Forward and Inverse Virtual Metrology for Phototransistor Gain: A Hierarchical, Uncertainty-Aware Approach for Small Production Datasets

Forward and Inverse Virtual Metrology for Phototransistor Gain: A Hierarchical, Uncertainty-Aware Approach for Small Production Datasets Mahshid Amirabgir 1,2 , Lorenza Ferrario 1 , Paolo Conci Mahdieh Amirabgir 4 , Giancarlo Orengo 2*

3

,

arXiv:2608.11868v1 [cs.LG] 12 Aug 2026

1 Micro Nano Facility (MNF), Center for Sensors and Devices, Fondazione

Bruno Kessler (FBK), Via Sommarive 18, Povo (Trento), 38123, Italy. 2 Department of Electronic Engineering, University of Rome Tor Vergata, Via del Politecnico 1, Rome, 00133, Italy. 3 Microfab Solutions, Via Sommarive 18, Povo (Trento), 38123, Italy. 4 Department of Civil and Environmental Engineering (DICA), Politecnico di Milano, Piazza Leonardo da Vinci 32, Milan, 20133, Italy.

*Corresponding author(s). E-mail(s): orengo@ing.uniroma2.it; Contributing authors: mahshid.amirabgir@students.uniroma2.eu; ferrario@fbk.eu; paolo.conci@microfabsolutions.com; mahdieh.amirabgir@mail.polimi.it; Abstract The customization, optimization and stabilization of the process flow of a silicon bipolar phototransistor commits months of cleanroom time before a finished device can be measured, so a model that predicts device gain from process parameters before a run has value out of proportion to its accuracy. We study this problem on a real fabrication history, thirteen to fourteen process runs of a single device: a small-sample, hierarchically structured setting unlike the largecorpus regime of conventional virtual metrology. Decomposing the variance of device gain, we find that roughly half of it lies between process runs rather than within them, so recipe-only prediction is bounded by construction. Building on these findings we provide a forward gain predictor with a relative, uncertaintyaware signal, an inverse search that returns recipes for a target gain, and, as the foundation for all of it, a multi-level data-quality assessment tailored to the

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nested physical entities of fabrication (batch, wafer, die) with an explicit crosslevel linkage score. The normalized dataset and analysis code are released for full reproducibility. Keywords: Virtual metrology, Semiconductor manufacturing, Hierarchical modelling, Uncertainty quantification, Data quality, Small-sample learning

1 Introduction In the strategic panorama of semiconductors, supported by the EU Commission with investments and research frameworks such as the Chips Act and the recently announced Chips Act 2, a significant role is played by research institutions actively contributing to the “lab to fab” path. Moving from pure research and development to the market requires several steps, often corresponding to a gradual increase in Technology Readiness Level (TRL), where highly innovative laboratory devices are optimized to become part of application systems based on stabilized, repeatable fabrication process flows. In addition to improving the quality of a manufacturing process, specific customization may be necessary in order to adapt the process itself to a specific application. Two examples explain this concept: silicon photomultipliers (SiPM) and silicon bipolar phototransistors. SiPM can be adapted to different light wavelengths by working on the components known as “optical windows”, allowing SiPM for the ultraviolet (UV) or near-infrared–ultraviolet (NIR–UV) ranges, or, combined with germanium, for the infrared (IR) spectrum. Silicon bipolar phototransistors, the device this paper focuses on, can be addressed to industrial applications such as decoders, which require a specified collector–emitter breakdown voltage (BVCE ), or to applications requiring controlled, high response speed. Simulation, design and fabrication of a silicon phototransistor customized to a specific application (Dalla Betta et al., 1998) commits months of cleanroom time and irreversible material before a single finished device can be measured. A process engineer choosing an emitter implant dose or an oxidation step is therefore making a decision whose consequence, the current gain of the resulting device, will not be known until a full fabrication cycle has completed. This asymmetry, between the speed of a decision and the cost of learning whether it was right, is what makes predictive modelling of fabrication outcomes valuable: a model that forecasts gain from process parameters before a run, even imperfectly, can convert blind iteration into informed choices and remove clearly infeasible recipes before they consume fab time. Predicting product quality from process parameters rather than measuring it directly is the established domain of virtual metrology (Maitra, Su, & Shi, 2024). That literature, however, has developed largely in high-volume processes where tens of thousands of wafers are available, and where natural groupings in the data (chambers, tools, products) are commonly handled by training a separate model for each group, an approach that limits generalization across groups (Breidung et al., 2025).

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The setting of this paper is the opposite on both counts. The data come from a real but limited fabrication history, on the order of fourteen process runs of a single device. This situation is typical in pilot lines, where new processes are repeated until controlled repeatability is reached and are then transferred to production, or when small productions are required to cover niche markets. The problem is therefore not how to exploit a large corpus but how to model honestly when batch-level data are scarce. And rather than fit a separate model per run, we treat the grouping structure explicitly, which turns out to be not merely a modelling convenience but the route to the paper’s central scientific result. That result is a finding about where gain variation lives. Decomposing the variance of device gain shows that roughly half of it sits between process runs rather than within them: the run, not only the recipe, governs the outcome. A direct consequence is that any model using only recipe parameters, which are fixed within a run, is bounded from above in what it can predict, and we report that bound honestly rather than obscuring it. A second, subtler finding concerns the effect of a wafer’s position within its run. During fabrication, wafers are loaded into horizontal furnaces to activate the formation of a thin layer of silicon dioxide. Typically 25 wafers are loaded on a quartz carrier which enters the furnace. A wafer’s index within its run has an effect on gain that is substantial within individual runs, but its direction is not shared across runs, so a single pooled model averages it toward a small, unrepresentative value and understates how much position matters within any given run. We do not claim this index corresponds to a physical position in the thermal chamber, as the load order is not recorded (Section 4.3); we treat it as within-run sequence order throughout. Recovering this requires modelling each run’s behaviour explicitly, and we confirm it through several independent statistical views that agree. These findings rest on data whose trustworthiness must itself be established, and fabrication data poses a quality problem that flat, single-table assessment does not address: it is organized in strict physical strata, batch, wafer and die, and the failures that matter most are failures of the correspondence between levels. We therefore develop a multi-level data-quality assessment tailored to these physical entities, with an explicit cross-level linkage score, as the foundation the modelling stands on. The main contributions of this paper are three, and Fig. 1 summarizes how they fit together as an end-to-end methodology. First, a prediction and recipe-design capability for device gain: a forward model that predicts gain from process parameters with leave-one-batch-out honest accuracy and a relative uncertainty signal, an inverse search that returns the recipes capable of reaching a target gain, and a Gaussian-process uncertainty layer that tells an engineer how far to trust each prediction. We are explicit that the cross-run predictive accuracy is modest, a direct and expected consequence of the variance structure above, and frame the models as an honest screening aid rather than a replacement for fabrication. Second, a scientific finding about the process: that between-run variance dominates recipe effects (the intraclass correlation, in the sense of Nakagawa, Johnson, and Schielzeth (2017), is near one-half), and that the within-run position effect is real but sign-inconsistent across runs, a structure invisible to any single pooled model and recovered only by modelling the run explicitly.

