Potential Loss Analysis (PLA) in Semiconductors

• Potential Loss Analysis (PLA) is a sequential framework that defines potential loss by estimating the best achievable downstream defect outcomes for wafer processing trajectories.
• It improves on Partial Trajectory Regression by replacing zero-baseline counterfactuals with optimal dynamic programming continuations, ensuring non-negative, monotonic attribution scores.
• PLA is validated on real semiconductor data, achieving a correlation of 0.87 in defect prediction and identifying specific process steps that significantly contribute to defect density.

Searching arXiv for the cited PLA paper and closely related sequential defect-attribution work. Potential Loss Analysis (PLA) is a framework for cross-process wafer defect root-cause analysis in semiconductor manufacturing. In the formulation introduced in “Cross-Process Defect Attribution using Potential Loss Analysis” (Idé et al., 27 Jul 2025), PLA attributes observed high wafer defect densities to upstream processes by comparing the best possible downstream outcomes available from partial processing trajectories. It is presented as a significant enhancement of partial trajectory regression (PTR): instead of explaining a defective wafer by zeroing out downstream processing information, PLA defines a value-like defect score over partial wafer states and interprets process responsibility as the incremental increase in that score along the route.

1. Conceptual definition and scope

PLA addresses a sequential attribution problem. A wafer traverses a long, heterogeneous process route; defect density is measured only after many upstream operations; and the analytical task is to determine which earlier process steps are responsible for a poor final outcome. The framework therefore treats wafer history as a variable-length trajectory rather than as a fixed tabular feature vector.

The central concept is the potential loss of a partial wafer state. In this usage, “loss” refers to defect-related process outcome, specifically log defect density in the reported experiments, not to financial loss. PLA asks a counterfactual question: from the current partial state, what is the best defect outcome that could still be achieved if the downstream continuation were chosen optimally? Process attribution is then defined by how much that best achievable outcome deteriorates after each process step.

This formulation changes the semantics of explanation. In PTR, attribution is based on the difference

$\alpha_k(\xi)=f(z_k)-f(z_{k-1}),$

which compares the current prefix to a representation in which downstream contributions are effectively absent. PLA replaces that baseline with an optimal feasible continuation. This makes the attribution problem inseparable from sequential prediction: the same learned function that estimates best achievable downstream defect outcome also supplies process-level defect scores.

2. Motivation and relation to partial trajectory regression

PLA is motivated by limitations of existing cross-process diagnosis methods in semiconductor manufacturing (Idé et al., 27 Jul 2025). The route of a wafer is long and heterogeneous, the number and type of process steps vary across wafers, and classical linear virtual-metrology models do not adequately represent complex nonlinear dependencies. PTR improves on static regression by modeling the wafer as a sequence of partial trajectories, but its attribution mechanism uses a counterfactual that the PLA paper characterizes as questionable.

The specific criticism is that PTR attribution,

$\alpha_k(\xi)=f(z_k)-f(z_{k-1}),$

implicitly compares the observed route to a continuation in which downstream embeddings are set to zero. If the learned process embeddings are not naturally centered at zero, the resulting attribution can be biased. PLA therefore introduces a different reference: compare the observed partial state not to a zeroed continuation but to the optimal downstream continuation from that state.

This reframing has two consequences. First, attribution becomes explicitly counterfactual in a dynamic-programming sense rather than in a feature-ablation sense. Second, the explanation is constrained to be non-negative at the step level, because each process can only worsen the best achievable downstream outcome once it has been applied. A plausible implication is that PLA is designed to trade some of the local flexibility of generic additive explanation schemes for route-level interpretability that is specific to manufacturing trajectories.

3. Trajectory representation and process embedding

PLA inherits its sequential state construction from the earlier PTR line of work (Idé et al., 27 Jul 2025). The training data are written as

$$D \triangleq \{ (y^{(n)}, \xi^{(n)}) \mid n=1,\ldots,N\},$$

where $y^{(n)}$ is the observed process outcome metric and $\xi^{(n)}$ is a variable-length process route. In the experiments, $y^{(n)}$ is log defect density. Each route consists of process embeddings $x_k \in \mathbb{R}^D$ and timestamps $t_k$.

