

Equations A.17 and A.18 allow us to derive the following inequality

$$|C_w({\tau }_{\max })-C_w({\tau }_{\min })|<C_w^{\max }{e}^{-a_w{P}_w{\tau }_{\min }}$$ A.19

from inequalities (A.6–A.8), we can derive the value of the difference in film thickness at maximum and minimum values of reactant step times

$$\{\begin{array}{l}|\mathrm{\Delta }d[p]|<N\frac{M}{{\gamma }_w\rho }|C_w({\tau }_{\max })-C_w({\tau }_{\min })|<N\frac{M}{{\gamma }_w\rho }{C}_w^{\max }{e}^{-a_w{P}_w{\tau }_{\min }}\\ a_w{P}_w<-\frac{\ln \left(\frac{\rho {\gamma }_w}{C_w^{\max }NM}|\mathrm{\Delta }d[p]|\right)}{{\tau }_{\min }}\end{array}$$ A.20

where: |Δd[p]|the difference in film thicknesses formed as a result of the reaction of the adsorbed reagent with the precursor vapors, obtained at the minimum and maximum reactant step times, τ_minminimum value of reactant step time. For values of ρ ∼10⁴ kg/m³, M ∼1 kg/mol, γ _w ∼1, |Δd[p]|∼10^–10 m, τ _min ∼10^–3 s, N ∼1000, C _w ^max ∼5 × 10^–5 mol/m², the value of a_wP_w is less than ∼10⁴ s^–1.

The obtained limits for determining the values of adsorption and desorption constants are rough estimates. The actual limits are narrower on the one hand, on the other hand, due to the complex dependence of the film thickness on the constants, the limits for one constant depend on the value of other constants. It is worth noting that from the practical application point of view, the obtained values of the constants and the accuracy of their determination should make it possible to predict the thickness within acceptable limits. At the same time, the acceptable prediction limits are at least limited by the accuracy of the experimental determination of the film thickness.

Appendix B

B. Simulation of experimental data and comparison parameters

B.1. Emulated experimental film thickness

To simulate experimental data, a normally distributed random variable with a mean of 0 and a specified variance was added to the known exact theoretical film thickness values

$$\{\begin{array}{l}{d}_e=d_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}+\xi (\mu =0,\sigma )\\ \mu =0\end{array}$$ B.1

where: ξ­(μ,σ)a random variable that follows a normal distribution, μmathematical expectation, σstandard deviation, d _initthe film thickness value calculated with the initial values of the kinetic constants, d _e emulated experimental value of film thickness.

B.2. MaxRD and MeanRD

The accuracy of determining the constants should ensure acceptable accuracy of predicting the film thickness. An estimate of the accuracy with which the obtained constant values can predict the film thickness can be determined using the maximum relative difference (MaxRD) and mean relative difference (MeanRD) between the thickness obtained from the initial constants and that obtained from the calculated constants as a function of the time of the ALD cycle steps

$$\{\begin{array}{l}\max RD=\underset{{\tau }_p,{\tau }_w,{\tau }_f,{\tau }_d}{\max }\mathrm{R}\mathrm{D}({\tau }_p,{\tau }_w,{\tau }_f,{\tau }_d)·100\%\\ \mathrm{M}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{R}\mathrm{D}=\sum _i\sum _j\sum _k\sum _l\frac{\mathrm{R}\mathrm{D}({\tau }_i,{\tau }_j,{\tau }_k,{\tau }_l)}{n_{\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{s}}}·100\%\\ \mathrm{R}\mathrm{D}({\tau }_p,{\tau }_w,{\tau }_f,{\tau }_d)=\frac{|d_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}}(\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}.\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.)-d_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}}(\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}.\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.)|}{d_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}}(\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}.\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.)}\end{array}$$ B.2

where: d _theor(init.const.)film thickness as a function of τ _p , τ _w , τ _f , τ _d for fixed initial values of kinetic constants, d _theor(calc.const.)film thickness as a function of τ _p , τ _w , τ _f , τ _d for fixed calculated values of kinetic constants, τ _i , τ _j , τ _k , τ _l values of τ _p , τ _w , τ _f , τ _d at the points for which the average relative difference is calculated, n _pointsthe number of points by which the average value of the relative difference is calculated.

