Improving Ti Thin Film Resistance Deviations in Physical Vapor Deposition Sputtering for Dynamic Random-Access Memory Using Dynamic Taguchi Method, Artificial Neural Network and Genetic Algorithm

Abstract

Many dynamic random-access memory (DRAM) manufacturing companies encounter significant resistance value deviations during the PVD sputtering process for manufacturing Ti thin films. These resistance values are influenced by the thickness of the thin films. Current mitigation strategies focus on adjusting film thickness to reduce resistance deviations, but this approach affects product structure profile and performance. Additionally, varying Ti thin film thicknesses across different product structures increase manufacturing complexity. This study aims to minimize resistance value deviations across multiple film thicknesses with minimal resource utilization. To achieve this goal, we propose the TSDTM-ANN-GA framework, which integrates the two-stage dynamic Taguchi method (TSDTM), artificial neural networks (ANN), and genetic algorithms (GA). The proposed framework requires significantly fewer experimental resources than traditional full factorial design and grid search method, making it suitable for resource-constrained and low-power computing environments. Our TSDTM-ANN-GA framework successfully identified an optimal production condition configuration for five different Ti thin film thicknesses: Degas temperature = 245 °C, Ar flow = 55 sccm, DC power = 5911 W, and DC power ramp rate = 4009 W/s. The results indicate that the deviation between the resistance values and their design values for the five Ti thin film thicknesses decreased by 86.8%, 94.1%, 95.9%, 98.2%, and 98.8%, respectively. The proposed method effectively reduced resistance deviations for the five Ti thin film thicknesses and simplified manufacturing management, allowing the required design values to be achieved under the same manufacturing conditions. This framework can efficiently operate on resource-limited and low-power computers, achieving the goal of real-time dynamic production parameter adjustments and enabling DRAM manufacturing companies to improve product quality promptly.

1. Introduction

Deposition for industrial products refers to the accumulation of high-performance metal or compound thin films on low-cost or specialized substrates, aiming to achieve product features such as appearance, wear resistance, and electrical conductivity requirements. Typically, techniques like Chemical Vapor Deposition (CVD) or Physical Vapor Deposition (PVD) are employed to achieve the film deposition. Due to the issue of toxic gas generation at high temperatures in CVD, PVD is currently predominantly used as the primary coating process.

PVD primarily uses the phase transition phenomenon of materials to heat or stimulate them, enabling the process of depositing thin film. PVD uses the physical energy from arcs or plasmas to bombard the surface of the cathode target material. This imparts kinetic energy to the target’s atoms, causing them to escape and adhere to the substrate intended for coating. As a result, cathode targets are divided into arc targets and sputtering targets (plasma targets). Arc targets involve the excitation of atoms through arcs, known as Arc Ion Plating (AIP), allowing direct interaction with gases. They are predominantly used for surface hardening coatings, such as TiN, TiC, TiCN, and more. On the other hand, sputtering targets use plasma-excited atoms, also referred to as Magnetron Sputtering. This method finds extensive application in various optoelectronic and semiconductor industries. Sputtered film layers exhibit fine smoothness, precise control over film thickness, and excellent repeatability. They can serve as targets for a wide range of materials, from conductors to insulators. In the context of semiconductors, the purpose of metal sputtering is to create metal wires that transmit information on chips. This involves the gradual deposition of atomic layers from the target material’s surface onto the semiconductor chip. Subsequently, the film is etched to create metal wires at the nanometer scale, connecting billions of micro-sized transistors within the chip. This interconnection facilitates the transmission of signals.

PVD can be categorized into three types: Vacuum Evaporation, Ion Plating, and Sputtering [1,2,3].

• Vacuum Evaporation: When metals are heated within a vacuum, they transform into gas and evaporate. Vacuum Evaporation exploits this principle. Processing is commonly carried out in a vacuum environment. Both metals and various compounds can be used as coating materials. Its applications include mirrors, reflective surfaces, plastic components, and more. However, it is rarely used for purposes like surface hardening of metals; instead, it is mainly employed for decorative objects.
• Ion Plating: This technique involves bombarding the target material with arcs, causing the atoms of the target material to be excited. These excited atoms then react with reactive gases, resulting in the deposition of compounds onto the surface of the workpiece. After achieving high vacuum within the chamber, inert gas is introduced and biased pressure is applied to generate argon ions (Ar⁺) and negatively charged electrons (e⁻). The positively charged argon ions will collide with the negatively biased substrate, effectively cleaning its surface. Subsequently, reactive gases are introduced, creating a plasma between the target material and the substrate, initiating the coating process. This method offers rapid film deposition and improved adhesion, making it suitable for coating cutting tools and similar applications.
• Sputtering: Sputtering involves the phenomenon where high-energy particles bombard a target material, causing atoms within the target to be ejected. In this principle, the target material acts as the cathode, the substrate as the anode, and when a high voltage is applied, ionized Ar⁺ ions near the cathode collide with it. The molecules or atoms knocked out by the impact of Ar⁺ ions are then deposited onto the substrate, forming a thin film. Sputtering PVD boasts advantages like strong adhesion, high uniformity, rapid deposition speed, and the ability to coat a wide range of materials, including alloys, metals, and insulators. Nowadays, sputtering deposition processes have become a pivotal technology in various applications, such as semiconductors, LCD panels, solar panels, optical components, and more.

Under standard environmental conditions, the resistance value of a sputtered Ti thin film with the same thickness should be consistent. However, in actual production processes, variations in process conditions can lead to discrepancies between the actual and design resistance values. To enhance the resistance stability of the final product, when maintaining the same thin film thickness, it’s desirable to minimize the differences between the actual resistance value and the designed resistance value.

The case study is derived from a dynamic random-access memory (DRAM) manufacturing company located in the Hsinchu Science Park, Taiwan. Dynamic Random-Access Memory (DRAM) is a type of volatile memory used in computers and electronic devices for storing data temporarily. It stores each bit of data in a separate capacitor within an integrated circuit. DRAM requires periodic refreshing of stored data to maintain integrity. Known for its high density and cost-effectiveness compared to static RAM (SRAM), DRAM is widely used in applications where large amounts of memory are needed, such as in main system memory for computers and smartphones. Its performance is critical for efficient data processing and multitasking. In the PVD sputtering process conducted by the case company, the Ti thin film produced experiences a significant resistance deviation due to suboptimal levels of the control variables being set.

The resistance of the Ti thin film is primarily determined by the thickness of the Ti thin film and the quality of the film produced during the sputtering process. Currently, the case company deals with abnormal Ti thin film resistance mainly by adjusting the film thickness to achieve the desired design resistance. For instance, if the Ti thin film resistance is too high, a process engineer would increase the thickness of the Ti thin film to meet the design resistance. However, excessive adjustments to the Ti thin film thickness can lead to deviations from the customer’s intended design structure in the DRAM devices, resulting in an increase in customer complaints. Furthermore, the case company applies various Ti thin film thicknesses in different product structures. If different Ti thin film resistances require distinct control variable settings, it would also escalate the complexity of manufacturing management. The purpose of this study is to enhance the resistance consistency of the Ti thin film, allowing it to achieve the desired design values for different film thicknesses.

The ideal function of this study is for the relationship between Ti thin film resistances and designed resistances to be directly proportional, representing a dynamic quality characteristic, as shown in Figure 1. The current status of the case company in terms of Ti thin film resistances and their corresponding design resistances for different Ti thin film thicknesses is presented in

Table 1. The summary table of resistance in different thickness.

Thickness (Å)	Ideal Resistance Value (Ω)	Resistance before Improvement (Ω)
100	42	162
150	28	88.6
200	21	55.1
300	14	23.1
500	8.4	9.12

. From Table 1, it is evident that when the Ti thin film thickness is 100 Å, 150 Å, 200 Å, 300 Å, and 500 Å, their respective ideal Ti thin film resistance (designed resistance value) should be 42 Ω, 28 Ω, 21 Ω, 14 Ω, and 8.4 Ω. However, the actual Ti thin film resistances for the case company are measured at 162 Ω, 88.6 Ω, 55.1 Ω, 23.1 Ω, and 9.12 Ω. Figure 2 shows the deviations between the current resistances and the designed resistance values for various thicknesses.