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Third, a multi-level data-quality methodology that extends per-level quality scoring to the nested physical entities of fabrication data and introduces an explicit cross-level linkage score, with each level’s dimensions chosen, and each exclusion justified, rather than applied uniformly. A reproducible public release of the normalized wafer-level dataset is deposited on Zenodo under a CC-BY-4.0 licence (Amirabgir, Ferrario, Conci, Amirabgir, & Orengo, 2026) and the analysis code on GitHub (https://github.com/mahshid-amirabgir/ phototransistor-virtual-metrology), with a dual-mode pipeline that produces identical results on the released normalized data and the internal raw data, so every result here is reproducible from the public artifacts. Beyond these contributions, we operationalize the prediction capability through an engineer-facing natural-language assistant (Section 6). A local large language model interprets an engineer’s question and calls the validated forward, inverse and uncertainty models to answer it, but is constrained so that it never generates a gain value or a recipe itself; every number it returns originates from the models and is shown with its provenance. This is a deployment layer rather than a new modelling result: its purpose is to make the validated models usable by a non-specialist without exposing the modelling machinery, and to run entirely on a local machine so that no proprietary recipe data leaves the institute. The remainder of the paper is organized as follows. Section 2 reviews related work in phototransistor fabrication, data-driven process modelling, hierarchical models, Gaussian-process uncertainty and data quality. Section 3 describes the device, the process and the dataset. Section 4 presents the multi-level data-quality assessment. Section 5 presents the models and findings, the variance hierarchy and position effect, then the forward, inverse and uncertainty models. Section 6 presents the engineer-facing natural-language assistant. Section 7 discusses implications and generalization.

2 Related work 2.1 Phototransistor fabrication and prior FBK work The silicon bipolar phototransistor is a mature optoelectronic device: a bipolar junction transistor with an enlarged base–collector junction that collects light and amplifies the resulting photocurrent through transistor action, yielding responsivities well above those of comparable photodiodes. The device studied here descends directly from the optical-encoder phototransistor developed at the institute (then IRST, now Fondazione Bruno Kessler) and optimized by Dalla Betta et al. (1998), who used coupled process and device simulation to define a fabrication sequence achieving current gains of several hundred and responsivities above 0.4 A/W, and established that the emitter implant is the dominant lever on gain. That work is the foundation for the present device and process. The phototransistor remains an active research target, though recent work has moved largely toward novel materials and structures rather than the conventional implanted-silicon device. High-performance variants have been demonstrated using indium phosphide nanopillars grown on silicon for optical interconnects (Ko et 4

data artifact

method step

Raw fabrication data batch / wafer / die strata ∼9.3M die measurements

pivot finding

deployment

Two-arm corpus

Multi-level data quality

Arm A strict (260 w) Arm B sensitivity (285 w)

9 dimensions per level cross-level linkage score

Variance decomposition ICC ≈ 0.5 : between-run ≈ within-run

Forward model

Inverse search

mixed-effects + ridge recipe for target gain LOBO honest R2 feasibility map

GP uncertainty Matérn kernel trust tiers H/M/L

Engineer-facing deployment layer natural-language query over validated models

Reproducible release Zenodo (CC-BY-4.0) + GitHub · dual-mode pipeline

Fig. 1 End-to-end methodology. Raw multi-level fabrication data is assessed by the multi-level data-quality framework, including a cross-level linkage score, yielding a two-arm corpus. Variance decomposition (intraclass correlation approximately 0.5) is the pivot finding that motivates the modelling choices: a forward mixed-effects model, an inverse recipe search and a Gaussian-process uncertainty layer. The validated models are exposed through a deployment layer, and the normalized dataset and dual-mode pipeline are released for reproducibility. Boxes are keyed by outline style and fill as shown in the legend: data artifact, method step, pivot finding and deployment

al., 2016), thin InGaAs films on silicon waveguides reaching responsivities of order 106 A/W for photonic-circuit power monitoring (Ochiai et al., 2022), reconfigurable silicon-nanowire phototransistors operating in both n- and p-type polarities for broadband UV–visible–NIR optical sensing on CMOS-compatible photonic platforms (Sharma, Kumar, Gupta, & Das, 2026), and GeSn/Ge heterojunction bipolar phototransistors on silicon for optical-network applications (Neha, Kumar, & Kumar, 2026). These advances target device performance and integration. Our contribution is orthogonal to this device-physics line. The conventional silicon bipolar process is established and deployed, and its limiting practical problem is no longer device design but the cost and variability of customization and of fabrication. We therefore bring a data-driven modelling approach to the established process, predicting and inverting gain from process parameters across many real fabricated batches, rather than proposing a new device. 5

