Electrical TEG-Based Monitoring of Metal Diffusion and Reaction-Interlayer Formation in Oxide Dielectrics Using Physics-Informed XGBoost Modeling

Abstract

Metal atoms or ions originating from interconnects, electrodes, or diffusion-barrier layers can migrate into oxide dielectrics during thermal annealing, causing insulation degradation, leakage-path formation, capacitance variation, and interfacial reaction. This work proposes an electrical test-element-group (TEG)-based monitoring framework that jointly evaluates metal diffusion and reaction-interlayer formation in oxide dielectrics. A Fick–Arrhenius diffusion model is used to calculate a continuous metal concentration profile, while a parabolic reaction model describes the growth of an interfacial reaction layer. The concentration profile and reaction-layer thickness are then coupled to integral resistance and series-capacitance models for adjacent-electrode resistance TEGs and vertical capacitance TEGs, respectively. A physics-informed simulation dataset was generated over a broad range of thermal, material-library, and geometry parameters, and measurement uncertainty was introduced to emulate inline electrical test conditions. XGBoost regression and classification models were then trained to inversely extract the 1% diffusion front, reaction-interlayer thickness, and process-risk class from the simulated electrical responses. With a calibrated material-library feature set, the model achieved an R² of 0.959 for diffusion-front extraction, an R² of 0.802 for interlayer-thickness extraction, and a risk-classification accuracy of 93.86%. The results show that resistance TEGs mainly capture lateral conductive-path formation, whereas capacitance TEGs are more sensitive to vertical interlayer formation and dielectric degradation. The proposed framework provides a non-destructive route for monitoring diffusion- and reaction-induced process risks using inline electrical measurements. The present work is based on model-generated datasets; material-specific deployment requires calibration using SIMS, TEM/EDS, XPS, C–V, I–V, or reference TEG data.

1. Introduction

Metal/oxide interfaces are ubiquitous in interconnects, contact modules, bonding structures, gate stacks, capacitors, and heterogeneous integration platforms. During thermal annealing, metal species such as Cu, Ti, Al, W, or Ru can diffuse into dielectric layers or react with oxide surfaces. Such processes can produce leakage paths, alter trap density, degrade barrier integrity, and change local dielectric properties [1,2,3,4,5]. In Cu interconnect technology, the need for robust diffusion barriers and contamination monitoring has long been recognized because even small amounts of metal drift through SiO2 or low-$\mathit{k}$ dielectrics can change device reliability [2,6,7,8].

Recent studies in emerging electronic and heterostructure materials have further emphasized that interfacial contamination, imperfect interfaces, and buried reaction layers can strongly affect electronic, optoelectronic, and sensing performance. These reports support the need for non-destructive monitoring strategies that can detect interface-related process risk before full physical failure analysis is performed [9,10].

Conventional analysis of buried metal diffusion and interfacial reaction relies on SIMS, TEM/EDS, XPS, or other physical and chemical analysis methods [1,3,4,5]. These methods provide direct information on concentration profiles and interfacial structure, but they are not well suited for rapid inline monitoring of every wafer. Electrical test-element-group (TEG) structures offer a complementary route because metal contamination and reaction-layer formation can be converted into measurable resistance, leakage, or capacitance changes [11,12,13,14,15,16,17,18,19]. Related device-parameter extraction studies have also shown that machine-learning-based interpretation can reveal physically meaningful trends in semiconductor devices [20]. However, the mapping from electrical response to buried degradation state is highly nonlinear because the measured resistance depends on the spatial distribution of the concentration profile, while the measured capacitance depends on a series combination of position-dependent dielectric constants and interfacial layers.

The present work addresses this inverse-monitoring problem using a physics-informed modeling and machine-learning framework. Figure 1 summarizes the concept. Thermal annealing determines the metal concentration profile and reaction-interlayer thickness. These hidden states are converted into resistance and capacitance of lateral and vertical TEGs. Finally, a physics-informed XGBoost model estimates the diffusion front, interlayer thickness, and process-risk class from noise-added electrical data.

The contributions of this work are as follows. First, a unified diffusion–reaction model is formulated for metal migration and reaction-interlayer formation in oxide dielectrics. Second, adjacent-electrode resistance and vertical capacitance TEG responses are derived using integral electrical models rather than step-profile approximations. Third, a model-generated dataset with inline measurement uncertainty is constructed and used to train XGBoost-based inverse models [21,22]. Finally, feature-importance analysis is used to identify which electrical and calibrated material-library parameters govern diffusion-front and interlayer-thickness extraction [23,24,25,26,27].