Confronting this issue, the company formed a dedicated group of process engineers. Employing a trial-and-error methodology, these engineers worked towards improving the resistance characteristics of the sputtered Ti thin film. Regrettably, this trial and error strategy yielded only marginal improvements, proving both time and resource-intensive. Given the limitations, the case company couldn’t feasibly carry out extensive experimentation for control variable adjustments using this method. Consequently, we introduce an innovative technique in this study to finely adjust control variables and optimize sputtered Ti thin film resistance value, all while minimizing the need for exhaustive on-site trials.

Our suggested approach TSDTM-ANN-GA involves the application of a two-stage dynamic Taguchi method (TSDTM), integrating an artificial neural network (ANN) and a genetic algorithm (GA). This framework is designed to mitigate the shortcomings of the traditional full factorial design and grid search method, ultimately streamlining the process of optimizing sputtered Ti thin film resistance.

2. Methodologies

A.

Taguchi method

Genichi Taguchi introduced the Taguchi method in the 1950s. This experimental design technique consistently produces favorable results and has remained effective over time. Over time, the Taguchi method has gained widespread application in the industrial sphere, predominantly for quality control purposes.

The Taguchi method proposes the utilization of an orthogonal array for experimental design. This choice streamlines the experimental process by minimizing the necessary number of trials to achieve a comprehensive and dependable dataset. Moreover, the employment of an orthogonal array highlights the reproducibility of the experiments [4].

The pivotal idea behind experimental design in the Taguchi method involves configuring control variable values using an orthogonal array (OA). An OA is represented as $L_a\left(b^c\right).$ The meanings of the symbols within the OA $L_a\left(b^c\right)$ are as follows:

L: stands for “Latin square”

a: represents the overall count of treatments

b: indicates the quantity of levels, encompassing distinct values for each control variable.

c: represents the upper limit of control variables that can be organized.

Table 2. L₈(2⁷) orthogonal array.

Treatment	A	B	C	D	E	F	G
1	1	1	1	1	1	1	1
2	1	1	1	2	2	2	2
3	1	2	2	1	1	2	2
4	1	2	2	2	2	1	1
5	2	1	2	1	2	1	2
6	2	1	2	2	1	2	1
7	2	2	1	1	2	2	1
8	2	2	1	2	1	1	2

shows the L₈(2⁷) OA. Columns A through G denote the control variables. The table descripts the allocation of various levels for each control variable, with levels 1 and 2 corresponding to differing values of their respective variables. The L₈(2⁷) OA consists of seven columns and eight rows, indicating an experimental design that involves seven production control variables and eight unique treatments.

Taguchi suggests that minimizing quality loss happens when a quality feature is in line with a designated target value. Greater quality corresponds to reduced quality loss. In order to quantify this loss, it is crucial to evaluate the connection between a quality characteristic and the resultant quality loss [4]. Taguchi strongly promoted the utilization of the signal-to-noise (S/N) ratio as a means to evaluate the effectiveness of Taguchi experimental design. Measured in decibels, the S/N ratio gauges the quality of control variable settings [5].

The goal of the dynamic Taguchi method is to treat the system’s quality feature as a function and optimize it to generate a range of outputs. Unlike static cases with a single output value as the optimization objective, dynamic scenarios involve multiple outputs to optimize. Dynamic Taguchi method is suitable when there’s an ideal relationship between input factors within a certain range and the output. Thus, dynamic factors (signal factors) are adjustment factors chosen based on engineering knowledge before conducting experiments. This is where dynamic situations significantly differ from the typical static case.

Various functional relationships might exist between the output and signal factors, determining the type of dynamic problem. The most common and effective form is the linear form, where signal factors act as proportionality factors. The zero-point proportional form is the primary type.

In the case of the zero-point proportional form, a simple linear relationship exists between the output (y) and the signal factor (M). Ideal experimental data points fall on a straight line passing through the origin:

$$y=\beta M$$

(1)

where

$$\beta ={\displaystyle \frac{{\sum }_{i=1}^n{M}_i{y}_i}{{\sum }_{i=1}^n{M}_i^2}}$$

(2)

The symbol β represents the slope of the line, and M represents the signal factor from repeated experiments under various conditions.

The closer the ideal behavior of the dynamic quality feature matches $y=\beta M$, the better. The formula for using the S/N ratio is outlined as follows:

$$S/N=-10log\left({\displaystyle \frac{S^2}{{\beta }^2}}\right)$$

(3)

where

$$S^2={\displaystyle \frac{\sum _{i=1}^n{\left(y_i-\beta M_i\right)}^2}{\mathrm{n}-1}}$$

(4)

J. Sun and J. Kainz (2023) adopted the Taguchi method to explore how four parameters in the hybrid pulse power characteristic (HPPC) test influence the performance of the equivalent circuit model (ECM) for Li-ion batteries, identifying optimal parameter combinations for predictable charge/discharge patterns [6]. S. Antoniou et al. (2023) used the Taguchi method to analyze the effect of leading-edge slat design parameters on high-angle-of-attack stability in a Blended-Wing-Body UAV [7]. K. S. Garud et al. (2023) applied the Taguchi method and computational fluid dynamics, this study explores the impact of non-Newtonian fluid models and factors like geometry and operation on shear stress in microfluidic cerebrovascular channels [8]. R. Kruzel et al. (2023) investigates spark plasma sintering (SPS) for sinters production, widely adopted in practical industries, optimizing a Si₃N₄–Al₂O₃–ZrO₂ composite using the Taguchi method [9]. Presently, within Taiwan, the Taguchi method is employed by several semiconductor companies for quality control purposes.

B.

Artificial Neural Network

The artificial neural network (ANN) is a method of data analysis that facilitates the generation of mathematical models for the purposes of prediction, decision-making, and diagnosing engineering and medical challenges. This technique involves using a dataset that includes input parameters and output variables to construct and refine these models [10].

The training process of an ANN entails iterative modifications to the weights of the network’s connections, symbolizing the intensity of connections between neurons. Elevated weight values indicate a higher significance for the output variables, as they influence the likelihood of activation. The amalgamation of multiple neurons gives rise to the construction of an ANN [11]. Shown in Figure 3 is an ANN consisting of three layers of neurons. The fundamental objective of ANN training is to ensure accurate correspondence between each input parameter and its corresponding output variable. The training process aims to minimize the disparity between desired target values and the output values generated by the ANN. Convergence is attained when this disparity stabilizes, indicating the completion of training. For assessing the ANN’s performance, validation using a distinct set of untrained samples is imperative to gauge the error between predicted values and known output variables [12]. A pivotal element in ANN training is the learning rate, a crucial hyperparameter influencing the convergence pace of the ANN. A higher learning rate accelerates convergence, while a lower rate decelerates the process [13,14]. It’s apparent that hyperparameters significantly impact the prediction model’s effectiveness.

Artificial neural network has been extensively employed in various studies for modeling and prediction purposes. N. Matera et al. (2023) employed a network of artificial neural network to train and predict the global hourly electrical power generated by various PV modules [15]. H. Azgomi et al. (2023) proposed an automatic apple disease diagnosis using a neural network, achieving 73.7% accuracy with color and texture features [16]. T. Calisir et al. (2023) introduced an artificial neural network model with 8 design parameters as inputs, achieving high accuracy in predicting heat transfer and weight for panel radiator production [17]. L. P. Lingamdinne et al. (2023) applied artificial neural networks to investigate the adsorption mechanism, pH effect, and desorption studies, yielding promising predictions for Pb(II) and As(V) removal [18].

C.

Genetic Algorithm

In 1975, John Holland introduced the genetic algorithm (GA), which has since undergone significant refinement. Demonstrating its prowess, the GA has emerged as a potent search technique for identifying globally optimal solutions. Its foundation lies in the principle of natural selection, favoring the survival of the most adept entities in biological systems. Leveraging this concept, the GA selects the most appropriate population for continued existence, culminating in a self-adapting optimization methodology that efficiently explores intricate, high-dimensional spaces.

The GA finds extensive utility in research involving hyperparameter optimization in machine learning models and engineering control variable refinement [19]. The evolutionary trajectory of the GA involves three primary stages: chromosome reproduction and duplication, crossover, and mutation. Operating within the solution domain of engineering quandaries, the GA, through numerous iterations, identifies the most versatile solution that fulfills all stipulated constraints. This corresponds to the chromosome exhibiting the highest fitness function value, signifying the globally optimal solution [20]. The algorithmic procedure of the GA is outlined as follows:

(1)

Population Initialization: Generate a set of potential solutions at random, forming the initial population state.

(2)

Evaluate Population: Assess the performance metric for each potential solution, denoted as the fitness function value.