Because semiconductor processes are heterogeneous, PLA first maps process records into a shared vector space. The construction begins with MES attributes such as equipment ID, recipe ID, tool type, photo layer ID, and route ID. These are concatenated into a synthetic token,

$$(\text{process token}) = eqp \oplus recipe \oplus tool\_type \oplus photo\_layer \oplus route \oplus \cdots$$

A vocabulary of such tokens is formed, and a kernel matrix $K\in\mathbb{R}^{V_d\times V_d}$ is computed using a substring-kernel variant. Spectral decomposition then yields process embeddings:

$\alpha_k(\xi)=f(z_k)-f(z_{k-1}),$0

The wafer state after the first $\alpha_k(\xi)=f(z_k)-f(z_{k-1}),$1 processes is updated sequentially. The lightweight linear PTR-style recurrence emphasized in the paper is

$\alpha_k(\xi)=f(z_k)-f(z_{k-1}),$2

with time-gap weighting

$\alpha_k(\xi)=f(z_k)-f(z_{k-1}),$3

This representation is important because wait times are part of the state evolution rather than auxiliary metadata. The later empirical interpretation of attribution jumps at specific tools with unusually long wait times follows directly from this design.

4. Dynamic-programming formulation

The distinctive mathematical step in PLA is the introduction of a value-like function over partial wafer states (Idé et al., 27 Jul 2025). For a chosen downstream continuation, PLA defines

$\alpha_k(\xi)=f(z_k)-f(z_{k-1}),$4

Here $\alpha_k(\xi)=f(z_k)-f(z_{k-1}),$5 denotes the wafer state at step $\alpha_k(\xi)=f(z_k)-f(z_{k-1}),$6, $\alpha_k(\xi)=f(z_k)-f(z_{k-1}),$7 is the instantaneous loss, and transitions are given by $\alpha_k(\xi)=f(z_k)-f(z_{k-1}),$8. Terminal states are absorbing, and the instantaneous loss is realized only at the terminal state:

$\alpha_k(\xi)=f(z_k)-f(z_{k-1}),$9

The optimal expected cumulative loss is then

$$D \triangleq \{ (y^{(n)}, \xi^{(n)}) \mid n=1,\ldots,N\},$$0

This is the minimum achievable loss if, from state $$D \triangleq \{ (y^{(n)}, \xi^{(n)}) \mid n=1,\ldots,N\},$$1, one can choose the best possible downstream route. The paper shows that $$D \triangleq \{ (y^{(n)}, \xi^{(n)}) \mid n=1,\ldots,N\},$$2 satisfies a Bellman equation:

$$D \triangleq \{ (y^{(n)}, \xi^{(n)}) \mid n=1,\ldots,N\},$$3

Under deterministic transitions, if $$D \triangleq \{ (y^{(n)}, \xi^{(n)}) \mid n=1,\ldots,N\},$$4 is the next state after action $$D \triangleq \{ (y^{(n)}, \xi^{(n)}) \mid n=1,\ldots,N\},$$5, the learned model is constrained through the Bellman inequality

$$D \triangleq \{ (y^{(n)}, \xi^{(n)}) \mid n=1,\ldots,N\},$$6

The learning problem is written as

$$D \triangleq \{ (y^{(n)}, \xi^{(n)}) \mid n=1,\ldots,N\},$$7

with $$D \triangleq \{ (y^{(n)}, \xi^{(n)}) \mid n=1,\ldots,N\},$$8 the empirical state distribution. This approximate-dynamic-programming perspective is the formal core of PLA: identifying the best possible downstream outcome is reduced to a Bellman equation, and attribution is derived from the resulting value function.