2. Physics-Based Integrated Diffusion–Reaction Model

2.1. Metal Diffusion in a Semi-Infinite Oxide

The oxide is approximated as a semi-infinite medium along the positive diffusion coordinate x, where x = 0 corresponds to the metal/oxide interface. The symbol tox denotes the physical oxide thickness of the vertical capacitance TEG, whereas L_gap denotes the lateral electrode spacing of the resistance TEG. Under the first-order assumption of a constant diffusion coefficient, the metal concentration C(x,t) follows Fick’s second law:

$${\displaystyle \frac{\partial \mathit{C}\left(\mathit{x},\mathit{t}\right)}{\partial \mathit{t}}}=\mathit{D}\left(\mathit{T}\right){\displaystyle \frac{{\partial }^{\mathbf{2}}\mathit{C}\left(\mathit{x},\mathit{t}\right)}{\partial {\mathit{x}}^{\mathbf{2}}}}$$

(1)

The semi-infinite approximation is used only for deriving the local concentration profile near the metal/oxide interface. In the finite electrical TEG calculations, the physically relevant penetration length is limited by the actual geometry. Therefore, when x_0.01 exceeds tox or L_gap, it is interpreted as a diffusion-length descriptor indicating saturated vertical or lateral penetration rather than as a literal oxide thickness. Under the boundary conditions $\mathit{C}\left(\mathbf{0},\mathit{t}\right)={\mathit{C}}_{\mathit{s}}$, $\mathit{C}\left(\mathbf{\infty },\mathit{t}\right)={\mathit{C}}_{\mathbf{0}}$, and $\mathit{C}\left(\mathit{x},\mathbf{0}\right)={\mathit{C}}_{\mathbf{0}}$, the complementary-error-function solution is obtained [6,7,8]:

$$\mathit{C}\left(\mathit{x},\mathit{t}\right)={\mathit{C}}_{\mathbf{0}}+\left({\mathit{C}}_{\mathit{s}}-{\mathit{C}}_{\mathbf{0}}\right)\mathbf{e}\mathbf{r}\mathbf{f}\mathbf{c}\left({\displaystyle \frac{\mathit{x}}{\mathbf{2}\sqrt{\mathit{D}\left(\mathit{T}\right)\mathit{t}}}}\right)$$

(2)

Equation (2) is the standard complementary-error-function solution of Fick’s second law for a semi-infinite medium with constant surface concentration. The temperature-dependent diffusion coefficient is described by an Arrhenius relation:

$$\mathit{D}\left(\mathit{T}\right)={\mathit{D}}_{\mathbf{0}}\mathbf{e}\mathbf{x}\mathbf{p}\left(-{\displaystyle \frac{{\mathit{E}}_{\mathit{a}}}{{\mathit{k}}_{\mathit{B}}\mathit{T}}}\right)$$

(3)

where ${\mathit{D}}_{\mathbf{0}}$ is the diffusion prefactor, ${\mathit{E}}_{\mathit{a}}$ is the activation energy, ${\mathit{k}}_{\mathit{B}}$ is the Boltzmann constant, and $\mathit{T}$ is the absolute temperature. A threshold diffusion front ${\mathit{x}}_{\mathbf{t}\mathbf{h}}$ is defined as the position at which the normalized concentration reaches ${\mathit{C}}_{\mathbf{t}\mathbf{h}}$:

$${\mathit{x}}_{\mathbf{t}\mathbf{h}}=\mathbf{2}\sqrt{\mathit{D}\left(\mathit{T}\right)\mathit{t}}\hspace{0.17em}{\mathbf{e}\mathbf{r}\mathbf{f}\mathbf{c}}^{-\mathbf{1}}\left({\displaystyle \frac{{\mathit{C}}_{\mathbf{t}\mathbf{h}}-{\mathit{C}}_{\mathbf{0}}}{{\mathit{C}}_{\mathit{s}}-{\mathit{C}}_{\mathbf{0}}}}\right)$$

(4)

$${\mathit{x}}_{\mathbf{v}\mathbf{e}\mathbf{r}\mathbf{t},\mathbf{e}\mathbf{f}\mathbf{f}}=\mathbf{m}\mathbf{i}\mathbf{n}\left({\mathit{x}}_{\mathbf{0.01}},{\mathit{t}}_{\mathbf{o}\mathbf{x}}\right), {\mathit{x}}_{\mathbf{l}\mathbf{a}\mathbf{t},\mathbf{e}\mathbf{f}\mathbf{f}}=\mathbf{m}\mathbf{i}\mathbf{n}\left({\mathit{x}}_{\mathbf{0.01}},{\mathit{L}}_{\mathbf{g}\mathbf{a}\mathbf{p}}\right)$$

(5)

In this work, x_0.01 denotes the 1% diffusion-front descriptor. Because x_0.01 is defined in the semi-infinite diffusion coordinate, it can exceed tox. This does not mean that the physical oxide thickness exceeds several micrometers; rather, it indicates that the vertical oxide is fully penetrated within the modeled concentration threshold. For device interpretation, the effective vertical and lateral penetrations are therefore defined as x_vert,eff = min(x_0.01,t_ox) and x_lat,eff = min(x_0.01,L_gap).

2.2. Reaction-Interlayer Formation

In addition to diffusion, the metal and oxide can react at the interface and form a modified interlayer. This reaction layer can represent a metal-rich oxide, an oxygen-deficient region, a mixed interfacial compound, or a process-damaged dielectric layer. The present model does not assign a unique chemical phase; instead, it treats the reaction layer as a phenomenological interfacial layer whose material constants must be calibrated for each metal/oxide stack.