(3)

Parent Selection: Choose solutions with superior fitness values as parents. Commonly, methods like roulette wheel selection or elite preservation are employed to determine parent selection.

(4)

Crossover: Randomly pick two parents for crossover, yielding new chromosomes through the process.

(5)

Mutation: Apply mutations to offspring with a designated probability, preventing convergence into local optima.

(6)

Next Generation Selection: Elect a subset of offspring and parents to form the subsequent population generation.

(7)

Iterate Steps 2 to 6: Continuously repeat steps 2 to 6 until the convergence criterion is met or the maximum iteration limit is achieved.

(8)

Final Selection of Optimal Solution: Ultimately, the global optimal solution will be identified.

Genetic algorithms have been widely used in numerous studies for modeling and prediction purposes. A. Neumann et al. (2023) reviewed Genetic Algorithm (GA) use in Engineer-To-Order (ETO) planning and scheduling problems, highlighting key ETO characteristics, constraints, and objectives. Common encoding formats and genetic operators are identified, emphasizing multi-objective approaches [21]. M. G. Altarabichi et al. (2023) proposed a two-stage surrogate-assisted evolutionary approach using Genetic Algorithm (GA) for efficient feature selection in large datasets [22]. R. M. Aziz et al. (2023) proposed a novel approach using Genetic Algorithm (GA) and deep learning for reliable detection of fraudulent transactions in Ethereum smart contracts, achieving high accuracy and outperforming various popular techniques [23]. R. Ghezelbash et al. (2023) adopted the genetic algorithm to improve mineral prospectivity mapping through unsupervised clustering and supervised machine learning methods [24].

D.

The Suggested approach

This paper introduces the utilization of Taguchi method’s OAs to facilitate an efficient experimental plan for the case company. Our focus is on employing the dynamic Taguchi method to optimize the control variable settings for attaining an elevated yield ratio in the manufacturing of DRAM devices. Initially, the first-stage dynamic Taguchi method is employed to gather valuable experimental data and identify significant control variables. Subsequently, the second-stage dynamic Taguchi method is employed to determine a favorable setting for these control variables. Nonetheless, this setting, restricted by predetermined levels of distinct control variables, might not fulfill the ultimate optimal control variable setting.

To address this, the pursuit of the global optimal setting entails exploring all potential combinations of control variable values within their respective ranges. To accomplish this, an artificial neural network is established to create a predictive model for the sputtered Ti thin film resistance across various control variable values. The predictive model forms the core of a GA, facilitating the exploration of the finest global control variable setting. This is accomplished through the evaluation of the fitness function linked to the anticipated Ti thin film resistance. The suggested approach’s flowchart is depicted in Figure 4.

3. Case Study

This study centers on a semiconductor company that specializes in the production of dynamic random-access memory (DRAM) devices. However, the observed discrepancy between the sputtered Ti thin film resistance and the intended designed resistance values of 8.4 Ω, 14 Ω, 21 Ω, 28 Ω, and 42 Ω has prompted the need for a solution in quality control. To address this challenge, we present an approach comprising a two-stage dynamic Taguchi method, an ANN, and a GA. The goal is to minimize the variance between the Ti thin film resistances of the DRAM devices and their intended target designed resistance values.

In this study, the concept of ideal sputtered Ti thin film resistance (M_i) is introduced as the signal factor, while the corresponding measured Ti thin film resistance (y_i) serves as the response variable. In the most ideal scenario, the slope of the line in Figure 1 is 1 (i.e., β = 1). To align with this, we consider the ideal Ti thin film resistance (M_i) as the signal factor, with the respective ideal values set as M₁ = 8.4 Ω, M₂ = 14 Ω, M₃ = 21 Ω, M₄ = 28 Ω, and M₅ = 42 Ω.

A.

Details of the Suggested approach

(1)

Adopting first-stage dynamic Taguchi method to choose the important control variables and determine the optimal directions for setting their levels

The resistance of the Ti thin film is influenced by the sputtering process, as shown in Figure 5. This figure encompasses various sub-processes, each associated with its respective production control variables. The production control variables for each sub-process are shows in

Table 3. Variables in the sputtering process.

Sub-Process	Variable
FOUP	L/L in
FI robot	FI Robot transfer
Aligner	ORT
Buffer/transfer robot	Robot transfer
Degas recipe	Temperature
	Pressure
	Time
Ti recipe	Ar gas stable flow
	Ar gas stable time
	Ar flow
	DC power
	DC power ramp rate
	Cryo pump 1st temperature
	Cryo pump 2nd temperature
Buffer/transfer robot	Robot transfer
FI robot	FI Robot transfer
Wafer input	L/L out

.

Drawing from engineering expertise, the team of process engineers has identified seven control variables for improving the Ti thin film resistance. These control variables encompass Degas temperature, Degas pressure, Ar flow, DC power, DC power ramp rate, Cryo pump 1st temperature, and Cryo pump 2nd temperature. The chosen control variables, along with their respective levels for the first-stage dynamic Taguchi method, are presented in

Table 4. Variables for the first-stage dynamic Taguchi method: L₈(2⁷).

Variable	Degas Temp. (°C)	Degas Pressure (Torr)	Ar Flow (sccm)	DC Power (W)	DC Power Ramp Rate (W/s)	Cryo Pump 1st Temp. (°C)	Cryo Pump 2nd Temp. (°C)
	A	B	C	D	E	F	G
Level 1	250	6	30	2000	2000	85	9
Level 2	350	8	60	4000	4000	100	11

. To uphold stringent quality control, the semiconductor company conducted Ti thin film resistance sampling from ten predetermined positions on a wafer, shown in Figure 6. Given the intricacy of our experimental design challenge involving seven control variables and two levels, we have opted for the L₈(2⁷) orthogonal array.

Process engineers were tasked with executing the first-stage dynamic Taguchi method to identify pivotal control variables. During this phase, a total of eight experiments were conducted, involving varied levels of settings for each control factor. The sputtered Ti thin film resistance values of ten DRAM devices for each experiment, structured in the Taguchi orthogonal array L₈(2⁷), are displayed in

Table 5. Results of L8(27) orthogonal array for the first-stage dynamic Taguchi method.