2.2 Data-driven process modelling in semiconductor fabrication Predicting product quality from process and equipment data, rather than measuring it directly, is the established paradigm of virtual metrology (VM) and advanced process control. A recent systematic review (Maitra et al., 2024) defines VM as the use of machine-learning conjecture models that take tool-state, recipe and sampled metrology variables as inputs to predict fabrication quality targets, and surveys its application across the major fab process steps; the motivation throughout is that physical metrology is costly and slow, so a reliable prediction reduces the measurement burden and tightens control. This is precisely the motivation of the present work, and our forward gain model is a VM model in this sense. Two features of the mainstream VM literature distinguish it from the setting here. First, it is typically a large-corpus enterprise: VM is well developed in high-volume, high-throughput processes such as chemical-mechanical planarization (CMP), plasma etch, and chemical vapor deposition (CVD), where large datasets and dense sensor traces are available to train flexible models (Dailey, Bertelson, Kim, & Djurdjanovic, 2024). Second, where the data contain natural groupings (chambers, products), it is common to train separate models per group, which a recent study on CMP nonuniformity prediction notes often limits cross-group generalizability (Breidung et al., 2025); the broader use of machine learning across CMP is surveyed comprehensively by Winkler et al. (2025). That CMP study also illustrates the arc this paper follows, repurposing a forward predictive model toward inverse recipe optimization. A common practical convention in this literature, which we also adopt, is to normalize features and withhold absolute recipe values for confidentiality; while many studies simply discard samples with missing values, an unpublished CVD virtual-metrology study instead benchmarks imputation as preprocessing and reports that it improves accuracy (Xie & Stearrett, preprint arXiv:2107.05071). Our contribution sits deliberately apart from large-corpus VM on two axes. The data are small where modelling decisions are made, on the order of fourteen fabrication runs, so the question is not how to exploit a large corpus but how to model honestly when the corpus is small. And rather than train separate models per run to handle the grouping, we model the grouping explicitly through hierarchical random effects, which both uses the data more efficiently and turns the between-group structure from a nuisance into the central scientific finding (Section 5).

2.3 Mixed-effects and hierarchical models for process data When data are organized in nested groups, a model that ignores the grouping treats correlated observations as independent and misstates both effects and their uncertainty. Mixed-effects (multilevel, hierarchical) models address this by adding random effects for the grouping factors alongside the fixed effects of interest, partitioning variance into between-group and within-group components rather than pooling them. This structure fits semiconductor fabrication data naturally, where batch, wafer and die form intrinsic nested levels. A recent review of regression and predictive modelling in wafer manufacturing notes that such multi-level data structures and correlated measurements are common in high-volume manufacturing, and that hierarchical and 6

mixed-model approaches are used for yield and wafer-map prediction (Chen & Chen, 2025). Directly relevant to the present setting, a hierarchical variability model for silicon-photonic circuits separates systematic from random process variation across the wafer, die and device levels (Xing, Dong, Khan, & Bogaerts, 2023), consistent with the nested lot–wafer–die grouping our analysis relies on, where lots contain at most twenty-five wafers and often fewer. The statistics that make such models interpretable are well established. The intraclass correlation coefficient (ICC), also called the variance partition coefficient, quantifies the share of total variance attributable to a grouping factor; for a randomintercept model it is the between-group variance divided by the total (Nakagawa et al., 2017). Marginal and conditional R2 , generalized to generalized linear mixed models by Nakagawa and Schielzeth (2013) and extended to random-slope models by Johnson (2014), separate the variance explained by the fixed effects alone (marginal) from that explained by fixed and random effects together (conditional); their difference reflects the variance attributable to the random effects. Nakagawa et al. (2017) revisit and expand these measures to further distributional families. When no fixed effects are fitted, the ICC equals the gap between conditional and marginal R2 , the same quantity viewed two ways. These tools are central to this paper. Section 5 begins by partitioning gain variance with the ICC, finds that roughly half is between-run, and uses exactly the marginalversus-conditional contrast to show that the within-run position effect is real but run-specific. Because that effect is a random slope, not merely a random intercept, the random-slope extensions above are what make the reported R2 values appropriate. What such models do not by themselves provide is a calibrated, point-by-point predictive uncertainty for a new recipe; for that we turn to Gaussian processes.

2.4 Gaussian-process uncertainty quantification in manufacturing A mixed-effects model partitions variance and yields confidence intervals on its parameters, but it does not by itself provide a point-by-point predictive uncertainty that widens as a query moves away from the observed data. Gaussian-process regression (GPR) does. A Gaussian process (GP) is a non-parametric Bayesian model that places a distribution over functions and returns, for every input, a predictive mean and variance (Rasmussen & Williams, 2006); the variance grows in regions far from the training data, so the model reports not only a prediction but where it lacks confidence. Two properties make GPR well suited to the present setting. It performs well with small datasets, since it conditions directly on the available observations rather than estimating many parameters, and its uncertainty is intrinsic to the probabilistic framework rather than bolted on; an unpublished tutorial emphasizes precisely that GPs accommodate the uncertainty arising from insufficient data and make explicit where the model is unsure (Li & Wang, preprint arXiv:2502.03090). Its behaviour is governed chiefly by the covariance kernel, with the squared-exponential and Matérn families the common choices, so the kernel is a modelling decision rather than a default. GPR has been used in semiconductor virtual metrology to attach predictive uncertainty to a quality target. It has been applied to predict material-removal rate in 7

chemical-mechanical planarization with quantified uncertainty (Cai et al., 2020) and, closest to the setting here, in a recent online Gaussian-process method that operates from minimal initial metrology data and uses the predictive uncertainty itself to decide where measurement is needed (Han et al., 2025). In both, the predictive variance is a deliberate output rather than a by-product; in the latter it is acted on directly, the logic this paper adopts in turning GP variance into engineer-facing trust tiers. In Section 5 we use a Gaussian process over the recipe features, with a Matérn kernel, as the uncertainty layer over the forward model: its predictive standard deviation is translated into engineer-facing trust tiers, and its divergence between interpolation and leave-one-run-out prediction is read directly as the signature of the between-run variance established in Section 5.1. How the trustworthiness of the underlying data is itself assessed, the precondition for any of this modelling, is the subject of the data-quality literature.

