2.1 Data preparation

The HfO₂ thin films used to construct the annealing process–thin film property relationship were grown on Si substrates using a commercial PEALD device (Picosun Advanced R200, Finland) with an integrated remote plasma source. HfO₂ thin films were grown by alternating exposure to the precursor tetrakis-ethylmethylamino hafnium (Hf(N(CH₃)(CH₂CH₃))₄, TEMAH) and O₂/Ar gas mixture plasma reactant at a deposition temperature of 150°C. The number of deposition cycles was 500, and the pulse sequence for each HfO₂ growth cycle was as follows: TEMAH feeding (1.6 s), N₂ purging (19 s), Ar/O₂ mixture feeding (11 s) and Ar purging (10 s). The samples were then annealed in quartz tube annealing equipment (RS 80/300/11, Nabertherm) for 3 h. The annealing process included a combination of three atmospheres (vacuum, O₂ and N₂) and six annealing temperatures (300°C to 800°C in 100°C increments). For vacuum annealing, the pressure in the tubular annealing chamber was approximately 1 × 10^–4 Pa. For O₂ and N₂ atmosphere annealing, the gas flow rate was 150 SCCM for both O₂ and N₂. The HfO₂ thin films were measured using an ellipsometer (Horiba Uvisel 2), and the thicknesses and refractive indices were extracted using the Tauc-Lorentz model in DeltaPsi2 software, neglecting the extinction coefficient (k). The O/Hf ratio of the HfO₂ thin films was analyzed using X-ray photoelectron spectroscopy (XPS) (Thermo Scientific) with a monochromatic Al Kα (1486.6 eV) X-ray source. The data used to construct the annealing process–thin film property relationship consisted of 19 samples, including 1 as-deposited sample and 18 annealed samples.

The HfO₂ thin film data used for LIDT modeling and prediction come from Ref. [Reference Lin, Zhu, Song, Liu, Yin, Zeng and Shao26], including 12 samples treated by different annealing process parameters. Among them, six samples were annealed in an O₂ atmosphere, and the other six samples were annealed in a N₂ atmosphere. The annealing temperature ranged from 300°C to 800°C.

The SiO₂ thin film data used for property modeling and prediction come from Ref. [Reference Yin, Zhu, Zeng, Song, Chai, Shao, Zhang, Zhao, Li and Shao27], including 10 samples grown by different deposition process parameters. Among them, four samples were grown at different temperatures ranging from 50°C to 200°C, and six samples were grown with different precursor source exposure times ranging from 0.2 to 0.7 s.

Table 1 lists the detailed parameters of the datasets used to model and predict the properties of HfO₂ and SiO₂ thin films, including the refractive index, thickness and stoichiometric ratio. As the annealing temperature increases, the thickness of the HfO₂ thin film decreases and the refractive index increases. In a vacuum environment, O₂ environment and N₂ environment, the thickness of HfO₂ thin films annealed at different temperatures changes in the range of 34.7–42.7, 38.5–49.1 and 36.3–46.7 nm, respectively, while the refractive index (at 355 nm) of HfO₂ thin films annealed at different temperatures changes in the range of 1.99–2.24, 1.83–1.97 and 1.88–2.00, respectively. This means that the packing density of the HfO₂ thin film increases with increasing annealing temperature^[Reference Lin, Zhu, Song, Liu, Yin, Zeng and Shao^26]. In addition, the O/Hf ratio of HfO₂ thin films annealed in an O₂ environment fluctuates slightly around the ideal value of 2.0. However, the O/Hf ratio of HfO₂ thin films annealed in vacuum and N₂ environments decreases with increasing annealing temperature.

Table 1

Datasets for property prediction of HfO₂ and SiO₂ thin films.

^*Note: 0, 1, 2 and 3 represent the as-deposited sample, O₂, N₂ and vacuum, respectively.