No		#1	#2	#3	#4	#5	#6	#7	#8
Variables	A	1	1	1	1	2	2	2	2
	B	1	1	2	2	1	1	2	2
	C	1	1	2	2	2	2	1	1
	D	1	2	1	2	1	2	1	2
	E	1	2	1	2	2	1	2	1
	F	1	2	2	1	1	2	2	1
	G	1	2	2	1	2	1	1	2
M1 = 8.4 Ω	N1	8.65	8.62	8.72	8.68	8.73	8.62	8.82	8.72
	N2	8.76	8.65	8.54	8.54	8.71	8.65	8.61	8.71
	N3	8.68	8.73	8.52	8.64	8.64	8.55	8.72	8.68
	N4	8.69	8.59	8.66	8.60	8.69	8.65	8.65	8.65
	N5	8.68	8.62	8.60	8.64	8.87	8.56	8.53	8.67
	N6	8.53	8.65	8.68	8.69	8.85	8.68	8.70	8.52
	N7	8.54	8.68	8.62	8.69	8.74	8.79	8.78	8.67
	N8	8.77	8.63	8.60	8.62	8.58	8.82	8.98	8.62
	N9	8.77	8.68	8.54	8.64	8.54	8.65	8.72	8.80
	N10	8.62	8.65	8.59	8.67	8.79	8.78	8.79	8.65
M2 = 14 Ω	N1	18.80	16.48	18.76	17.51	18.85	19.19	19.72	19.19
	N2	18.21	18.69	18.33	18.15	19.33	17.31	19.19	18.51
	N3	18.34	17.01	18.32	17.34	18.36	20.31	19.01	18.58
	N4	17.79	18.04	17.01	17.82	18.50	17.82	18.68	19.83
	N5	16.98	18.27	17.73	18.98	19.08	18.05	20.01	19.98
	N6	19.03	19.55	18.14	17.46	19.29	18.30	18.83	18.33
	N7	18.28	17.44	18.50	17.98	19.56	18.87	18.21	18.81
	N8	17.82	17.64	18.45	17.22	18.86	18.60	19.91	19.09
	N9	17.96	18.08	18.26	16.90	18.86	18.75	17.16	19.08
	N10	19.14	18.27	17.85	17.71	20.30	19.80	19.33	17.73
M3 = 21 Ω	N1	35.61	37.25	33.77	33.18	37.15	37.09	39.51	38.09
	N2	36.11	34.03	35.12	32.65	39.27	37.50	40.14	37.37
	N3	35.36	36.93	36.20	33.56	39.12	37.33	39.18	38.28
	N4	36.08	33.20	33.80	32.18	38.25	35.01	38.07	38.68
	N5	35.83	32.92	35.76	36.16	37.36	36.68	38.33	37.78
	N6	37.18	34.47	33.96	34.25	39.72	32.90	38.58	36.59
	N7	35.59	34.08	33.39	33.69	40.24	35.48	42.01	37.90
	N8	36.19	33.85	35.04	34.94	38.27	37.58	36.92	37.18
	N9	38.16	33.93	32.77	35.13	38.60	35.07	36.97	38.40
	N10	35.76	34.21	36.13	35.69	39.39	34.25	39.32	40.16
M4 = 28 Ω	N1	55.71	53.92	52.60	51.35	62.90	54.54	59.84	58.11
	N2	50.65	51.27	54.22	52.00	59.28	57.21	57.38	61.27
	N3	55.04	53.24	51.54	50.68	58.68	55.91	59.47	58.75
	N4	54.43	55.00	53.12	49.95	58.30	57.70	59.20	58.79
	N5	53.59	52.20	52.85	50.23	61.67	54.40	60.92	55.50
	N6	55.47	52.36	51.34	49.09	57.59	57.44	63.24	57.22
	N7	54.17	50.33	50.67	51.09	56.62	56.64	60.96	54.97
	N8	52.37	52.31	51.36	51.79	60.35	56.67	59.64	54.46
	N9	53.77	50.47	52.78	49.38	59.61	55.96	61.10	52.40
	N10	53.43	53.34	52.01	51.31	61.41	55.79	57.97	55.67
M5 = 42 Ω	N1	94.15	79.96	85.26	86.16	103.86	90.64	101.63	104.93
	N2	90.93	90.08	98.76	79.51	99.62	99.02	99.73	107.68
	N3	88.66	89.93	86.68	88.54	99.10	98.98	100.98	90.54
	N4	104.77	93.45	88.83	86.06	98.70	92.64	103.30	108.69
	N5	100.73	91.93	104.93	84.36	92.55	92.33	113.62	101.77
	N6	97.91	87.18	97.60	87.27	101.38	85.75	103.85	102.01
	N7	102.20	92.85	87.66	86.37	99.36	99.52	110.76	80.51
	N8	106.64	74.03	94.62	80.51	100.34	89.87	101.71	97.61
	N9	87.16	95.73	81.07	84.46	106.39	101.50	99.97	108.55
	N10	86.79	86.53	87.05	91.42	98.79	100.02	105.43	99.70
M1 = 8.4 Ω	Average	8.67	8.65	8.61	8.64	8.71	8.67	8.73	8.67
M2 = 14 Ω	Average	18.23	17.95	18.13	17.70	19.10	18.70	19.00	18.91
M3 = 21 Ω	Average	36.19	34.49	34.59	34.14	38.74	35.89	38.90	38.04
M4 = 28 Ω	Average	53.86	52.44	52.25	50.69	59.64	56.23	59.97	56.72
M5 = 42 Ω	Average	95.99	88.17	91.25	85.47	100.01	95.03	104.10	100.20
β		2.04	1.91	1.95	1.86	2.16	2.04	2.21	2.13
S		8.95	7.45	8.20	6.58	8.57	8.28	9.59	9.60
S/N ratio		−12.86	−11.83	−12.48	−10.99	−11.98	−12.15	−12.74	−13.08

. In this phase, employing a full factorial design to ascertain optimal settings for seven control variables, each featuring two levels, would necessitate a significant 128 experiments (2⁷ = 128). Nevertheless, through the utilization of the L₈(2⁷) orthogonal array, this study adeptly executed a compact set of only 8 experiments, effectively capturing a wide range of control variable settings.

Given the primary emphasis of this study on enhancing the resistance value of DRAM devices—a dynamic quality feature—the signal-to-noise ratio (S/N) for each experiment was computed using Equation (3). The resulting S/N ratios are tabulated in the final row of Table 5.

Table 6. Variable response table of S/N ratio for L₈(27).

Level	A	B	C	D	E	F	G
Level 1	−12.04	−12.21	−12.63	−12.52	−12.64	−12.23	−12.18
Level 2	−12.49	−12.32	−11.90	−12.01	−11.88	−12.30	−12.34
Effect	0.45	0.11	0.73	0.50	0.76	0.07	0.16
Rank	4	6	2	3	1	7	5

shows the outcomes of the variable response table, focusing on the S/N ratio of DRAM devices. In parallel, Figure 7 presents the results through a variable response chart. Additionally,

Table 7. ANOVA table of S/N ratio for L8(27).

Source	Degree of Freedom	Sum of Square	Mean of Square	F Value	p Value
A	1	0.40288	0.40288	14.36	0.032
B	1	0.02605 *
C	1	1.05424	1.05424	37.57	0.009
D	1	0.50887	0.50887	18.14	0.024
E	1	1.14844	1.14844	40.93	0.008
F	1	0.00937 *
G	1	0.04876 *
error	(3)	(0.08418)	(0.02806)
Total	7	3.19861
				R-Sq	R-Sq(adj)
				0.974	0.939

* pooling into the error term.

offers a comprehensive ANOVA (analysis of variance) breakdown for the same dataset. Analyzing the findings in Table 6, the impact of each production control variable on the S/N ratio follows this sequence, descending from high to low: E(0.76) > C(0.73) > D(0.50) > A(0.45) > G(0.16) > B(0.11) > F(0.07). Moving to Table 7, it is evident that the p-values associated with production control variables A, C, D, and E fall below the 0.05 threshold. This signifies that these four control variables play a significant role in shaping the S/N ratio. They emerge as crucial factors that wield notable influence on the S/N ratio. Guided by the insights from Figure 7, a better setting for the control variables surfaces: A setting of A = 250 °C, C = 60 sccm, D = 4000 W, and E = 4000 W/s stands out as a favorable combination to enhance the S/N ratio.

Based on the data detailed in

Table 8. Variable response table of slope β for L₈(27).

Level	A	B	C	D	E	F	G
Level 1	1.938	2.037	2.072	2.089	2.040	2.045	2.038
Level 2	2.136	2.038	2.002	1.985	2.035	2.030	2.037
Effect	0.198	0.001	0.070	0.104	0.005	0.016	0.001
Rank	1	7	3	2	5	4	6

, the descending order of the impact of control variables on the slope β of DRAM devices can be elucidated as follows: A(0.198) > D(0.104) > C(0.070) > F(0.016) > E(0.005) > G(0.001) > B(0.001). Adding to this insight,

Table 9. ANOVA table of slope β for L8(27).

Source	Degree of Freedom	Sum of Square	Mean of Square	F Value	p Value
A	1	0.078322	0.078322	585.30	0.000
B	1	0.000001 *
C	1	0.009890	0.009890	73.91	0.001
D	1	0.021669	0.021669	161.94	0.000
E	1	0.000050 *
F	1	0.000483 *
G	1	0.000001 *
error	(4)	(0.000535)	(0.000134)
Total	7	0.1104
				R-Sq	R-Sq(adj)
				0.995	0.992

* pooling into the error term.

furnishes the p values, highlighting that control variables A, C, and D all reside below the critical threshold of 0.05. As a result, it is evident that these three control variables wield substantial influence on the slope β of DRAM devices. They emerge as vital contributors that significantly shape β. Taking these discoveries into account, a more optimal setting for the control variables becomes evident: A setting of A = 250 °C, C = 60 sccm, and D = 4000 W for β, as presented in Figure 8.

To delve deeper into finding the most effective control variable setting to improve the resistance value of the DRAM devices, an additional Taguchi experimental design, denoted as L₉(3⁴), is set to be undertaken. Building on the analysis outcomes of the SN ratio and β from the preceding experiment, the focus narrows down to four pivotal control variables: A (Degas temperature), C (Ar flow), D (DC power), and E (DC power ramp rate). This new round of experimentation aims to recollect pertinent data within a range of experiments that might include the optimal solution, with the intent of pinpointing the optimal levels for control variables as suggested by the first-stage dynamic Taguchi method.