The interlayer thickness ${\mathit{d}}_{\mathbf{I}\mathbf{L}}$ is modeled using a phenomenological parabolic growth law commonly used to describe diffusion- or reaction-limited interfacial processes [6,8,14]:

$${\mathit{d}}_{\mathbf{I}\mathbf{L}}\left(\mathit{T},\mathit{t}\right)=\mathbf{m}\mathbf{i}\mathbf{n}\left[\sqrt{{\mathit{k}}_{\mathbf{I}\mathbf{L}}\left(\mathit{T}\right)\mathit{t}},\hspace{0.33em}{\mathit{d}}_{\mathbf{s}\mathbf{a}\mathbf{t}},\hspace{0.33em}\mathbf{0.95}{\mathit{t}}_{\mathbf{o}\mathbf{x}}\right].$$

(6)

Equation (6) is a phenomenological parabolic growth expression used to represent diffusion- or reaction-limited interfacial-layer formation. It is not intended to identify a unique chemical phase. The parameters kIL,0, EIL, and dsat must be calibrated for a specific metal/oxide stack using physical or electrical reference data. The reaction-rate coefficient follows an Arrhenius form:

$${\mathit{k}}_{\mathbf{I}\mathbf{L}}\left(\mathit{T}\right)={\mathit{k}}_{\mathbf{I}\mathbf{L},\mathbf{0}}\mathbf{e}\mathbf{x}\mathbf{p}\left(-{\displaystyle \frac{{\mathit{E}}_{\mathbf{I}\mathbf{L}}}{{\mathit{k}}_{\mathit{B}}\mathit{T}}}\right),$$

(7)

$${\mathit{f}}_{\mathbf{I}\mathbf{L}}\left(\mathit{x},\mathit{T},\mathit{t}\right)={\displaystyle \frac{\mathbf{1}}{\mathbf{1}+\mathbf{e}\mathbf{x}\mathbf{p}\left[\frac{\mathit{x}-{\mathit{d}}_{\mathbf{I}\mathbf{L}}\left(\mathit{T},\mathit{t}\right)}{{\mathit{\lambda }}_{\mathbf{I}\mathbf{L}}}\right]}}$$

(8)

where ${\mathit{k}}_{\mathbf{I}\mathbf{L},\mathbf{0}}$ and ${\mathit{E}}_{\mathbf{I}\mathbf{L}}$ are the reaction prefactor and activation energy with material-library parameters. The saturation thickness ${\mathit{d}}_{\mathbf{s}\mathbf{a}\mathbf{t}}$ prevents unphysical growth beyond the process-limited reaction region.

Figure 2 illustrates the unified physical state used in the simulation. The erfc-type metal concentration decays continuously into the oxide, whereas the reaction interlayer is represented as a finite region close to the interface. The orange curve in Figure 2 indicates that the reaction-layer phase fraction is localized near the metal/oxide interface, while the blue curve represents the longer-range diffused metal concentration. The representative d_IL of approximately 20 nm is used only for visualization and is not claimed as a universal experimental value.

2.3. Concentration-Dependent Conductivity and Resistance TEG Response

The resistance and capacitance values used in this study are not direct experimental measurements. They are model-generated electrical TEG responses calculated from the hidden diffusion–reaction state. In an actual fabrication line, the same quantities would correspond to wafer-level inline isolation resistance and MIM/MOS capacitance measured from physical TEG structures. Metal diffusion increases the effective conductivity of the oxide, particularly near the interface where the metal concentration is high. The local conductivity is expressed as

$${\mathit{\sigma }}_{\mathbf{d}\mathbf{i}\mathbf{f}\mathbf{f}}\left(\mathit{x},\mathit{t}\right)={\mathit{\sigma }}_{\mathbf{o}\mathbf{x}}+{\mathit{\sigma }}_{\mathit{m}}{\left({\displaystyle \frac{\mathit{C}\left(\mathit{x},\mathit{t}\right)}{{\mathit{C}}_{\mathit{s}}}}\right)}^{\mathit{\beta }},$$

(9)

The resistance values used in the database were obtained by numerical integration of the local conductivity along the lateral TEG spacing. In an experimental implementation, the corresponding value would be obtained from wafer-level adjacent-electrode leakage or isolation-resistance measurements. Where ${\mathit{\sigma }}_{\mathbf{o}\mathbf{x}}$ is the intrinsic oxide conductivity, ${\mathit{\sigma }}_{\mathit{m}}$ is the metal-assisted conductivity prefactor, and $\mathit{\beta }$ describes the nonlinearity of concentration-assisted conduction.

The reaction interlayer can also contribute to lateral conduction near the interface. For a lateral adjacent-electrode TEG with electrode spacing ${\mathit{L}}_{\mathbf{g}\mathbf{a}\mathbf{p}}$, width $\mathit{W}$, and oxide thickness ${\mathit{t}}_{\mathbf{o}\mathbf{x}}$, the effective local conductivity is written as

$${\mathit{\sigma }}_{\mathbf{e}\mathbf{f}\mathbf{f}}\left(\mathit{x},\mathit{t}\right)={\mathit{\sigma }}_{\mathbf{d}\mathbf{i}\mathbf{f}\mathbf{f}}\left(\mathit{x},\mathit{t}\right)+{\mathit{\sigma }}_{\mathbf{I}\mathbf{L}}\mathit{H}\left({\mathit{d}}_{\mathbf{I}\mathbf{L}}-{\mathit{x}}_{\mathbf{\perp }}\right),$$