2.5 Data-quality methodologies for fabrication data Data quality has been formalized through dimension-based frameworks that decompose fitness for use into measurable axes. Wang and Strong (1996) derived data-quality dimensions empirically from data consumers and grouped them into intrinsic, contextual, representational and accessibility categories, establishing the dimensional vocabulary the field still reuses; recent surveys of quality metrics and tooling for machine learning organize dimensions along the same lines and note that few available tools are tailored to machine-learning tasks (Zhou, Tu, Sha, Ding, & Chen, 2024). Several recent frameworks address this fragmentation: a materials data quality governance framework arranges nine dimensions across a machine-learning lifecycle, pairing data-driven methods with domain knowledge to detect and correct issues (Liu, Yang, et al., 2025); an extension of the ISO/IEC 25012 standard consolidates the literature’s many terms into a hierarchical model and adds governance, usefulness, quantity and semantics dimensions (Miller et al., 2024); and others score up to ten dimensions and combine them through a fuzzy comprehensive evaluation model with weights derived by the analytic hierarchy process (AHP), demonstrated on tabular datasets such as KDDCup99 (Zhang et al., 2024). Two threads are directly relevant here. On small, high-dimensional data, where the feature-space dimension is large relative to the sample size, Liu et al. (2023) argue the feature-to-sample ratio should itself be a governance target, cautioning that generative augmentation may produce samples that are not physically valid for a given material system and advocating domain-knowledge constraints to keep generated data within reasonable value ranges; we adopt the ratio-based view in scoring quantity and, in our setting, forgo augmentation altogether. On anomaly detection, Liu, Ma, et al. (2025) encode domain knowledge as symbolic rules spanning individual descriptor values, inter-descriptor correlations and inter-sample similarity; we adapt the individual-descriptor-value rule for the accuracy dimension. Broader reviews map data quality against the machine-learning lifecycle (Priestley, O’Donnell, & Simperl, 2023) and catalog manufacturing-adapted metrics (Peixoto, Oliveira, Oliveira, & Ribeiro, 2025), while a recent production framework embeds real-time scoring and 8

drift detection in continuous-stream pipelines (Bayram, Ahmed, & Hallin, 2026), the latter premised on a data stream rather than the fixed historical corpus assessed here. A smaller line of work measures quality at more than one structural level, and it is the closest prior work. Multi-level tools compute dimensions at the schema level of an information system (Ehrlinger & Wöß, 2019), or across nested aggregation levels (attribute, concept, data source, system) with per-level scoring under user-adjustable per-dimension weights (Illescas Martinez, Ehrlinger, & Wöß, 2021), and a recent survey of measurement tools reports that cross-level aggregation is supported by almost no existing tool and never above the table level (Ehrlinger & Wöß, 2022). In every case, however, the levels are strata of a database or information system. No prior work targets the nested physical entities of semiconductor fabrication (batch, wafer, die), nor, to our knowledge, formalizes a quantitative cross-level linkage score between adjacent levels. Those two gaps are the contributions of the multi-level data-quality assessment developed in Section 4.

3 Data This section describes the device and its fabrication, the process parameters used as model inputs, how device gain is defined, and the hierarchical structure of the dataset on which the models of Section 5 are built.

3.1 Phototransistor fabrication overview The device studied here is an npn silicon bipolar phototransistor fabricated by a planar silicon process, in which the emitter and base are defined by ion implantation while the collector is provided by an n-type epitaxial layer on the substrate rather than by implantation. Figure 2 shows a schematic cross-section of the device, a fabricated wafer carrying the full die array, and the on-wafer test structures used to monitor the process. The device descends from the optical-encoder phototransistor developed at the institute (then IRST, now Fondazione Bruno Kessler) and optimized through coupled process and device simulation by Dalla Betta et al. (1998), who showed that increasing the emitter implant dose raises the current gain and used it to bring the gain to specification. The present work is the data-driven counterpart roughly three decades on: rather than defining the process by physics-based simulation and verifying it on fabricated devices, we model gain empirically across many fabricated wafers. Fabrication is carried out in a semiconductor cleanroom, where silicon wafers1 pass through a long sequence of material depositions, selective removals, thermal and doping steps, and metrology, interspersed with lithography steps that transfer the device layout onto the wafer. At the end of fabrication, every wafer is electrically characterized before it is diced into individual devices. Because gain is set principally at the emitter, this work focuses on the process steps that define and control it. The screen oxide is grown first: a layer (target thickness 100 nm) formed by silicon oxidation in an O2 -based furnace atmosphere, which controls the quantity and depth profile of the implanted phosphorus and limits crystal damage 1 A wafer is a thin, polished disc of crystalline silicon that serves as the substrate on which many devices are fabricated together, before being separated (“diced”) into individual chips.

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from the incident ions. Its thickness is monitored on dedicated test wafers and on some process wafers using an optical interferometer that maps 16 points per wafer; the mean and standard deviation are recorded and compared against control limits, with wet-etch thinning or further oxidation applied where the layer is out of range. Phosphorus is then ion-implanted to form the n-type emitter and electrically activated by a high-temperature drive-in (above 1000 ◦ C). Finally, the activated emitter sheet resistivity is measured by a four-point probe on dedicated test wafers.

metal contact n++ emitter contact

n emitter

silicon dioxide

p type base

n type epi layer

n type substrate, collector

metal contact

(c)

(a) (b)

Fig. 2 The npn silicon bipolar phototransistor. a Schematic cross-section showing emitter, base, n-type epitaxial layer, substrate acting as collector, contacts and screen oxide. b Photograph of a fabricated wafer carrying the full die array. c On-wafer test-strip structures used to monitor sheet resistivity, oxide thickness and contact resistivity

3.2 Process parameters and recipe features The models use three primary process parameters as recipe features, each corresponding to one of the emitter-control steps above: the screen-oxide thickness, the emitter implant dose and the emitter sheet resistivity after activation. A fourth feature, the within-batch resistivity spread, captures the uniformity of resistivity control. Table 1 gives the exact dataset column names, meanings and units. All four features are released in normalized form: each is expressed as a ratio to a per-arm reference (a robust within-arm central value), with the raw values replaced in place in the published dataset and the reference constants held internally under the institute’s data-sharing policy. This normalization does not affect any modelling result: each model either is invariant under per-feature rescaling or is fit within a pipeline that standardizes features on the training data before fitting, so a fixed linear rescaling of any feature leaves the fitted predictions unchanged. It is described further in Section 4.4.