(2)

Adopting second-stage dynamic Taguchi method to gather dataset and determine the local optimal control variable setting

In this stage, we employ the second dynamic Taguchi method to identify the local optimal control variable configuration. It’s essential to emphasize that the term “local optimal setting” specifically denotes the most optimal setting attained exclusively through the implementation of the second-stage dynamic Taguchi method. The specific control variables and their respective levels earmarked for second-stage dynamic Taguchi method are shown in

Table 10. Variables for the second-stage dynamic Taguchi method: L₉(34).

Variable	Degas Temp. (°C)	Ar Flow (sccm)	DC Power (W)	DC Power Ramp Rate (W/s)
	A	C	D	E
Level 1	245	55	4000	4000
Level 2	250	60	5000	5000
Level 3	255	65	6000	6000

. For the purpose of this second-stage dynamic Taguchi method, the L₉(3⁴) orthogonal array was chosen as the framework for the experimental design. In this stage, using a full factorial design to determine the local optimal setting for four control variables, each characterized by three levels, would demand a considerable 81 experiments (3⁴ = 81). However, by harnessing the L₉(3⁴) orthogonal array, this study efficiently conducted a concise series of 9 experiments, encompassing a diverse spectrum of control variable settings.

Table 11. Results of L9(34) orthogonal array for the second-stage dynamic Taguchi method.

No		#1	#2	#3	#4	#5	#6	#7	#8	#9
Variables	A	1	1	1	2	2	2	3	3	3
	C	1	2	3	1	2	3	1	2	3
	D	1	2	3	2	3	1	3	1	2
	E	1	2	3	3	1	2	2	3	1
M1 = 8.4 Ω	N1	8.36	8.34	8.37	8.42	8.39	8.38	8.40	8.35	8.38
	N2	8.41	8.39	8.42	8.38	8.35	8.37	8.34	8.40	8.36
	N3	8.39	8.38	8.35	8.36	8.38	8.38	8.40	8.41	8.38
	N4	8.37	8.35	8.41	8.41	8.34	8.37	8.33	8.37	8.42
	N5	8.35	8.37	8.36	8.40	8.29	8.39	8.32	8.36	8.37
	N6	8.36	8.40	8.39	8.39	8.36	8.34	8.35	8.31	8.39
	N7	8.39	8.34	8.41	8.33	8.39	8.37	8.38	8.32	8.40
	N8	8.37	8.38	8.40	8.39	8.43	8.36	8.35	8.43	8.35
	N9	8.34	8.37	8.40	8.38	8.38	8.42	8.38	8.35	8.34
	N10	8.35	8.37	8.37	8.39	8.41	8.38	8.35	8.36	8.37
M2 = 14 Ω	N1	13.68	13.30	14.13	13.94	13.37	13.98	13.43	13.65	13.43
	N2	13.75	14.46	14.08	14.17	14.58	14.54	14.31	14.26	12.91
	N3	13.38	14.10	13.46	13.08	13.97	13.83	13.88	13.54	13.86
	N4	12.93	13.95	13.44	13.44	13.88	12.70	13.29	14.38	13.69
	N5	13.64	13.34	13.62	14.24	13.73	13.61	13.53	13.73	13.58
	N6	13.81	13.79	13.59	13.64	13.49	13.04	13.62	14.13	14.23
	N7	13.30	14.99	14.44	13.67	14.02	14.40	13.77	14.74	13.51
	N8	13.12	12.93	13.73	13.36	12.89	13.22	14.01	13.91	13.29
	N9	13.17	13.04	14.19	13.32	13.35	13.87	13.12	14.91	12.80
	N10	13.39	15.00	13.86	14.07	13.60	14.20	13.65	13.19	13.68
M3 = 21 Ω	N1	20.98	19.13	21.18	21.15	20.73	21.15	22.76	20.20	19.81
	N2	21.38	20.23	22.85	22.57	21.40	19.44	19.11	23.54	20.64
	N3	19.86	22.00	20.07	17.11	20.49	17.38	19.75	21.24	21.03
	N4	18.23	20.21	22.23	21.30	17.82	19.68	19.41	23.99	22.10
	N5	23.70	20.28	23.20	18.30	22.76	23.49	21.08	21.11	20.68
	N6	20.76	22.23	22.14	20.47	23.64	20.89	21.12	18.46	19.89
	N7	20.59	19.96	19.75	17.43	20.05	21.19	16.60	21.66	24.08
	N8	19.06	21.86	22.72	21.94	26.27	20.38	18.81	16.15	20.68
	N9	22.29	18.79	22.02	21.85	20.36	21.86	24.52	22.42	21.10
	N10	20.30	20.85	22.40	21.26	21.16	22.03	21.65	19.97	19.44
M4 = 28 Ω	N1	28.28	30.02	29.05	31.15	27.74	28.41	25.94	24.44	27.02
	N2	27.49	30.26	28.24	24.01	26.70	30.47	27.14	28.11	28.08
	N3	27.94	30.84	25.35	26.90	29.43	37.02	32.24	33.23	30.52
	N4	25.87	27.26	29.15	32.87	31.15	31.23	29.20	29.17	26.56
	N5	26.52	28.71	28.90	25.53	27.60	28.41	28.75	27.26	27.46
	N6	27.25	28.22	29.63	31.61	28.55	25.05	30.20	33.51	29.26
	N7	26.08	27.59	28.61	28.88	29.65	33.60	27.11	26.57	25.93
	N8	31.83	35.58	27.60	31.88	27.83	34.11	31.59	25.09	25.04
	N9	28.48	28.46	28.01	27.86	23.21	28.62	27.05	27.14	30.41
	N10	26.86	27.26	29.58	24.39	24.81	27.64	30.78	27.40	30.96
M5 = 42 Ω	N1	41.78	53.16	31.01	39.42	52.49	41.03	32.27	44.12	57.58
	N2	45.93	54.75	36.17	35.87	43.09	18.35	36.20	37.18	52.83
	N3	48.30	40.96	59.32	40.01	50.97	44.75	42.73	40.41	43.89
	N4	38.60	36.63	52.88	55.50	45.25	45.00	40.62	44.27	52.42
	N5	28.05	68.80	34.94	50.08	35.93	40.87	43.20	52.10	35.85
	N6	44.13	59.24	53.08	40.52	43.07	40.49	62.11	46.18	44.47
	N7	46.84	40.54	61.46	33.42	50.24	48.37	54.89	38.08	54.73
	N8	45.23	45.75	50.81	38.39	42.20	29.22	40.69	41.46	46.81
	N9	45.98	51.00	54.86	36.30	46.01	31.56	40.96	32.58	37.08
	N10	38.51	35.37	53.44	55.60	37.83	55.17	41.74	31.66	47.89
M1 = 8.4 Ω	Average	8.37	8.37	8.39	8.38	8.37	8.37	8.36	8.37	8.38
M2 = 14 Ω	Average	13.42	13.89	13.86	13.69	13.69	13.74	13.66	14.04	13.50
M3 = 21 Ω	Average	20.71	20.56	21.86	20.34	21.47	20.75	20.48	20.87	20.95
M4 = 28 Ω	Average	27.66	29.42	28.41	28.51	27.67	30.45	29.00	28.19	28.12
M5 = 42 Ω	Average	42.33	48.62	48.80	42.51	44.71	39.48	43.54	40.80	47.35
β		1.00	1.09	1.10	1.01	1.03	0.99	1.02	0.99	1.07
S		2.79	5.11	5.02	3.89	2.88	5.08	4.02	3.18	3.70
S/N ratio		−8.94	−13.39	−13.22	−11.76	−8.91	−14.25	−11.87	−10.18	−10.80

provides information on the resistances of sputtered Ti thin films and their corresponding S/N ratios across various settings. In

Table 12. Variable response table of S/N ratio for L₉(34).

Level	A	C	D	E
1	−11.85	−10.86	−11.12	−9.55
2	−11.64	−10.83	−11.98	−13.17
3	−10.95	−12.75	−11.33	−11.72
Effect	0.90	1.93	0.86	3.62
Rank	3	2	4	1

, we focus on the S/N ratio for DRAM devices, as shown in the variable response table. Complementing these tables, Figure 9 represents the data through a variable response chart. Additionally,

Table 13. ANOVA table of S/N ratio for L₉(34).