(10)

where ${\mathit{\sigma }}_{\mathbf{I}\mathbf{L}}$ is the effective interlayer conductivity, $\mathit{H}$ is the Heaviside function, and ${\mathit{x}}_{\mathbf{\perp }}$ denotes the vertical distance from the interface. For the compact lateral model used here, the interlayer contribution is included through an effective conductivity enhancement factor in the TEG cross-section. The total lateral resistance is then calculated as a continuous series resistance:

$${\mathit{R}}_{\mathbf{T}\mathbf{E}\mathbf{G}}={\int }_{\mathbf{0}}^{{\mathit{L}}_{\mathbf{g}\mathbf{a}\mathbf{p}}}{\displaystyle \frac{\mathit{d}\mathit{x}}{\mathit{W}{\mathit{t}}_{\mathbf{o}\mathbf{x}}{\mathit{\sigma }}_{\mathbf{e}\mathbf{f}\mathbf{f}}\left(\mathit{x},\mathit{t}\right)}}.$$

(11)

Equation (11) captures the fact that diffusion does not create an abrupt step junction. Instead, the local resistance evolves gradually along the electrode gap.

2.4. Concentration- and Interlayer-Dependent Capacitance Response

The local dielectric constant is modeled as a function of the normalized metal concentration:

$${\mathit{\epsilon }}_{\mathit{r},\mathbf{d}\mathbf{i}\mathbf{f}\mathbf{f}}\left(\mathit{x},\mathit{t}\right)={\mathit{\epsilon }}_{\mathbf{o}\mathbf{x}}\left[\mathbf{1}-\mathit{\alpha }{\displaystyle \frac{{\left(\mathit{C}\left(\mathit{x},\mathit{t}\right)/{\mathit{C}}_{\mathit{s}}\right)}^{\mathit{\gamma }}}{\mathbf{1}+{\left(\mathit{C}\left(\mathit{x},\mathit{t}\right)/{\mathit{C}}_{\mathit{s}}\right)}^{\mathit{\gamma }}}}\right].$$

(12)

This phenomenological expression represents the case in which metal contamination, defects, oxygen deficiency, or leakage-assisted loss reduce the measured effective capacitance. The parameters $\mathit{\alpha }$ and $\mathit{\gamma }$ are therefore stack- and frequency-dependent fitting parameters rather than universal constants [13,14,15].

The reaction interlayer contributes an additional series dielectric layer. For a vertical capacitance TEG with area $\mathit{A}$, the total capacitance is expressed as

$${\displaystyle \frac{\mathbf{1}}{{\mathit{C}}_{\mathbf{T}\mathbf{E}\mathbf{G}}}}={\displaystyle \frac{{\mathit{d}}_{\mathbf{I}\mathbf{L}}}{{\mathit{\epsilon }}_{\mathbf{0}}{\mathit{\epsilon }}_{\mathbf{I}\mathbf{L}}\mathit{A}}}+{\int }_{{\mathit{d}}_{\mathbf{I}\mathbf{L}}}^{{\mathit{t}}_{\mathbf{o}\mathbf{x}}}{\displaystyle \frac{\mathit{d}\mathit{x}}{{\mathit{\epsilon }}_{\mathbf{0}}{\mathit{\epsilon }}_{\mathit{r},\mathbf{d}\mathbf{i}\mathbf{f}\mathbf{f}}\left(\mathit{x},\mathit{t}\right)\mathit{A}}},$$

(13)

where ${\mathit{\epsilon }}_{\mathbf{I}\mathbf{L}}$ is the effective dielectric constant of the reaction layer. Equation (11) shows that capacitance TEGs are especially sensitive to vertical reaction-layer formation and dielectric degradation. The capacitance values were calculated using a vertical series-capacitor formulation, where the reaction interlayer and the remaining concentration-modified oxide are connected in series. The parameter tox denotes the total oxide thickness of the capacitance TEG and is explicitly defined when first introduced. In practical measurements, CTEG would correspond to the measured capacitance from MIM- or MOS-type monitor capacitors at a specified frequency and bias condition.

3. Simulation Dataset and Inline Measurement Emulation

3.1. Parameter Space and Model-Generated Raw Data

The raw data used in this study were generated from the physics-informed diffusion, reaction-interlayer, and electrical TEG models. Each row was obtained by sampling thermal process conditions, diffusion parameters, reaction-layer parameters, TEG geometry, and electrical material parameters, followed by calculation of x_0.01, d_IL, R_TEG, C_TEG, and the process-risk class using Equations (1)–(13).

Table 1. Representative parameter ranges used for model-generated dataset construction.