MeanRD was calculated from 1234321 points for each calculated constants set (τ _p from 0.05 to 10 s with a step of 0.0995 s, thus 101 points for τ _p , τ _w from 0.05 to 10 s with a step of 0.0995 s, thus 101 points for τ _w , τ _f from 1 to 10 s with a step of 0.9 s, thus 11 points for τ _f , and τ _d from 3 s to 15 with a step of 1.2 s, thus 11 points for τ _d and a total of 101 × 101 × 11 × 11 = 1234321 points).

B.3. OGRRE and OCTRE

A practical application of the theory could be, for instance, finding the optimal parameters τ _p , τ _w , τ _f and τ _d (optimal cycle times, OCT) for the maximum film growth rate per ALD cycle time. The OCT in this case can be obtained by finding the maximum of the growth rate function (GRF), which can be expressed as follows

$$\{\begin{array}{l}\mathrm{G}\mathrm{R}\mathrm{F}({\tau }_p,{\tau }_w,{\tau }_f,{\tau }_d,N,\mathrm{k}\mathrm{i}\mathrm{n}.\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.)=\frac{\sum _{i=1}^N\mathrm{\Delta }{f}_i}{{\tau }_p+{\tau }_w+{\tau }_f+{\tau }_d}\\ \mathrm{\Delta }{f}_i=\mathrm{\Delta }{f}_i[w]+\mathrm{\Delta }{f}_i[p]\end{array}$$ B.3

For the initial kinetic constants (see Table ), the maximum GRF, subject to the constraints of τ _p from 0.05 to 10 s, τ _w from 0.05 to 10 s, τ _f from 1 to 10 s and τ _d from 3 s to 15, is achieved at τ _p = 1.345 s, τ _w = 0.186 s, τ _f = 1 s, τ _d = 3. Finding constants also allows one to obtain OCT that, however, doesn’t exactly match the optimal parameters obtained with the initial constants. To assess the accuracy of OCT determination, the constants previously calculated for σ = 0.1 nm, σ = 0.5 nm and σ = 1 nm (16 points) were used to find OCT. The obtained OCT data were used to calculate the film thickness using the initial kinetic constants to estimate the relative error in determining the film thickness at the optimal growth rate (optimal growth rate relative error, OGRRE).

$$\mathrm{O}\mathrm{G}\mathrm{R}\mathrm{R}\mathrm{E}=\frac{|d({\mathrm{O}\mathrm{C}\mathrm{T}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}},\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}.\mathrm{k}\mathrm{i}\mathrm{n}.\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.)-d({\mathrm{O}\mathrm{C}\mathrm{T}}_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}},\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}.\mathrm{k}\mathrm{i}\mathrm{n}.\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.)|}{d({\mathrm{O}\mathrm{C}\mathrm{T}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}},\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}.\mathrm{k}\mathrm{i}\mathrm{n}.\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.)}·100\%$$ B.4

where: τ _opt,initfilm thickness calculated using initial kinetic constants and using OCT obtained for initial kinetic constants, d(OCT_calc,init.kin.const.)Film thickness calculated using initial kinetic constants and using OCT obtained for the calculated kinetic constants.

To estimate the error in OCT determination, the relative errors of the optimal cycle times (optimal cycle times relative error, OCTRE) were calculated using the following equation

$$\mathrm{O}\mathrm{C}\mathrm{T}\mathrm{R}\mathrm{E}=\frac{|{\tau }_{\mathrm{o}\mathrm{p}\mathrm{t},\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}-{\tau }_{\mathrm{o}\mathrm{p}\mathrm{t},\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}}|}{{\tau }_{\mathrm{o}\mathrm{p}\mathrm{t},\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}}·100\%$$ B.5

where: τ _opt,initthe sum of the optimal values of τ _p , τ _w , τ _f , τ _d for the initial values of the kinetic constants (τ _opt,init = 1.345 + 0.186 + 1 + 3 = 5.531 s), τ _opt,calcthe sum of the optimal values of τ _p , τ _w , τ _f , τ _d for the calculated values of the kinetic constants.

The authors declare no competing financial interest.