Table 2 lists the detailed parameters of the datasets used for LIDT modeling and prediction. Compared with PEALD-HfO₂ thin films, PEALD-SiO₂ thin films have lower absorption and impurity content. Furthermore, properties such as absorption, impurity content and stoichiometric ratio influence each other. Detailed relationships are described in Refs. [Reference Lin, Zhu, Song, Liu, Yin, Zeng and Shao26,Reference Yin, Zhu, Zeng, Song, Chai, Shao, Zhang, Zhao, Li and Shao27]. The LIDT was tested in one-on-one mode according to ISO 21254 using a Gaussian-shape 3ω neodymium-doped yttrium aluminum garnet (Nd:YAG) laser (355 nm, 7.8 ns). The LIDT test was performed under normal incidence, and the maximum laser fluence with zero damage probability was determined as the LIDT. It is worth mentioning that the LIDT of HfO₂ thin films is lower than that of SiO₂ thin films, which is attributed to the fact that the bandgap of HfO₂ is lower than that of SiO₂.

Table 2

Datasets for LIDT prediction of HfO₂ and SiO₂ thin films.

^*Note: 1 and 2 represent HfO₂ samples and SiO₂ samples, respectively.

2.2 Models

Six models, namely four BPNN models with different numbers of hidden layers (single-hidden-layer BPNN, double-hidden-layer BPNN, three-hidden-layer BPNN and four-hidden-layer BPNN), a support vector machine regression (SVR) model^[ Reference Cortes and Vapnik ^{28 ]} using a Gaussian kernel function and a linear regression (LR) model^[ Reference Wang, Wu, Zheng, Zeng, Ding and Zhang ^{29 ]}, were used to establish the correlation between the annealing process and the refractive index, layer thickness and O/Hf ratio of PEALD-HfO₂ thin films. Except for the LR model, which belongs to the category of linear regression fitting, the other models belong to the category of nonlinear regression fitting. All models performed regression fitting by training on a training set, tuning the modeling parameters to achieve the highest accuracy (i.e., lowest error) and then validating on a validation set. When constructing the relationship between the annealing process and the properties of the PEALD-HfO₂ thin films, 6 samples were randomly selected as the validation set, and the remaining 13 samples (12 annealed samples and 1 as-deposited sample) were used as the training set. When predicting the LIDT of PEALD-grown thin films, 6 samples (3 HfO₂ samples and 3 SiO₂ samples) were randomly selected as the validation set, and the remaining 16 samples (9 HfO₂ samples and 7 SiO₂ samples) were used as the training set. When predicting the properties of PEALD-SiO₂ thin films, the leave-one-out cross-validation method was adopted owing to limited data. For each test, one sample was used as a validation set, and the remaining samples were used as a training set until every sample was used as a validation set. Subsequently, the average performance deviation was calculated for each model.

Figure 1 shows a schematic of the THL-BPNN model, including an input layer (layer 0), three hidden layers (layers 1–3) and an output layer (layer 4), with each layer containing one or more neurons. The number of neurons in the input and output layers was determined by the number of input and output variables in the dataset, whereas the number of neurons in the hidden layers was initially determined using Equation (1) (an empirical formula) and finally determined by a global traversal search:

(1) $$\begin{align}l=\sqrt{u+v}+a,\end{align}$$

where u, v and l are the numbers of neurons in the input, output and hidden layers, respectively, and a is a random number between 1 and 10.

Figure 1

THL-BPNN model with all neurons in adjacent layers connected, where x = [x ₁; x ₂], y ₁ and h _ij represent the input, output and intermediate processing signals, respectively.

The neurons receive input signals from the previous layer and generate output signals for the next layer^[ Reference Xu, Zhang, Fu and Liu ^{30 ,} Reference Ma, Li, Liu, Zhang, Zhang, Zheng and Lu ^{31 ]}. For example, the first neuron in layer 1 (from top to bottom), the circle where h ₁₁ is located, receives input signals, x = [x ₁; x ₂], from layer 0. Then x undergoes linear transformation to get the weighted sum, z, which is expressed as follows:

(2) $$\begin{align}z={\boldsymbol{w}}^{\mathrm{T}}\boldsymbol{x}+b,\end{align}$$

where w = [w ₁; w ₂]  $\in$  R is a weight vector between the neurons, and b  $\in$  R is a bias.