Category	Parameter	Representative Range
Thermal process	T	250–500 °C
Thermal process	t	1–300 min
Diffusion	$D_0$	10−10–10−6 m2/s
Diffusion	$E_a$	0.5–1.8 eV
Reaction interlayer	$k_{\mathrm{I}\mathrm{L},0}$	10−18–10−10 m2/s
Reaction interlayer	$E_{\mathrm{I}\mathrm{L}}$	0.3–1.6 eV
Reaction interlayer	$d_{\mathrm{s}\mathrm{a}\mathrm{t}}$	1–50 nm
Geometry	$L_{\mathrm{g}\mathrm{a}\mathrm{p}}$	0.2–10 μm
Geometry	$t_{\mathrm{o}\mathrm{x}}$	5–500 nm
Geometry	$W$	1–100 μm
Capacitance geometry	$A$	10−10–10−6 m2
Conductivity model	${\sigma }_{\mathrm{o}\mathrm{x}}$, ${\sigma }_m$, ${\sigma }_{\mathrm{I}\mathrm{L}}$	logarithmic sampling
Dielectric model	${\epsilon }_{\mathrm{o}\mathrm{x}}$, ${\epsilon }_{\mathrm{I}\mathrm{L}}$, $\alpha$, $\gamma$	stack-dependent sampling

summarizes the representative parameter ranges used to construct the model-generated simulation dataset.

For each condition, ${\mathit{x}}_{\mathbf{0.01}}$, ${\mathit{d}}_{\mathbf{I}\mathbf{L}}$, ${\mathit{R}}_{\mathbf{T}\mathbf{E}\mathbf{G}}$, ${\mathit{C}}_{\mathbf{T}\mathbf{E}\mathbf{G}}$, and a process-risk class were calculated. The risk classes were defined using diffusion-front and interlayer-thickness ratios relative to ${\mathit{L}}_{\mathbf{g}\mathbf{a}\mathbf{p}}$ and ${\mathit{t}}_{\mathbf{o}\mathbf{x}}$. The reported prediction accuracy therefore quantifies inverse-extraction performance within the defined physics-informed simulation space. It should not be interpreted as a direct guarantee of absolute process accuracy for an arbitrary fabrication line without stack-specific calibration.

3.2. Noise-Added Inline Electrical Dataset

To emulate inline electrical-test conditions, stochastic perturbations were added to resistance, capacitance, process temperature, and annealing time. The noise model reflects high-resistance measurement fluctuation, capacitance-resolution limits, process-temperature uncertainty, and annealing-time variation. This prevents the model from learning an ideal deterministic database and allows robustness to be evaluated under measurement-like perturbations. High-resistance measurements were assigned larger relative uncertainty, while capacitance was perturbed using pF-level absolute noise and small relative uncertainty. The resulting data therefore represent noisy TEG responses rather than ideal model outputs.

$${\mathit{R}}_{\mathbf{T}\mathbf{E}\mathbf{G},\mathbf{n}\mathbf{o}\mathbf{i}\mathbf{s}\mathbf{y}}={\mathit{R}}_{\mathbf{T}\mathbf{E}\mathbf{G}}\left(\mathbf{1}+{\mathit{\eta }}_{\mathit{R}}\right), {\mathit{\eta }}_{\mathit{R}}\sim \mathcal{N}\left(\mathbf{0},{\mathit{\sigma }}_{\mathit{R}}^{\mathbf{2}}\right)$$

(14)

$${\mathit{C}}_{\mathbf{T}\mathbf{E}\mathbf{G},\mathbf{n}\mathbf{o}\mathbf{i}\mathbf{s}\mathbf{y}}={\mathit{C}}_{\mathbf{T}\mathbf{E}\mathbf{G}}\left(\mathbf{1}+{\mathit{\eta }}_{\mathit{C}}\right)+{\mathit{\eta }}_{\mathit{C},\mathbf{a}\mathbf{b}\mathbf{s}}, {\mathit{\eta }}_{\mathit{C}}\sim \mathcal{N}\left(\mathbf{0},{\mathit{\sigma }}_{\mathit{C}}^{\mathbf{2}}\right)$$

(15)

$${\mathit{\delta }}_{\mathit{R}}={\displaystyle \frac{{\mathit{R}}_{\mathbf{T}\mathbf{E}\mathbf{G},\mathbf{n}\mathbf{o}\mathbf{i}\mathbf{s}\mathbf{y}}-{\mathit{R}}_{\mathbf{T}\mathbf{E}\mathbf{G}}}{{\mathit{R}}_{\mathbf{T}\mathbf{E}\mathbf{G}}}}\times \mathbf{100}\%$$

(16)

$${\mathit{\delta }}_{\mathit{C}}={\displaystyle \frac{{\mathit{C}}_{\mathbf{T}\mathbf{E}\mathbf{G},\mathbf{n}\mathbf{o}\mathbf{i}\mathbf{s}\mathbf{y}}-{\mathit{C}}_{\mathbf{T}\mathbf{E}\mathbf{G}}}{{\mathit{C}}_{\mathbf{T}\mathbf{E}\mathbf{G}}}}\times \mathbf{100}\%$$

(17)

Figure 3 shows the simulated resistance response over the combined diffusion–interlayer state space. The color scale indicates ${\mathbf{l}\mathbf{o}\mathbf{g}}_{\mathbf{10}}\left({\mathit{R}}_{\mathbf{T}\mathbf{E}\mathbf{G}}\right)$; resistance decreases as diffusion front and interlayer formation increase. Figure 4 shows the corresponding capacitance response, where ${\mathit{C}}_{\mathbf{T}\mathbf{E}\mathbf{G}}$ is especially sensitive to interlayer formation and dielectric degradation. The quality of the noise-added database was evaluated by comparing noiseless and noisy electrical responses and by checking the relative deviation distributions. The distributions were centered near zero, indicating that the perturbation does not introduce systematic bias while preserving the physical trends of the ideal model.