Source	Degree of Freedom	Sum of Square	Mean of Square	F Value	p Value
A	2	1.3363 *
C	2	7.3050	7.3050	3.65250	0.067
D	2	1.2032 *
E	2	19.9284	19.9284	9.96422	0.013
error	(4)	(2.54)	(0.6349)
Total	8	29.773
				R-Sq	R-Sq(adj)
				0.9147	0.8294

* pooling into the error term.

furnishes a comprehensive breakdown of the ANOVA results for the same dataset. Upon closer examination of the data presented in Table 6, the impact of each production control variable on the S/N ratio follows a descending order from high to low: E (3.62) > C (1.93) > A (0.90) > D (0.86). Transitioning to Table 13, it becomes evident that the p values associated with production control variables C and E are below the 0.05 threshold. This signifies the significant influence these four control variables exert on shaping the S/N ratio, establishing them as pivotal factors. Drawing insights from Figure 9, an better setting for the control variables emerges: C = 60 sccm and E = 4000 W/s. This setting is identified as favorable for enhancing the S/N ratio, guided by the representation of the data.

Based on the data provided in

Table 14. Variable response table of slope β for L9(34).

Level	A	C	D	E
1	1.062	1.009	0.989	1.033
2	1.008	1.038	1.056	1.035
3	1.026	1.050	1.051	1.029
Effect	0.054	0.041	0.066	0.006
Rank	2	3	1	4

, we can elucidate the order of influence of control variables on the slope β of DRAM devices as follows: D (0.066) > A (0.054) > C (0.041) > E (0.006). This insight is complemented by

Table 15. ANOVA table of slope β for L9(34).

Source	Degree of Freedom	Sum of Square	Mean of Square	F Value	p Value
A	2	0.004590	0.002295	95.94	0.010
C	2	0.002692	0.001346	56.27	0.017
D	2	0.008217	0.004108	171.74	0.006
E	2	0.000048 *
error	(2)	(0.000048)	(0.000024)
Total	8	0.015547
				R-Sq	R-Sq(adj)
				0.9969	0.9877

* pooling into the error term.

, which presents the p values. Notably, control variables A, C, and D all have p values below the critical threshold of 0.05. This confirms that these three control variables play a substantial role in shaping the slope β of DRAM devices, establishing them as crucial contributors. Considering these findings, a better setting for the control variables emerges: A = 250 °C, C = 55 sccm, and D = 4000 W for β, as indicated in Figure 10. It’s worth noting that Figure 9 reveals minimal disparity between the S/N ratios for C₁ and C₂ settings, suggesting that control factor C has a limited impact on the S/N ratio. Consequently, we identify the A₂C₁D₁E₁ setting as the locally optimal configuration for the control variables. Additionally, Table 13 highlights the insignificance of factors C and E on S/N ratios, while Table 15 confirms the lack of significance of factors A, C, and D on the slope β.

Following this, we aim to estimate the S/N ratio and β values using the local optimal control variables. Since variables A and D exert a negligible influence on the S/N ratio, we have excluded them from our S/N ratio prediction. The mean of nine S/N ratios is depicted in Table 11 as $\overline{SN}$, which equals −11.48 dB. We can also represent the mean of S/N ratio for C1 as ${\overline{C}}_1$, amounting to −10.86 dB, and the mean of S/N ratio for E1 as ${\overline{E}}_1$, which stands at −9.55 dB. Following dynamic Taguchi method, we calculate the predicted S/N ratio for the local optimal control variable setting can be computed as:

$$\begin{array}{l}\widehat{SN}=\overline{SN}+\left({\overline{C}}_1-\overline{SN}\right)+\left({\overline{E}}_1-\overline{SN}\right)={\overline{H}}_1+{\overline{I}}_1-\overline{SN}\\ =\left(-10.86\right)+\left(-9.55\right)-\left(-11.48\right)=-8.93\mathrm{d}\mathrm{B}.\end{array}$$

(5)

Likewise, since factors E have minimal impact on the slope β, their inclusion is disregarded in the prediction of β. The average of the nine β values is $\overline{\beta }=1.032$ ohms. The average β for A2 is ${\beta }_{A_2}=1.028$ ohms, for C1 it’s ${\beta }_{C_1}=1.009$ ohms, and for D1 it’s ${\beta }_{D_1}=0.989$ ohms. Thus, the predicted slope $\stackrel{̑}{\beta }$ for the local optimal control variable setting can be computed as:

$$\begin{array}{l}\stackrel{̑}{\beta }=\overline{\beta }+\left({\beta }_{A_2}-\overline{\beta }\right)+\left({\beta }_{C_1}-\overline{\beta }\right)+\left({\beta }_{D_1}-\overline{\beta }\right)={\beta }_{A_2}+{\beta }_{C_1}+{\beta }_{D_1}-2\times \overline{\beta }\\ =1.008+1.009+0.989-2\times 1.033=0.942\end{array}$$

(6)

To validate the precision of the projected S/N ratio and β values, the case company in question conducted a series of five confirmation trials. Table 16 presents a detailed summary of the results derived from these confirmation trials. To enhance the reliability of our predictions, we calculated confidence intervals at a 95% for both the predicted S/N ratio. This crucial step was accomplished by employing Equation (7) as detailed in reference [20].

$$CI=\sqrt{F_{\alpha ;1,\nu }\times V_e\times \left({\displaystyle \frac{1}{n_{obs}}}+{\displaystyle \frac{1}{r}}\right)}$$

(7)

where

$F_{\alpha ;1,{\nu }_2}$: the F statistic;

α: producer’s risk;

ν: degrees of freedom for the variance associated with the combined error;

V_e: variance of combined error;

n_obs: effective observation number

$={\displaystyle \frac{Totalnumberofexperiments}{1+\left(totaldegreeoffreedomassociatedwithitemsusedinestimatingmean\right)}}$; and

r: number of replicates.

In order to determine the confidence intervals at a 95% for both the S/N ratio and β, the subsequent calculations of the following components are carried out:

$${\left(n_{obs}\right)}_{S/N}={\displaystyle \frac{9}{1+{\nu }_C+{\nu }_E}}={\displaystyle \frac{9}{1+2+2}}=1.8$$

$$C{I}_{S/N}=\sqrt{F_{0.05;\mathrm{1,4}}\times V_e\times \left({\displaystyle \frac{1}{{\left(n_{obs}\right)}_{S/N}}}+{\displaystyle \frac{1}{r}}\right)}$$

$$=\sqrt{7.7086\times 0.6349\times \left({\displaystyle \frac{1}{1.8}}+{\displaystyle \frac{1}{3}}\right)}=2.09;$$

The confidence interval at a 95% for the S/N ratio ranges from −8.93 to −6.84. To confirm the better settings, the semiconductor company performed Ti thin film resistance sampling at three predetermined positions on a wafer, as shown in Figure 11. Analysis of the confirmation trial results, presented in Table 16, reveals a mean S/N ratio of −7.94 dB. Significantly, these value comfortably lie within its corresponding confidence intervals. This alignment between the experimental results and their respective confidence intervals provides strong evidence supporting the accuracy and reliability of our projections for the S/N ratio. It underscores the robustness of our conclusions, reaffirming the validity of the anticipated S/N ratio values derived from the analysis.

Table 16. Confirmation trials for two-stage dynamic Taguchi method.

No		Trial #1	Trial #2	Trial #3	Trial #4	Trial #5	Average
M1 = 8.4 Ω	N1	8.87	8.88	8.89	8.89	8.89
	N2	8.89	8.90	8.87	8.86	8.89
	N3	8.88	8.87	8.86	8.88	8.88
M2 = 14 Ω	N1	14.67	14.58	14.79	15.11	14.80
	N2	15.05	14.52	15.02	14.69	14.88
	N3	15.30	14.98	14.94	14.82	14.86
M3 = 21 Ω	N1	22.32	23.25	22.28	22.90	22.20
	N2	22.37	23.73	23.65	22.69	22.75
	N3	21.05	21.88	24.65	21.51	21.86
M4 = 28 Ω	N1	27.86	29.49	31.74	31.06	28.62
	N2	32.59	26.97	28.78	29.78	28.14
	N3	27.89	32.07	30.80	30.41	29.15
M5 = 42 Ω	N1	44.83	39.23	39.10	38.13	42.13
	N2	42.49	44.48	34.69	36.93	32.58
	N3	36.48	35.35	41.71	40.67	44.88
M1 = 8.4 Ω	Average	8.88	8.88	8.87	8.88	8.88	8.879
M2 = 14 Ω	Average	15.01	14.69	14.92	14.87	14.85	14.869
M3 = 21 Ω	Average	21.91	22.95	23.52	22.37	22.27	22.605
M4 = 28 Ω	Average	29.45	29.51	30.44	30.42	28.64	29.690
M5 = 42 Ω	Average	41.26	39.68	38.50	38.58	39.86	39.577
β		1.01	1.00	1.00	0.99	0.99	0.999
S		2.16	2.58	2.80	2.23	2.75	2.505
S/N ratio		−6.57	−8.24	−8.96	−7.06	−8.88	−7.940

Table 16. Confirmation trials for two-stage dynamic Taguchi method.