3.3 Gain measurement Device current gain hF E is the response variable for all models. After fabrication, every phototransistor on a wafer is measured to extract its gain, among other parameters. The wafer-level response used in modelling, gain median, is the median gain over 10

Table 1 Process parameters and recipe features used as model inputs. All recipe-defining quantities are released in normalized form Feature (prose name)

Dataset column

Meaning

Units

Form release

Screen-oxide thickness Emitter implant dose Emitter sheet resistivity Resistivity spread

batch thickness mean nm emitter dose

Mean screen-oxide thickness (per batch) Phosphorus implant dose Mean activated-emitter sheet resistance Within-batch resistivity standard deviation

nm

Normalized

ions cm−2

Normalized

Ω/sq

Normalized

Ω/sq

Normalized

resistivity mean ohm sq resistivity std

in

the dies that pass quality screening, those flagged GOOD and surviving the cleaning steps described in Section 4, which makes the response robust to die-level outliers and incomplete measurement coverage. Gain values are subject to a physical-validity filter of [100, 5000]: readings outside this window are treated as non-physical and removed before aggregation. This is an admissibility bound on the measurement, not a yield criterion. The device-acceptance range desired for the encoder application is [600, 1500]; this is a design target rather than a data filter, and enters the analysis only as the target range for the inverse recipe search of Section 5.5. It excludes no wafer from the dataset. Empirically, the modelling corpus spans [644.0, 1444.5], so the fabricated material sits inside the acceptance window without that window being imposed. In addition to the phototransistors, each wafer carries 16 test strips: monitoring structures placed among the devices that allow parameters such as metal resistivity, oxide thickness and contact resistivity to be checked at end-of-line. These independently confirm the in-process thickness and resistivity controls described in Section 3.1.

3.4 Dataset structure The data are hierarchical, with three strata: fabrication batch, wafer and die. Thirteen gain-bearing phototransistor batches enter the corpus, each comprising 16 to 25 wafers except batch 3 (45; see below), with more than 20,000 phototransistors and 16 test strips per wafer, on the order of 9.3 million die-level gain measurements in total across the corpus. Table 2 gives the per-arm totals and Table 3 the per-run breakdown; Figure 3 shows how the three physical strata nest and how they join as dataset records. The modelling unit, however, is the run rather than the physical batch, because one batch must be subdivided. Batch 3 was fabricated as 45 wafers, but a data-driven analysis (detailed in Sections 4.3 and 5) identifies two sub-populations within it: reindexing wafer position at wafer 26 reverses the position–gain correlation, a signature independent of where the boundary is placed, and a corroborating discontinuity in median gain appears at the same location. We therefore treat batch 3 as two runs, labelled b3a (wafers 1–25) and b3b (wafers 26–45), recorded in the data through a process run id field that equals the batch number for every batch except batch 3. 11

Two analysis arms are then defined, differing by a single documented decision about one batch. Batch 2 is missing its emitter-resistivity measurement. The strict arm (Arm A) excludes batch 2 entirely, yielding 13 runs over 260 wafers. The inclusive arm (Arm B) instead imputes the missing resistivity, on the reasoning that batch 2 retains valid thickness, gain and other data, and need not be discarded over a single absent feature, yielding 14 runs over 285 wafers. The 25-wafer difference between the arms is exactly the batch-2 cohort. We carry both arms through the modelling and treat the comparison between them as a sensitivity analysis on the imputation decision; whether including the imputed batch changes any conclusion is reported in Section 5. The two arms share roughly 91% of their wafers, so their agreement is a check on the single imputation decision rather than independent validation; we do not treat arm agreement as confirmation of a result by independent data. Note that run counts of 13 and 14 correspond to 12 and 13 distinct batch identifiers respectively, since the single physical batch 3 contributes two runs in each arm.

Table 2 Dataset summary: per-arm totals. Arm A is the strict, primary arm; Arm B is the inclusive arm used for sensitivity analysis Arm

Batches

Process runs

Wafers

Die measurements

12 13

13 14

260 285

8,597,910 9,292,910

A, strict (primary) B, inclusive (sensitivity)

Table 3 Per-run breakdown of the corpus, showing the batch-3 split into runs 3a and 3b and the batch-2 asymmetry between the two arms. A check mark indicates that the run is included in that arm Process run

Physical batch

1 2 3a 3b 4 5 6 7 8 9 10 11 12 13

1 2 3 3 4 5 6 7 8 9 10 11 12 13

Wafers 23 25 25 20 20 16 20 20 20 20 20 20 16 20

12

Arm A

Arm B

✓

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

WAFER

BATCH

DIE magnify one die emitter metal n++ contact n emitter

loaded into

wafer position along boat

front

NPHTA23 02 W3 one substrate · >20,000 dies + 16 test strips (flanks)

one furnace run (“process run”) 16–25 wafers

SiO2 (screen oxide) p-type base n-type epi n substrate (collector) metal contact

one phototransistor gain (hF E ) measured

Data linkage: how the physical levels join in the dataset BATCH record process run id recipe features

L1

L1 wafer→batch: each wafer joins exactly one recipe

WAFER record batch id, wafer id gain median

L2

DIE record wafer id, die x, die y gain, GOOD flag

L2 wafer→die: each wafer has a non-empty die set L3 count plausibility: die count within robust band

Fig. 3 The batch/wafer/die hierarchy and its mapping to the dataset. Upper panel: the physical nesting, in which a furnace run (batch) loads 16–25 wafers in a quartz boat, each wafer carries more than 20,000 dies plus 16 test strips, and each die is one npn phototransistor whose gain is measured. Lower panel: how the three physical levels join as dataset records, with the cross-level linkage checks L1 (wafer-to-batch), L2 (wafer-to-die) and L3 (die-count plausibility) used by the multi-level dataquality assessment of Section 4