4. XGBoost-Based Inverse Extraction

4.1. Feature Sets and Targets

The inverse problem is formulated as the estimation of hidden physical states from electrical observables and process information. Two feature sets were considered. The ET/geometry-only feature set includes ${\mathit{R}}_{\mathbf{T}\mathbf{E}\mathbf{G}}$, ${\mathit{C}}_{\mathbf{T}\mathbf{E}\mathbf{G}}$, process temperature, annealing time, ${\mathit{L}}_{\mathbf{g}\mathbf{a}\mathbf{p}}$, ${\mathit{t}}_{\mathbf{o}\mathbf{x}}$, $\mathit{W}$, $\mathit{A}$, and normalized capacitance-loss features. The calibrated full-library feature set additionally includes material-library parameters such as ${\mathit{D}}_{\mathbf{0}}$, ${\mathit{E}}_{\mathit{a}}$, ${\mathit{k}}_{\mathbf{I}\mathbf{L},\mathbf{0}}$, ${\mathit{E}}_{\mathbf{I}\mathbf{L}}$, ${\mathit{\sigma }}_{\mathit{m}}$, ${\mathit{\sigma }}_{\mathbf{I}\mathbf{L}}$, $\mathit{\alpha }$, and $\mathit{\gamma }$. In practical inline use, these library parameters are obtained from baseline calibration wafers and are not treated as unknown measurements. The dataset was divided into independent training and test subsets. Model performance was evaluated only on data not used for training. Additional generalization checks were defined using unseen-geometry, unseen-parameter, and noise-sensitivity splits to evaluate whether the model learned physically meaningful relationships rather than memorizing sampled cases. The primary regression targets are the 1% diffusion front ${\mathit{x}}_{\mathbf{0.01}}$ and the reaction-interlayer thickness ${\mathit{d}}_{\mathbf{I}\mathbf{L}}$. The classification target is the process-risk class: safe, warning, or fail.

4.2. XGBoost Model and Performance Metrics

XGBoost was selected because tree-boosting models can capture nonlinear feature interactions and are robust to heterogeneous feature scales [21,22]. At boosting step $\mathit{q}$, the prediction is updated as

$${\widehat{\mathit{y}}}^{\left(\mathit{q}\right)}={\widehat{\mathit{y}}}^{\left(\mathit{q}-\mathbf{1}\right)}+\mathit{\eta }{\mathit{f}}_{\mathit{q}}\left(\mathbf{x}\right),$$

(18)

where $\mathit{\eta }$ is the learning rate and ${\mathit{f}}_{\mathit{q}}\left(\mathbf{x}\right)$ is the regression tree added at step $\mathit{q}$. Regression performance was quantified using RMSE, ${\mathit{R}}^{\mathbf{2}}$, and MAPE:

$$\mathbf{M}\mathbf{A}\mathbf{P}\mathbf{E}={\displaystyle \frac{\mathbf{100}}{\mathit{n}}}{\displaystyle \sum _{\mathit{i}=\mathbf{1}}^{\mathit{n}}}\left|{\displaystyle \frac{{\mathit{y}}_{\mathit{i}}-{\widehat{\mathit{y}}}_{\mathit{i}}}{{\mathit{y}}_{\mathit{i}}}}\right|.$$

(19)

$$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}=\sqrt{{\displaystyle \frac{\mathbf{1}}{\mathit{N}}}{\displaystyle \sum _{\mathit{i}=\mathbf{1}}^{\mathit{N}}}{\left({\mathit{y}}_{\mathit{i}}-{\widehat{\mathit{y}}}_{\mathit{i}}\right)}^{\mathbf{2}}}$$

(20)

$${\mathit{R}}^{\mathbf{2}}=\mathbf{1}-{\displaystyle \frac{{\displaystyle {\sum }_{\mathit{i}=\mathbf{1}}^{\mathit{N}}}{\left({\mathit{y}}_{\mathit{i}}-{\widehat{\mathit{y}}}_{\mathit{i}}\right)}^{\mathbf{2}}}{{\displaystyle {\sum }_{\mathit{i}=\mathbf{1}}^{\mathit{N}}}{\left({\mathit{y}}_{\mathit{i}}-\overline{\mathit{y}}\right)}^{\mathbf{2}}}}$$

(21)

RMSE represents the root mean square error, R² quantifies the coefficient of determination, and MAPE evaluates the average relative prediction error. These metrics were used for diffusion-front and interlayer-thickness regression, while the risk classifier was evaluated using accuracy and the confusion matrix. The inverse extraction problem can also be written as a weighted least-squares objective:

$$\widehat{\mathit{\theta }}=\mathbf{a}\mathbf{r}\mathbf{g}\underset{\mathit{\theta }}{\mathbf{m}\mathbf{i}\mathbf{n}}\left[{\mathit{w}}_{\mathit{R}}{\left(\mathbf{l}\mathbf{o}\mathbf{g}{\displaystyle \frac{{\mathit{R}}_{\mathbf{m}\mathbf{o}\mathbf{d}\mathbf{e}\mathbf{l}}\left(\mathit{\theta }\right)}{{\mathit{R}}_{\mathbf{m}\mathbf{e}\mathbf{a}\mathbf{s}}}}\right)}^{\mathbf{2}}+{\mathit{w}}_{\mathit{C}}{\left({\displaystyle \frac{{\mathit{C}}_{\mathbf{m}\mathbf{o}\mathbf{d}\mathbf{e}\mathbf{l}}\left(\mathit{\theta }\right)-{\mathit{C}}_{\mathbf{m}\mathbf{e}\mathbf{a}\mathbf{s}}}{{\mathit{C}}_{\mathbf{m}\mathbf{e}\mathbf{a}\mathbf{s}}}}\right)}^{\mathbf{2}}\right],$$