No		Trial #1	Trial #2	Trial #3	Trial #4	Trial #5	Average
M1 = 8.4 Ω	N1	8.87	8.88	8.89	8.89	8.89
	N2	8.89	8.90	8.87	8.86	8.89
	N3	8.88	8.87	8.86	8.88	8.88
M2 = 14 Ω	N1	14.67	14.58	14.79	15.11	14.80
	N2	15.05	14.52	15.02	14.69	14.88
	N3	15.30	14.98	14.94	14.82	14.86
M3 = 21 Ω	N1	22.32	23.25	22.28	22.90	22.20
	N2	22.37	23.73	23.65	22.69	22.75
	N3	21.05	21.88	24.65	21.51	21.86
M4 = 28 Ω	N1	27.86	29.49	31.74	31.06	28.62
	N2	32.59	26.97	28.78	29.78	28.14
	N3	27.89	32.07	30.80	30.41	29.15
M5 = 42 Ω	N1	44.83	39.23	39.10	38.13	42.13
	N2	42.49	44.48	34.69	36.93	32.58
	N3	36.48	35.35	41.71	40.67	44.88
M1 = 8.4 Ω	Average	8.88	8.88	8.87	8.88	8.88	8.879
M2 = 14 Ω	Average	15.01	14.69	14.92	14.87	14.85	14.869
M3 = 21 Ω	Average	21.91	22.95	23.52	22.37	22.27	22.605
M4 = 28 Ω	Average	29.45	29.51	30.44	30.42	28.64	29.690
M5 = 42 Ω	Average	41.26	39.68	38.50	38.58	39.86	39.577
β		1.01	1.00	1.00	0.99	0.99	0.999
S		2.16	2.58	2.80	2.23	2.75	2.505
S/N ratio		−6.57	−8.24	−8.96	−7.06	−8.88	−7.940

Table 17. The improvement percentage following dynamic Taguchi method.

Comparison	Degas Temp. (°C)	Ar Flow (sccm)	DC Power (W)	DC Power Ramp Rate (W/s)	|M1-8.4| (Ω)	|M2-14| (Ω)	|M3-21| (Ω)	|M4-28| (Ω)	|M5-42| (Ω)
	A	C	D	E
Before improvement	350	30	3000	3000	0.72	9.10	34.10	60.60	120.00
TSDTM	250	55	4000	4000	0.479	0.869	1.605	1.690	2.423
Improvement					33.5%	90.5%	95.3%	97.2%	98.0%

presents a comparison between the outcomes achieved prior to and subsequent to the implementation of the dynamic Taguchi method. Notably, the differences observed between the average resistances of the sputtered Ti thin films and the designed resistance values have demonstrated considerable improvements of 33.5%, 90.5%, 95.3%, 97.2%, and 98.0% for five different thicknesses. This improvement is shown in Figure 12, which showcases the contrast between the scenarios before and after the application of the dynamic Taguchi method. It is evident from the figure that the resistance values of the sputtered Ti thin films post the dynamic Taguchi method exhibit a remarkable proximity to the idealized design resistance values.

(3)

Applying ANN for constructing a predictive model

Using the information from Table 13 and Table 15, pivotal control variables can be discerned, namely A (Degas temp.), C (Ar flow), D (DC power), and E (DC power ramp rate). These selected variables are slated to function as the input parameters for the ANN. The disparity between the sputtered Ti thin film resistance value and the designed target resistance value will be employed as the output variable. Additionally, the experimental data derived from the L₉(3⁴) setting in Table 11 will be harnessed as the dataset for training the artificial neural network.

In this research, an artificial neural network architecture featuring three hidden layers is put forth. To effectively train this network, an allocation of 80% of the dataset is arbitrarily designated as the training dataset, while the remaining 20% is earmarked for use as the test dataset. The pursuit of optimal hyperparameters entails employing a grid search methodology. This process encompasses the exploration of diverse settings for hyperparameters, including the solver, activation function, the quantity of nodes within each hidden layer, learning rate, and momentum rate. The solver is subjected to assessment using the options [lbfgs, adam, sgd], while the activation function is scrutinized with the choices [logistic, tanh, relu]. Furthermore, the learning rate undergoes evaluation using values of [0.1, 0.2, 0.3, 0.4], and the momentum rate is explored across values of [0.6, 0.7, 0.8, 0.9]. Simultaneously, the number of nodes situated within each hidden layer is preset within the range of [1 to 10].

The artificial neural network undergoes training over a span of 1000 iterations. The criterion employed to finalize the weights is based on the attainment of the lowest root mean square error (RMSE) across the test dataset. Following the execution of the grid search methodology, the optimal setting for the hyperparameters are as follows: the solver is configured as adam, the activation function is specified as tanh, the learning rate is determined as 0.4, the momentum rate is defined at 0.6, and the quantity of nodes integrated within the three hidden layers are set to 5, 3, and 10, respectively. Consequently, the optimal architecture for the neural network is represented as 4-5-3-10-1, as shown in Figure 13.

(4)

Applying GA to find the global optimal control variable setting

In the framework of GA, the four control variables are represented by binary strings that correspond to their values. The boundaries for each control factor are established in accordance with their respective ranges, outlined in Table 10. The fitness function, articulated through Equation (8), plays a pivotal role in the GA process. When implementing the GA, the values assigned to a given control variable are initiated via random generation within the predefined range. This range spans from the lowest to the highest levels associated with the particular control factor, as detailed in Table 10.

$$Fittnessvalue={\displaystyle \frac{1}{\left|Predictedvalue-designedvalue\right|}}$$

(8)

The GA commences by establishing its initial generation, where 30 initial populations are randomly generated. Each chromosome signifies a unique setting of the control variables. Subsequently, the GA proceeds to execute a sequence of crossover and mutation processes, thereby facilitating the evolutionary progression of the population. For the crossover operations, a rate of 0.85 is employed, while the probability of mutation is designated at 0.05. The iteration process continues until it reaches a maximum of 500 iterations. We also defined the allowable ranges of production variable using “Level 1” and “Level 3” as specified in Table 10.

In this paper, we replicated the experimentation process by conducting the genetic algorithm (GA) procedure a total of ten trials. The outcomes of these trials are outlined in

Table 18. The results of ten trials of the ANN-GA.

Trials	A	C	D	E	Fitness Value
#1 run	245	55	5544	4084	0.93346
#2 run	245	55	5947	4095	0.94256
#3 run	245	55	5925	4406	0.92252
#4 run	245	55	5911	4009	0.95459
#5 run	245	55	5632	4462	0.95004
#6 run	245	55	5887	4050	0.94155
#7 run	245	55	5774	4146	0.93257
#8 run	245	55	5236	4097	0.94324
#9 run	245	55	5547	4165	0.95031
#10 run	245	55	5319	4133	0.94248

. Notably, Table 18 presents the paramount global fitness value as 0.95459, accompanied by the corresponding variable settings of A = 245 °C, C = 55 sccm, D = 5911 W, and E = 4009 W/s. Furthermore, the statistical summaries of the fitness values across the experiments are shown in

Table 19. The summarized table for ten trials of the ANN-GA.

Item	Value
The maximum fitness value in 10 trials	0.95459
The minimum fitness value in 10 trials	0.92252
Average fitness value of 10 trials	0.94133
Standard deviation of 10 trials	0.00961

. This table shows the maximum, minimum, mean, and standard deviation values of the fitness outcomes. The considerably low standard deviation across the experiments underscores the stability and reliability of the GA results.

In GA, we conducted 30 initial population tests and 500 iterations, resulting in a total of 15,000 tests. In contrast, a grid search would require $10\times 10\times 2000\times 2000=4\times {10}^8$ tests, significantly reducing the number of modeling tests.

B.

Conducting the Confirmation Trials

To substantiate the authenticity of the global optimal control variable setting found by GA, the case company carried out a set of five confirmation trials using the GA-identified optimal setting. The outcomes of these trials are chronicled in

Table 20. Confirmation trials for ANN-GA.