4 Data quality The models in Section 5 predict device gain from a small set of process parameters that vary at the nanoscale: implant doses, an oxide thickness with a 100 nm target, a sheet resistivity. At that scale small data defects translate into real predictive error, since a single mis-keyed wafer, an unflagged measurement artefact, or a hidden change in process conditions can move a fitted coefficient more than the physical effect it is meant to capture. A large empirical study of how data-quality dimensions affect machine-learning performance across nineteen algorithms finds that completeness, feature accuracy and target accuracy degrade supervised models the most, while consistency and uniqueness matter comparatively little; and, relevant to the small, high-feature-count corpus here, that smaller datasets and those with many features relative to their sample size showed greater sensitivity along several of these dimensions, with the largest dataset consistently the most robust (Mohammed et al., 2025). For a small-sample fabrication corpus, then, the trustworthiness of the data is not a hygiene step but a precondition for the modelling. This section establishes that the dataset the models rest on is trustworthy, how missing and anomalous values were handled, and how the release is structured so the assessment can be repeated as future batches are added. 13

4.1 Why hierarchical data-quality assessment is needed for fabrication data The foundational dimensional vocabulary of Wang and Strong (1996), recent surveys of data-quality dimensions and tooling for machine learning (Zhou et al., 2024), and hierarchical consolidations of that vocabulary such as the ISO/IEC 25012 extension of Miller et al. (2024) all specify their dimensions for a single, undifferentiated dataset, with no mechanism for scoring distinct nested levels or the correspondence between them. Where such frameworks are themselves called hierarchical, the hierarchy organizes dimension categories (intrinsic, contextual, representational, and so on) rather than levels of the data. Fabrication data does not fit that shape. It is organized in strict physical strata: a batch is processed under one recipe, a wafer is one substrate processed within that batch, and a die is one measured device, and the same quantity behaves differently at each stratum. A recipe parameter is one value per batch but broadcasts across twenty-odd wafers on joining, while gain exists only at the die level and is a summary everywhere above. A single-level rubric must therefore either pick one level and discard the rest, or flatten the levels and double-count the broadcast values. Fabrication data needs both of the pieces identified as missing in Section 2.5, because the failures that matter most are cross-level integrity failures (a wafer tracing to no recipe, an implausible die count, a batch identifier concealing two process runs) that are properties of the links between levels. The corpus is also small where modelling decisions are made (twelve to thirteen batches), so distribution-dependent dimensions are meaningless at the batch level yet well-defined at the die level, where measurements number in the millions. The multi-level data-quality assessment (MLDQ) developed here responds to both constraints: it extends per-level scoring to the nested physical entities of fabrication data, with a dimension set tailored to each level, and adds an explicit cross-level linkage score as a first-class dimension.

4.2 MLDQ methodology MLDQ scores nine data-quality dimensions, summarized in Table 4: Traceability (records carrying batch and wafer identifiers), Completeness (non-missing fraction over in-scope columns), Accuracy (values within physical hard bounds), Consistency (column-name and unit conformance), Redundancy (penalizing near-duplicate columns at |r| > 0.95), Balance (gain-distribution shape and good-die ratio), Uniqueness (non-duplicate records on the natural key), Usefulness (leave-one-batch-out2 predictive R2 of a wafer-level model), and Quantity (sample-to-feature ratio). The dimension set is rooted in the intrinsic and contextual data-quality tradition established by Wang and Strong (1996). Most dimensions are taken from the materials-data governance framework of Liu, Yang, et al. (2025) (Traceability, Completeness, Consistency, Redundancy and Balance). Usefulness and Quantity follow the ISO/IEC 25012 extension of Miller et al. (2024), with the sample-to-feature framing of Quantity also 2 Leave-one-batch-out cross-validation (LOBO-CV): a validation scheme in which one entire batch (process run) is withheld as the test set while the model is trained on all remaining batches, repeated so that every batch is held out exactly once; unlike a random train–test split it measures how well the model predicts a run it has never seen.

14

drawing on Liu et al. (2023); Accuracy follows the single-descriptor rule of Liu, Ma, et al. (2025) detailed below; and Uniqueness is a standard referential-integrity dimension. The operational definitions used here, among them the |r| > 0.95 redundancy threshold and the leave-one-batch-out R2 used for Usefulness, are specific to MLDQ rather than inherited from these sources. The Accuracy dimension builds on Rule 1 (the value-and-grammar rule) of the single-descriptor accuracy detection model (SDAD) of Liu, Ma, et al. (2025), which checks each descriptor value against a declared empirical range together with type and unit conformance. We adopt that symbolic range rule alone, encoding the bounds in a machine-readable rules file; we deliberately do not apply the statistical-detector ensemble that S-DAD intersects with the rule. At this sample size, distribution-based detectors flag physically valid extremes as anomalies (a false-positive mode the S-DAD authors document on small, high-diversity data), and, because a value can violate a hard physical bound while still falling within the bulk of an otherwise unremarkable distribution, the range rule on its own is the more reliable guard against physically impossible values here. Timeliness is omitted throughout, as the data are static. The contribution is not running this rubric three times but deciding which dimensions are meaningful at which granularity, and stating which are not and why. Each level scores a tailored subset (Table 4), and every exclusion carries a reason rather than a silent omission. At the batch level (recipe space, N = 12–13), Quantity applies as a deliberately low score: against roughly thirty features it scores around ten (10.2 in the baseline and Arm B, 9.4 in Arm A). We treat that low score as information rather than a defect to suppress: it is the quantitative signal that batch-level modelling is ill-supported. Reviews of small-sample materials machine learning observe that the feature-to-sample ratio, rather than feature count or sample count alone, is the quantity that governs this regime, reporting that the imbalance persists even when samples outnumber features roughly four-fold (Liu et al., 2023). Rather than respond with the synthetic augmentation those reviews catalog among the standard remedies, which for physics-grounded measurements would fabricate process conditions never observed in the fab, MLDQ surfaces the scarcity as a scored quantity and lets it justify two adaptations downstream: no synthetic augmentation, and analysis shifted to the denser wafer level under the hierarchical models of Section 5 rather than recipe-only fitting at the batch level. The batch-level exclusions each carry a reason: Redundancy because Pearson correlation over a dozen rows is meaningless, Balance because distribution shape is a die-level property, and Usefulness because the recipe-only model does not generalize across runs, its leave-one-batch-out R2 being at best marginal (itself a scientific finding, reported in Section 5.3); Uniqueness is inherited, batch identity being the trivially unique key. At the wafer level (N ≈ 285) all nine apply, with one coherence fix: recipe-origin columns are scored once at batch level and inherited at wafer level, so a single missing recipe is not counted as missing across the twenty wafers it broadcasts to (the W1 inheritance rule). At the die level (millions of rows) the perwafer scores aggregate to a distribution rather than a mean, surfacing outlier wafers. Die-level Accuracy uses the physical gain bound [100, 5000], an out-of-range value being an acquisition glitch, deliberately not the engineering acceptance range [600, 1500], which is a yield metric never folded into a quality score. Each level reports the