Subsequently, z passes through a nonlinear activation function $f\left(\cdotp \right)$ ^[ Reference Qiu ^{32 ]}, and the output signal h ₁₁ is generated as follows:

(3) $$\begin{align}{h}_{11}=f(z).\end{align}$$

These processes were performed for each neuron in each layer to form the final output signal, y ₁ ^[ Reference Wiecha, Arbouet, Girard and Muskens ^{33 ]}. Obviously, mapping from the input space to the output space is initially established through layer-by-layer information transfer.

To further improve the mapping accuracy, a training loss was constructed in the output layer, and an appropriate training algorithm is selected to update the relevant parameters (weights w and bias b) in combination with the chain rule^[ Reference LeCun, Bengio and Hinton ^{34 ]} until the loss or the number of iterations reaches the preset threshold^[ Reference Guo, Barrett, Wang and Lvovsky ^{35 ]}. The Levenberg–Marquardt algorithm^[ Reference Hagan and Menhaj ^{36 ]} was used to solve the nonlinear least squares problem. The hyperbolic tangent function was selected as the activation function for all hidden layers. The initialization state of each run was fixed to avoid interference from other factors.

2.3 Model specification and evaluation

2.3.1 Variable scaling

Considering that different distribution ranges of the input and output values may lead to biased assessments, Equation (4) is used to scale the input and output of the data to [–Reference Xu, Zhu, Chai, Roshanzadeh, Boyd, Rudolph, Zhao, Chen and Shao1, 1]:

(4) $$\begin{align}{X}_\mathrm{norm}=\frac{\left({Y}_\mathrm{max}-{Y}_\mathrm{min}\right)\left(X-{X}_\mathrm{min}\right)}{X_\mathrm{max}-{X}_\mathrm{min}}+{Y}_\mathrm{min},\end{align}$$

where X is the input or output vector; X _max and X _min are the maximum and minimum values of the input or output vector, respectively; and Y _max and Y _min are the maximum and minimum values after normalization, respectively.

2.3.2 Model evaluation metrics

The coefficient of determination (R ²)^[ Reference Chicco, Warrens and Jurman ^{37 ]} was used to evaluate the overall performance of each model. The average accuracy (AA) was used to evaluate the performance of each model on a validation set with only a single sample. The root mean square error (RMSE)^[ Reference Chai and Draxler ^{38 ]} was used to measure the deviation between the predicted and measured values:

(5) $$\begin{align}{R}^2&=1-{\frac{\sum \left({Y}_i-{T}_i\right)}{\sum {\left({Y}_i-\overline{Y}\right)}^2}}^2,\end{align}$$

(6) $$\begin{align}\mathrm{AA}&=\frac{1}{n}\sum \limits_{i=1}^n\left(1-\frac{\left|{Y}_i-{T}_i\right|}{\left|{Y}_i\right|}\right),\end{align}$$

(7) $$\begin{align}\mathrm{RMSE}&={\left[\frac{1}{n}\sum \limits_{i=1}^n{\left({Y}_i-{T}_i\right)}^2\right]}^{1/2},\end{align}$$

where n is the size of the dataset; Y_i and T_i are the measured and predicted values of the ith sample in the dataset, respectively; and $\overline{Y}$ is the average of the measured values. A lower RMSE (close to 0) and higher R ² and AA (close to 1) indicate smaller differences between the measured and predicted values.