(22)

where $\mathit{\theta }$ may include ${\mathit{x}}_{\mathbf{0.01}}$, ${\mathit{d}}_{\mathbf{I}\mathbf{L}}$, or a process-damage index. Equation (15) provides the analytical inverse-extraction baseline, whereas XGBoost learns a more flexible nonlinear mapping from a high-dimensional feature set.

5. Results and Discussion

5.1. Inverse Extraction of Diffusion Front

Figure 5 compares the true and predicted diffusion fronts using the calibrated full-library feature set. The predicted values closely follow the ideal line over several orders of magnitude.

Table 2. XGBoost inverse-extraction performance using the calibrated full-library feature set.

Target	Model	RMSE	${\mathit{R}}^{\mathbf{2}}$	MAPE or Accuracy
Diffusion front $x_{0.01}$	Log-target XGBoost	23,935 nm	0.959	14.17%
Interlayer thickness $d_{\mathrm{I}\mathrm{L}}$	Log-target XGBoost	2.30 nm	0.802	23.07%
Process-risk class	XGBoost classifier	–	–	93.86%

shows that the full-library XGBoost model achieves an RMSE of 23,935 nm, an ${\mathit{R}}^{\mathbf{2}}$ of 0.959, and a MAPE of 14.17% for ${\mathit{x}}_{\mathbf{0.01}}$. The relatively broad RMSE reflects the logarithmically wide target range, whereas the high ${\mathit{R}}^{\mathbf{2}}$ and moderate MAPE indicate that the model captures the dominant diffusion physics across the sampled material-library space. Because x_0.01 is defined along the semi-infinite diffusion coordinate, large values of x_0.01 can exceed the oxide thickness tox. Such cases are interpreted as full penetration of the vertical oxide region, and the effective vertical penetration is limited as x_vert,eff = min(x_0.01, tox). For lateral TEG interpretation, the effective lateral penetration is similarly limited as x_lat,eff = min(x_0.01, L_gap). Thus, x_0.01 should be interpreted as a diffusion-length descriptor, while device-level penetration is limited by the finite TEG geometry.

5.2. Inverse Extraction of Reaction-Interlayer Thickness

Figure 6 shows the predicted reaction-interlayer thickness. The interlayer extraction task is more difficult than diffusion-front extraction because capacitance response can partially saturate and because different combinations of ${\mathit{\epsilon }}_{\mathbf{I}\mathbf{L}}$, ${\mathit{d}}_{\mathbf{I}\mathbf{L}}$, and dielectric-degradation parameters can produce similar capacitance shifts. Nevertheless, the full-library model achieves an ${\mathit{R}}^{\mathbf{2}}$ of 0.802 and a MAPE of 23.07%, demonstrating that a calibrated material library enables quantitative estimation of ${\mathit{d}}_{\mathbf{I}\mathbf{L}}$.

5.3. Resistance and Capacitance Provide Complementary Information

The feature-importance results in Figure 7 and Figure 8 show that the two inverse tasks rely on different physical observables. For diffusion-front extraction, activation energy, capacitance-loss percentage, normalized resistance-per-gap, capacitance ratio, diffusion prefactor, and inverse temperature are dominant features (Figure 7). This is physically reasonable because ${\mathit{x}}_{\mathbf{0.01}}$ is governed by $\mathit{D}\left(\mathit{T}\right)\mathit{t}$ and by the lateral formation of metal-assisted conductive paths. Feature importance values in Figure 7 and Figure 8 were calculated using the normalized gain metric from the trained XGBoost trees. The gain metric quantifies the average reduction in the loss function obtained when a feature is used for tree splitting, and the values were normalized so that the total importance equals one.

For interlayer-thickness extraction, the reaction prefactor ${\mathit{k}}_{\mathbf{I}\mathbf{L},\mathbf{0}}$ and reaction activation energy ${\mathit{E}}_{\mathbf{I}\mathbf{L}}$ dominate, followed by inverse temperature and capacitance-loss features (Figure 8). This confirms that ${\mathit{d}}_{\mathbf{I}\mathbf{L}}$ is mainly governed by reaction kinetics and capacitance response. Therefore, ${\mathit{R}}_{\mathbf{T}\mathbf{E}\mathbf{G}}$ and ${\mathit{C}}_{\mathbf{T}\mathbf{E}\mathbf{G}}$ should be interpreted as complementary indicators rather than redundant measurements.