15

mean of its scored dimensions only, so the level means are comparable without inflation by trivial perfect scores. The exclusion-with-reason pattern is itself part of the contribution: it is the visible evidence the design is tailored to each stratum. The cross-level linkage dimension is the new scored quantity, measuring how cleanly the hierarchy joins through three sub-scores: L1 (wafer-to-batch), the fraction of wafers joining exactly one recipe; L2 (wafer-to-die), the fraction with a non-empty die set; and L3 (count plausibility), the fraction whose die count falls within a robust band around the corpus median. The combined cross-level linkage score is a weighted mean of the three sub-scores, Linkage = 0.40 L1 + 0.40 L2 + 0.20 L3 ,

(1)

where L1 is the wafer-to-batch referential integrity, L2 the wafer-to-die completeness, and L3 the die-count plausibility: the two hard structural-integrity checks carry equal high weight, while L3 is soft and carries less, since an unusual count may be valid. The L3 plausibility band around the median die count m is band = m ± max(3 MAD, 0.05 m) ,

(2)

where MAD is the median absolute deviation of the per-wafer die counts and the fivepercent-of-median floor prevents the band from collapsing when the MAD is zero, as occurs here (Section 4.5); using median and MAD rather than mean and standard deviation follows standard robust practice (Leys, Ley, Klein, Bernard, & Licata, 2013). All three sub-scores are reported alongside the combined number, and linkage is never folded into a level mean.

4.3 Handling missing and anomalous data: two-arm imputation and the batch-3 split Two data-handling decisions shape the modelling corpus, and rather than commit to one treatment of each, we carry the alternatives as explicit sensitivity questions, the principle being that with so few batches a single discretionary choice should never silently determine a result. The first is missing data. Batch 2’s sheet resistivity failed measurement, and batch 8 carries a structural gap in its resistivity spread from a single-spot protocol. The two arms defined in Section 3.4 differ exactly in the treatment of batch 2. Batch 8’s missing resistivity spread is filled in both arms, as it is the only resistivity-spread gap and does not depend on the batch-2 decision; the pre-imputation baseline scope leaves it unfilled. Imputed values are unweighted donor-pool means over the batches with a measured value, unweighted at batch level so larger batches cannot dominate, and every imputed cell carries a row-level provenance flag. After publication normalization, the imputed resistivity mean coincides with the corpus central value (a normalized ratio of one), and the imputed spread corresponds to normalized ratios of about 1.11 (Arm A) and 1.07 (Arm B). Because both imputed quantities are derived summary statistics rather than raw measurements, this gap-filling is categorically distinct from the synthetic data augmentation the framework declines. Crucially, we do not claim 16

Table 4 MLDQ dimensions, sources and per-level applicability. In the three right-hand columns, A denotes a dimension that applies and is scored, INH a dimension that is inherited or structural and is therefore reported but excluded from the level mean, and X a dimension excluded for the stated reason. Each level mean is taken over its A-dimensions only, and Timeliness is omitted globally because the data are static Dimension

Source

Scoring method

Traceability

Liu, Yang, et al. (2025) Liu, Yang, et al. (2025) Liu, Ma, et al. (2025), SDAD Rule 1 only Liu, Yang, et al. (2025) Liu, Yang, et al. (2025)

% records with batch id A and wafer id mean non-missing fraction, A in-scope columns 100 · (1 − violations/total) A against hard bounds

Completeness Accuracy

Consistency Redundancy

Balance Uniqueness

Usefulness

Quantity

Linkage

column-name and unit conformance 100 · (1 − redundant pairs/total), |r| > 0.95 Liu, Yang, et gain skewness and good-die al. (2025) ratio standard; cf. % unique on natural key Miller et al. (2024) Miller et al. LOBO-CV ridge R2 (2024)

Batch

A

Wafer

Die

A

INH

A

A

A

A (physical)

A

X (scored once) X (signal)

X (N =12– A 13, noise) X (die- A level) INH A

A A

X (nega- A X (no die tive R2 is a model) finding) Miller et al. min(1, N/(4·features))·100 A (deliber- A X (trivially (2024); Liu et ately low) 100) al. (2023) novel 0.40L1 + 0.40L2 + 0.20L3 cross-level

the imputation is harmless: the resistivity spread is a predictively relevant feature in the models of Section 5, so rather than assume its fill leaves conclusions untouched, we carry both arms and test that directly. As Section 4.5 shows, the quality scores themselves are by construction invariant to the imputation value, responding only to the more consequential decision of whether the batch is retained; whether any modelling conclusion shifts is reported in Section 5. The second is anomalous structure, and it was a discovery rather than a decision. Batch 3 contained 45 wafers, twice the corpus norm, and examining that anomaly revealed two independent lines of evidence for two sub-populations. The stronger, and the one our split rests on, is a reversal in the position effect: a Spearman correlation between wafer position and gain of +0.25 over all 45 wafers becomes −0.47 (p = 0.001) once position is re-indexed from the 26th wafer, with a coherent negative withinsegment slope in each half (b3b within-half ρ = −0.50, p = 0.025). This re-indexing signature does not depend on where the boundary is placed and is therefore unaffected by the boundary having been selected from a scan. A second, corroborating signature is a discontinuity in median gain of about 80 units at the same location. We report this gap with appropriate caution: because the W25/W26 boundary was selected as 17