3.1 Analysis of the number of hidden layers of the BPNN model

The influence of the number of hidden layers in the BPNN model on the modeling accuracy was studied using the measured data of the refractive index, layer thickness and O/Hf ratio of the PEALD-HfO₂ thin films treated with different annealing process parameters. The optimal number of neurons in each hidden layer was determined by a global traversal search on the training set corresponding to the lowest mean absolute error, and then the optimal model was applied to the validation set. For the refractive index and layer thickness datasets, the total number of neurons in the BPNN model with multiple hidden layers was consistent with that of the single-hidden-layer BPNN model. For the O/Hf ratio dataset, because the optimal number of neurons in the single-hidden-layer BPNN model is only five, this value is set as the maximum number of neurons in each hidden layer in the BPNN model with multiple hidden layers. The modeling and prediction accuracies are shown in Figure 2. Overall, as the number of hidden layers increased from one to three, the difference between the R ² and RMSE in the training and validation sets decreased, indicating that the model moved from inexact to exact fitting. However, as the number of hidden layers was further increased to four, the difference between the R ² and RMSE in the training and validation sets increased. This may be due to the fact that the combination of neurons in each layer grows exponentially with the number of hidden layers, which introduces the risk of overfitting while potentially obtaining better solutions. The only exception is the modeling of the refractive index, where a single-hidden-layer BPNN also exhibits good performance, which could be attributed to the small variation in the properties and the uncomplicated relationship between the input and output. With the three-hidden-layer BPNN model, the R ² values of the refractive index, layer thickness and O/Hf ratio were higher than 0.90 in both the training and validation sets. The THL-BPNN model was selected for the follow-up study.

Figure 2

Accuracy of BPNNs with one to four hidden layers based on (a) the refractive index (at 355 nm), (b) layer thickness and (c) O/Hf ratio of PEALD-HfO₂. The four columns in each subgraph represent the R ² values of the model in the training and validation sets and the RMSE values in the training and validation sets, respectively. The table indicates the number of neurons in each hidden layer of each model.

3.2 Comparison of the THL-BPNN model with other models

The performance of the THL-BPNN model was further evaluated and compared with the LR and SVR models. The refractive index, layer thickness and O/Hf ratio of the HfO₂ thin films predicted by the three models were compared with the measured values, as shown in Figure 3 and Table 3. As shown in Figures 3(a), 3(d) and 3(g), the poor performance of the LR model on all three datasets indicates a nonlinear relationship between the annealing process and the thin film properties. As shown in Figures 3(b), 3(e) and 3(h), the SVR model obtains a better fit than the LR model on the layer thickness and O/Hf ratio datasets, but it still does not perform well enough on the refractive index dataset. As shown in Figures 3(c), 3(f) and 3(i), the predicted and measured values of most samples are in good agreement, particularly for the refractive index dataset, indicating that the THL-BPNN model has a high accuracy in modeling and predicting the relationship between the annealing process parameters and HfO₂ thin film properties.

Figure 3

Measured and predicted (a)–(c) refractive index, (d)–(f) layer thickness and (g)–(i) O/Hf ratio of HfO₂ thin films. The data in the left-hand, middle and right-hand columns are predicted by the LR model, SVR model and THL-BPNN model, respectively. The blue line (with a slope of 1) serves as a guideline for perfect prediction.

Table 3

Evaluation of the LR, SVR and THL-BPNN models.

Table 3 lists the specific performance of all models on the training and validation sets. The THL-BPNN model performs best among the three regression models, with R ² values not lower than 0.90 for the refractive index, layer thickness and O/Hf ratio datasets. High R ² values and low RMSE values indicate that the THL-BPNN model can capture the patterns and extend them to unknown data. In short, the THL-BPNN model shows good stability in constructing the relationship between the annealing process and HfO₂ thin film properties under several conditions.