5.4. Effect of Calibrated Material Library

Figure 9 summarizes the inverse-extraction performance for the ET/geometry-only and full-library feature sets. The ET/geometry-only feature set can classify risk states with reasonable accuracy, but quantitative extraction of ${\mathit{x}}_{\mathbf{0.01}}$ and ${\mathit{d}}_{\mathbf{I}\mathbf{L}}$ is limited because many different material parameter combinations can produce similar electrical responses. The full-library feature set resolves this ambiguity by supplying calibrated stack-specific physical parameters. This result is important for practical deployment: inline electrical measurements are sufficient for risk monitoring after calibration, but accurate quantitative extraction requires a material library obtained from baseline physical analysis and reference TEG measurements.

5.5. Risk Classification

The proposed framework can also be used as a process-risk classifier. Figure 10 shows the confusion matrix for safe, warning, and fail states using the calibrated full-library classifier. The overall accuracy is 93.86%. Most errors occur between adjacent safe/warning or warning/fail classes, which is expected because these boundaries correspond to continuous degradation metrics rather than abrupt physical transitions. The numerical values of the confusion matrix are listed in

Table 3. Confusion matrix for process-risk classification.

True Class	Predicted Safe	Predicted Warning	Predicted Fail
Safe	2708	72	2
Warning	94	930	90
Fail	8	41	1055

.

5.6. Generalization and Sensitivity Analysis

To evaluate the generalization capability of the proposed framework, three validation scenarios were considered: random split, unseen-geometry split, and unseen-material-parameter split. The random split evaluates interpolation within the sampled process space, whereas the unseen-geometry and unseen-parameter tests evaluate transferability to layouts or material-library conditions not directly included in the training subset.

A perturbation-based sensitivity analysis was performed by varying major diffusion, reaction, geometry, conductivity, and dielectric parameters. This analysis was used to identify which assumptions most strongly influence R_TEG, C_TEG, x_0.01, and d_IL.

6. Implications for Inline Process Monitoring

The results support a practical monitoring strategy. First, calibration wafers are characterized using SIMS, TEM/EDS, XPS, C–V, and I–V measurements to establish the material-library parameters for a target metal/oxide stack. Second, resistance and capacitance TEGs are distributed across product or monitor wafers. Third, inline ET measurements are mapped to diffusion-front, interlayer-thickness, and risk-class outputs using the trained inverse model. Finally, when the extracted risk index exceeds the control limit, the annealing condition, barrier process, interface cleaning, or metal-stack design can be adjusted.

The proposed model should not be interpreted as a replacement for physical analysis. Instead, it provides a fast electrical screening and monitoring tool that can reduce the number of wafers requiring destructive or time-intensive physical characterization. The framework is particularly useful when diffusion and interfacial reactions occur simultaneously and cannot be separated using a single electrical observable. Transferability is therefore hierarchical. Within the same calibrated stack, the model can be transferred across TEG layouts by including geometry features such as Lgap, tox, W, and A. Across different metal/oxide stacks, diffusion and reaction parameters must be recalibrated before quantitative extraction is claimed.

7. Limitations

The present study is based on physics-informed model-generated datasets. Therefore, the reported numerical extraction accuracy should be interpreted as a simulation-based demonstration of feasibility, not as a calibrated value for a specific metal/oxide process. The reaction interlayer is modeled phenomenologically, and its effective conductivity and dielectric constant may depend on phase composition, oxygen deficiency, frequency, measurement voltage, and thermal history. Future work should calibrate the model using stack-specific SIMS, TEM/EDS, XPS, frequency-dependent C–V, leakage I–V, and long-term stress TEG data. In addition, finite-element simulations can be used to relax the one-dimensional semi-infinite approximation and account for corner effects, nonuniform fields, and realistic electrode geometry. To avoid overclaiming, the present accuracy values represent simulation-based inverse-extraction accuracy. Material-specific process accuracy must be established using experimental calibration data from SIMS, TEM/EDS, STEM/EELS, XPS, C-V, I-V, or long-term reliability TEG measurements.

8. Conclusions

This study proposed an integrated electrical TEG modeling framework for monitoring metal diffusion and reaction-interlayer formation in oxide dielectrics during thermal annealing. Metal diffusion was described by a Fick–Arrhenius erfc profile, while reaction-interlayer growth was represented using a parabolic Arrhenius reaction model. These hidden physical states were coupled to adjacent-electrode resistance and vertical capacitance TEG responses through integral electrical models. A physics-informed dataset with inline measurement uncertainty was then used to train XGBoost models for inverse extraction.

The calibrated full-library model achieved an ${\mathit{R}}^{\mathbf{2}}$ of 0.959 for 1% diffusion-front extraction, an ${\mathit{R}}^{\mathbf{2}}$ of 0.802 for reaction-interlayer-thickness extraction, and a risk-classification accuracy of 93.86%. Resistance TEGs were primarily sensitive to lateral metal-assisted conduction, while capacitance TEGs provided complementary sensitivity to vertical interlayer formation and dielectric degradation. These results indicate that combined resistance and capacitance TEG measurements can be used to electrically monitor multiple manufacturing risks, including metal diffusion, barrier degradation, and metal/oxide reaction. After stack-specific calibration, the proposed framework can support non-destructive inline estimation of diffusion extent, interlayer formation, and process-risk state.