No		Trial #1	Trial #2	Trial #3	Trial #4	Trial #5	Average
M1 = 8.4 Ω	N1	8.510	8.498	8.487	8.477	8.493
	N2	8.515	8.507	8.507	8.515	8.473
	N3	8.482	8.510	8.484	8.499	8.491
M2 = 14 Ω	N1	14.474	14.754	14.579	14.763	14.495
	N2	14.518	14.447	14.586	14.749	14.648
	N3	14.292	14.645	14.332	14.286	14.509
M3 = 21 Ω	N1	21.941	23.931	21.521	22.152	22.490
	N2	23.879	22.980	23.012	20.611	22.064
	N3	21.167	21.752	22.740	22.930	22.759
M4 = 28 Ω	N1	26.566	28.493	30.413	29.116	28.770
	N2	31.403	29.063	29.107	29.014	29.803
	N3	31.025	27.248	26.983	31.696	27.914
M5 = 42 Ω	N1	46.310	41.946	38.212	44.069	40.562
	N2	44.768	38.007	43.068	36.078	39.299
	N3	36.322	39.984	37.381	41.392	40.705
M1 = 8.4 Ω	Average	8.502	8.505	8.493	8.497	8.486	8.497
M2 = 14 Ω	Average	14.428	14.615	14.499	14.599	14.550	14.538
M3 = 21 Ω	Average	22.329	22.888	22.425	21.898	22.438	22.395
M4 = 28 Ω	Average	29.664	28.268	28.835	29.942	28.829	29.108
M5 = 42 Ω	Average	42.467	39.979	39.554	40.513	40.189	40.540
β		1.031	0.991	0.987	1.006	0.996	1.002
S		2.415	1.593	1.917	2.100	1.256	1.856
S/N ratio		−7.395	−4.118	−5.766	−6.391	−2.020	−5.138

. Moreover, a comprehensive juxtaposition of the average resistances of the sputtered Ti thin films prior to and subsequent to the implementation of the suggested approach (ANN_GA) is presented in

Table 21. The improvement percentage following suggested approach.

Comparison	Degas Temp. (°C)	Ar Flow (sccm)	DC Power (W)	DC Power Ramp Rate (W/s)	|M1-8.4| (Ω)	|M2-14| (Ω)	|M3-21| (Ω)	|M4-28| (Ω)	|M5-42| (Ω)
	A	C	D	E
Before improvement	350	30	3000	3000	0.72	9.10	34.10	60.60	120.00
TSDTM	250	55	4000	4000	0.48	0.87	1.61	1.69	2.42
ANN-GA	245	55	5911	4009	0.10	0.54	1.40	1.11	1.46
Improvement					86.6%	94.1%	95.9%	98.2%	98.8%

. This table reveals that the difference between the average resistances and the designed resistance values are 0.10 Ω, 0.54 Ω, 1.40 Ω, 1.11 Ω, and 1.46 Ω, respectively. These differences have been further improved by 86.8%, 94.1%, 95.9%, 98.2%, and 98.8% for five different thicknesses. Figure 14 serves as a representation of the contrast between the outcomes prior to and subsequent to the adoption of the suggested approach. After applying the suggested approach, the resistance values of the sputtered Ti thin films exhibit a significant convergence towards the ideal resistance values.

4. Conclusions

Due to resistance value deviations in Ti thin films produced during the PVD process for DRAM, this study aims to minimize these deviations across multiple film thicknesses with minimal resource utilization. To achieve this goal, we propose the TSDTM-ANN-GA framework.

In the first-stage dynamic Taguchi method, we utilized the L₈(2⁷) orthogonal array to conduct experiments and identify the key control variables affecting the resistance values of Ti thin films. Typically, a full factorial experiment with seven control variables and two levels would require 128 experiments. However, using the L₈(2⁷) orthogonal array effectively reduces this to just 8 experiments. The first-stage dynamic Taguchi method drastically curtails the experimental workload, achieving an 16X reduction relative to the full factorial experiment. Through the use of variable response tables and analysis of variance (ANOVA), we identified four key control variables: A (Degas temperature), C (Ar flow), D (DC power), and E (DC power ramp rate).

Next, the second-stage dynamic Taguchi method was executed using the L₉(3⁴) orthogonal array to re-collect experimental data for the four key control variables. Traditionally, a full factorial experiment with four control variables at three levels each would require 81 experiments. However, the L₉(3⁴) orthogonal array simplifies this to just 9 experiments. The second-stage dynamic Taguchi method significantly reduces the experimental workload, achieving a 9-fold reduction compared to a full factorial experiment. Using the L₉(3⁴) array, the optimal combination settings for the four key control variables were determined to be A (Degas temperature) = 250 °C, C (Ar flow) = 55 sccm, D (DC power) = 4000 W, and E (DC power ramp rate) = 4000 W/s. This optimal combination significantly reduced the resistance value deviations in the Ti thin films. For the five different Ti film thicknesses, the resistance deviation percentages were reduced by 33.5%, 90.5%, 95.3%, 97.2%, and 98.0%, respectively.

The dataset obtained from the L₉(3⁴) experiments was used as training and testing data for the artificial neural network (ANN). In constructing the network, the four control variables were used as inputs. Hyperparameters were optimized using a grid search method, resulting in a learning rate of 0.4, a momentum rate of 0.6, and three hidden layers with 5, 3, and 10 nodes, respectively. The final ANN architecture was 4-5-3-10-1. Subsequently, a genetic algorithm (GA) was employed to determine the global optimal control variable settings in the parameter space. In the GA, a total of 15,000 tests were conducted. In contrast, a grid search would require 4 × 10⁸ tests, significantly reducing the modeling effort and making it suitable for low-power computers. The globally optimal solution identified was A (Degas temperature) = 245 °C, C (Ar flow) = 55 sccm, D (DC power) = 5911 W, and E (DC power ramp rate) = 4009 W/s. Final validation experiments showed that the proposed TSDTM-ANN-GA framework reduced the deviations between the Ti thin film resistance values and their design values by 86.8%, 94.1%, 95.9%, 98.2%, and 98.8%, respectively.

The proposed method effectively reduced resistance deviations for the five Ti thin film thicknesses and simplified manufacturing management, allowing the required design values to be achieved under the same manufacturing conditions. This framework can efficiently operate on resource-limited and low-power computers, achieving the goal of real-time dynamic production parameter adjustments and enabling DRAM manufacturing companies to improve product quality promptly.

This study presents an integrated framework that combines the Taguchi method, neural networks, and genetic algorithms. This framework demonstrates considerable flexibility, allowing for potential integration with other machine learning techniques (such as decision trees, SVM, etc.) or heuristic algorithms (like particle swarm optimization, simulated annealing, etc.) in future studies.

Further optimization can be achieved by refining each method individually. For example, the Taguchi method can be enhanced by considering the interaction effects among variables. Additionally, the ANN structure can be improved to more accurately predict the impact of process parameters on resistance stability, and the GA can be used to more effectively search for optimal parameter combinations. Incorporating self-adjusting algorithms could also be considered, enabling the TSDTM-ANN-GA framework to dynamically adapt and learn in response to process variations, thereby further enhancing the resistance stability of the thin films.

To adapt the framework for real-time dynamic production parameter adjustments, it is essential to enhance its data processing and feedback mechanisms to enable real-time monitoring and response. Integrating Internet of Things (IoT) technology could facilitate instant data collection and analysis. Additionally, faster algorithms may need to be developed or implemented to ensure timely and accurate parameter adjustments during the manufacturing process. Furthermore, the models within the framework, such as ANN and GA, should be capable of quickly adapting to changes in production conditions, thereby supporting real-time dynamic adjustments in various environments.

The framework has the potential for scalability and can be applied to other semiconductor manufacturing processes. However, modifications may be necessary, including adjusting input variables and response variables to meet the specific requirements of different processes. Additionally, the training data for the ANN and the objective functions for the GA may need to be redesigned and calibrated according to the new process goals to ensure the accuracy and relevance of the optimization results.

The primary limitation of this research is that relying on a single dataset may restrict the generalizability of the results. Therefore, future studies should incorporate independent datasets from various semiconductor manufacturing processes to enhance the framework’s validity and reliability.

The proposed framework can also be extended to deep learning-based image product defect detection tasks. Notably, its suitability for deployment in manufacturing edge artificial intelligence (edge AI) devices is highlighted, offering more efficient training and accurate inference results for image defect detection tasks in resource-constrained environments.