5. Learning objective, non-negative attribution, and interpretive semantics

The Bellman constraint is embedded in a penalty objective of time-difference type (Idé et al., 27 Jul 2025):

$$D \triangleq \{ (y^{(n)}, \xi^{(n)}) \mid n=1,\ldots,N\},$$9

After simplification over trajectories, the paper states that the final objective combines two effects. The first is terminal regression through

$y^{(n)}$0

which fits terminal defect density. The second is a penalty on successive differences, encouraging

$y^{(n)}$1

along the trajectory.

To make this monotone structure explicit, PLA parameterizes the stepwise increase in potential loss as

$y^{(n)}$2

where the ReLU output ensures non-negativity. The process-level attribution score is then

$y^{(n)}$3

This definition gives PLA its characteristic semantics. The learned value $y^{(n)}$4 is the estimated best possible cumulative defect loss from state $y^{(n)}$5 onward. The score $y^{(n)}$6 is the incremental potential loss introduced by process $y^{(n)}$7: it measures how much the best achievable future outcome worsens once that process is included. Unlike PTR, the score is always non-negative and is computed without setting downstream steps to zero.

The cumulative attribution curve is correspondingly monotone:

$y^{(n)}$8

The paper reports that this monotonicity improves interpretability. A plausible implication is that PLA is intended less as a generic attribution method than as a route-diagnostic instrument: upward jumps are to be read as irreversible degradations in the best still-attainable outcome.

6. Empirical evaluation, practical interpretation, and terminological disambiguation

PLA is evaluated on a real FEOL wafer-history dataset from NY CREATES Albany NanoTech (Idé et al., 27 Jul 2025). The reported dataset contains $y^{(n)}$9 wafers, each with hundreds of process steps, and defect density is measured at a process-limited yield evaluation point. The implementation uses Python and PyTorch 2.3.1 together with the kernel embedding method for process and route embeddings. The paper reports that t-SNE visualizations of process embeddings show clear clusters, that route embeddings form micro-clusters, and that the highest-defect routes are concentrated in certain regions.

In predictive terms, a linear PTR-style baseline achieved a cross-validated correlation of $\xi^{(n)}$0 between predicted and actual log defect density, whereas PLA with a two-hidden-layer neural network in $\xi^{(n)}$1 achieved a correlation coefficient of $\xi^{(n)}$2. In attribution terms, the reported cumulative curves for PLA are stable and monotone, whereas PTR can produce noisy ups and downs and can be uninformative for wafers near the population mean. Large upward jumps in PLA attribution corresponded to unusually long wait times at specific tools, suspected to be a main contributor to the defect type. These findings place PLA simultaneously in the spaces of virtual metrology, sequence modeling, and root-cause analysis.

A recurring source of confusion is the acronym itself. On arXiv, “PLA” is used for multiple unrelated concepts. In the semiconductor context considered here it denotes Potential Loss Analysis, but other uses remain common.

PLA usage	Meaning	Representative arXiv id
Potential Loss Analysis	Cross-process wafer defect attribution	(Idé et al., 27 Jul 2025)
Physical Layer Authentication	Wireless authentication via physical-channel features	(Amin et al., 2024, Forssell et al., 2018)
PLA as a function space	Almost-everywhere convergent analytic power series on the circle	(Kozma et al., 2012)
Potential Loss Analysis in counterparty risk	Loss-based framing centered on Potential Future Loss	(Kenyon et al., 2017)
PLA as material abbreviation	Poly(lactic acid) in materials and biofabrication	(Daskalova et al., 2021, Marouazi et al., 2022, Masoumi et al., 13 Jan 2025)

This ambiguity is substantive rather than merely bibliographic. In semiconductor manufacturing, PLA refers to a Bellman-based sequential attribution framework for defect density; in wireless security it usually abbreviates physical layer authentication; in analysis it refers to power-series representability; and in materials science it commonly denotes poly(lactic acid). Any technical reading of “PLA” therefore depends on domain context.

Within its own domain, however, the meaning is precise. PLA is a unified model for defect density prediction and defect score attribution that replaces zero-baseline counterfactuals with optimal downstream counterfactuals, expresses best achievable outcome through a Bellman equation, and interprets each process step through the incremental worsening of that optimum.