3.3 Evaluation of the THL-BPNN model for other thin film applications

3.3.1 Prediction of the LIDT of PEALD-HfO₂ and PEALD-SiO₂ thin films

The LIDT value is a key specification for thin films used in laser systems^[ Reference Pu, Liu, Wang, Pan, Chen and Liu ^{39 ,} Reference Du, Zhu, Shi, Liu, Sun, Yi and Shao ^{40 ]}. Firstly, we analyzed the main factors affecting the LIDT. According to Ref. [Reference Lin, Zhu, Song, Liu, Yin, Zeng and Shao26], the main factors affecting the LIDT of HfO₂ thin films are the C impurity content, N impurity content, absorption and O/Hf ratio. Pearson’s correlation coefficient was used to further analyze the correlation between the main influencing factors and the LIDT. The results shown in Figure 4 indicate that, except for the O/Hf ratio, which is positively correlated with the LIDT, all other parameters are negatively correlated with the LIDT. The change in the C and N impurity contents can be represented by the total impurity content. Likewise, for SiO₂ thin films, factors affecting the LIDT include the total impurity contents, absorption and O/Si ratio. Then, we applied the THL-BPNN to the quantitative prediction of the LIDT based on these factors. The total impurity contents, absorption, stoichiometric ratio and type of thin film were fed into the THL-BPNN as input variables, and the LIDT was derived as the output variable.

Figure 4

Correlations between properties of HfO₂ thin films used in this section. Blue indicates a negative correlation, whereas red indicates a positive correlation. Darker colors and larger circles indicate higher correlations. The numbers inside the circles indicate the corresponding correlation coefficients of the two features.

Furthermore, the predicted LIDT and measured LIDT of each sample are shown in Figure 5. It is observed that the THL-BPNN model performs well in both training and validation sets with high accuracy and low error, which is smaller than the relative error of the LIDT. The relative error of damage probability is about ±15%, mainly due to the uncertainty of the nonuniformity among the samples (3%), the measurement of the laser spot area (5%) and the fluctuation of laser energy (5%)^[ Reference Liu, Wei, Wu, Yu, Cui, Yi and Shao ^{41 ]}. For the training set and validation set, the R ² values are 1.00 and 0.97, respectively, and the RMSE values are 0.48 and 2.32, respectively. The results show that the THL-BPNN model is effective for predicting LIDT values of HfO₂ and SiO₂ thin films.

Figure 5

Comparison of measured and predicted LIDT values on the (a) training set and (b) validation set.

3.3.2 Prediction of other properties of PEALD-SiO₂ thin films

SiO₂ is the most common low-refractive-index material used for laser thin films in the ultraviolet to near-infrared wavelength region. It is of great significance to study the correlation between the properties of SiO₂ thin films and the deposition parameters. Therefore, we applied the THL-BPNN model to evaluate the relationship between the deposition parameters and the properties of PEALD-SiO₂ thin films. Figure 6 shows the excellent performance of the THL-BPNN model in predicting the properties of PEALD-SiO₂ thin films on the validation set, including the refractive index, layer thickness and O/Si ratio. For most samples, the prediction deviation was smaller than the measurement error.

Figure 6

Comparison of measured and predicted values of (a) the refractive index (at 355 nm), (b) the layer thickness and (c) the O/Si ratio for SiO₂ thin films in the validation set.

Table 4 lists the R ², AA and RMSE values of the THL-BPNN model for SiO₂ thin film properties. Except for the average R ² value of the O/Si ratio on the training set of 0.81, the other values, including the average R ² value of the refractive index and layer thickness in the training set and the AA values of the three properties in the validation set, are higher than 0.98. Although the THL-BPNN model did not perform sufficiently well on the O/Si ratio training set, it still provided accurate predictions on the corresponding validation set. This could be attributed to the successful learning of correlations by the THL-BPNN model through training. Therefore, the THL-BPNN model can be used to construct the relationship between the deposition parameters and PEALD-SiO₂ thin film properties, thus proving the universality of the THL-BPNN model in studying the nonlinear relationship between the process parameters and thin film properties.

Table 4

Evaluation of the THL-BPNN model for SiO₂ thin film properties.