the sharpest of 36 candidate cuts (W5 to W40), the naive two-sample p-value (Welch t = −3.35, p = 1.8 × 10−3 ) overstates the evidence. A permutation test over the maximized discontinuity statistic, which corrects for the selection, gives p = 0.16; a gap of this size arises in roughly one in six datasets of this length under the null of no change-point. The gap is therefore consistent with the split but, on its own, not significant once the search is accounted for. The split is justified by the selectionindependent re-indexing signature, with the gap as corroboration. We therefore treat batch 3 as two consecutive process runs under one identifier (25 wafers then 20), encoded as a process-run label that leaves every source measurement unaltered. All results reported below use this split. The most plausible physical reading is that a 45-wafer batch spans two loading cassettes, since a standard cassette holds about 25 wafers, and the discontinuity falls where that boundary would lie; direct cassette records are not available, so this is offered as the likeliest explanation rather than a confirmed cause, and the statistical evidence for the split stands independently of it. Primary results are reported with the split, and the corresponding results without it are given in Online Resource 1. Figure 4 shows the die-level gain map for the first of the two resulting runs. The discovery is itself an argument for hierarchical assessment: one batch identifier concealed two physically distinct populations, and comparing structure across levels exposed what a flat assessment of the wafer table would not. The full modelling consequence of the split is taken up in Section 5.

run 3a, W20 (low GAIN, mean=944.0)

175

175

150

150

125

125

100

75

50

50

25

25 0

20

40

60

80 die_x

100 120 140

1400 1200

100

75

0

1600

gain

200

die_y

die_y

run 3a, W2 (high GAIN, mean=1250.2) 200

0

1000 800

0

20

40

60

80 die_x

100 120 140

Fig. 4 Die-level gain map for process run 3a, showing the spatial structure of gain across the wafer array

4.4 Data sharing, normalization and reuse The dataset is released to support reuse while respecting the confidentiality of the underlying recipes, and is structured so the quality assessment can be re-run as new batches arrive. The public release centres on the wafer-level table, per-wafer gain summaries, the derived process-run identifier, and the recipe and substrate features used for modelling, accompanied by the quality reports, the modelling outputs and figures, the analysis scripts, and a machine-readable schema documenting every column, its role and its validation bounds. The raw die-level records are retained internally: they 18

number in the millions, are the most sensitive layer, and are not required to reproduce any result here, since all modelling operates on wafer-level summaries. Because the schema encodes each column’s validation bounds and the pipeline is runnable end to end, appending a future batch re-runs the same per-level scores and linkage checks without rework; the assessment is designed to be repeated, not performed once. Recipe-defining quantities are normalized before release: absolute implant doses, resistivities and oxide thicknesses are replaced by values expressed relative to internally held references, preserving the relative structure the models depend on (orderings, ratios and relationships among parameters) while withholding the absolute specification. Every public-pipeline analysis runs identically on the normalized data, and the modelling results are numerically identical between the internal and published representations to within floating-point tolerance: the mixed-effects and Gaussian-process models are invariant under per-feature rescaling, and the ridge and random-forest models are fit within a pipeline that standardizes features on the training fold before fitting, so per-feature rescaling leaves their predictions unchanged. Normalization therefore changes the numerical representation of the recipe features without changing any modelling result. The normalized wafer-level dataset is deposited on Zenodo under a CC-BY-4.0 licence with a citable digital object identifier (Amirabgir et al., 2026), and the analysis code, trained models and the same normalized dataset are released together on GitHub, so every result here is reproducible from the public artefacts without access to internal data; the raw die-level records and the normalization constants remain internal. The release is organized along the FAIR (findable, accessible, interoperable, reusable) principles (Wilkinson, Dumontier, Aalbersberg, et al., 2016): findable through a persistent identifier and deposition in a searchable, indexed repository; accessible through that identifier over a standard, open protocol; interoperable through a documented, machine-readable schema; and reusable under the stated licence, with recorded provenance and per-column validation bounds.

4.5 MLDQ applied: scoring matrix, robustness and what linkage surfaces Applying MLDQ to the cleaned corpus produces a per-level scoring matrix across the three strata plus the linkage dimension, evaluated under three scopes: a pre-imputation baseline (285 wafers, batch 3 scored as one batch-level row, batch 2 carrying missing resistivity), Arm A (260 wafers, batch 2 dropped) and Arm B (285 wafers, batch 2 imputed). Comparing the matrix across scopes is itself a test of the method: a scheme whose verdicts swung with an arguable imputation choice would have limited diagnostic value. The headline result is that MLDQ is robust to the imputation choice by construction. The baseline and Arm B scopes produce identical scores at every level mean and all nine wafer-level dimensions, to three decimal places, because of the W1 inheritance rule: batch 2’s resistivity, missing in the baseline and imputed in Arm B, is scored at batch level and never enters the wafer-level Completeness numerator, so the single imputed value cannot move a wafer-level score. The only divergence comes from Arm A’s stricter rule, which removes batch 2 entirely, and even that stays under half a point at every level (batch mean +0.06, wafer −0.30, die +0.31, combined linkage 19

Table 5 Three-scope MLDQ comparison. The baseline and Arm B scopes are identical by the inheritance rule; only Arm A, which drops batch 2, diverges, and by at most half a point at any level mean. The two right-hand columns give the difference of each arm from the baseline. The batch, wafer and die level means are computed over 13, 285 and 285 records respectively in the baseline and Arm B scopes, and over 12, 260 and 260 records in Arm A Level or dimension

Baseline

Arm A

Arm B

∆A

∆B

Batch level mean Wafer level mean Die level mean Linkage, combined L1 wafer→batch L2 wafer→die L3 count plausibility

72.86 95.37 92.02 97.89 100.00 100.00 89.47

72.92 95.07 92.33 97.69 100.00 100.00 88.46

72.86 95.37 92.02 97.89 100.00 100.00 89.47

+0.06 −0.30 +0.31 −0.20 +0.00 +0.00 −1.01

+0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00

−0.20; Table 5). The batch level shows why this is stability rather than mere insensitivity: dropping batch 2 lowers Quantity by roughly 0.8 points (smaller N ) but raises Accuracy by roughly the same amount, because batch 2’s unmeasured resistivity had violated the hard minimum and that violation leaves the scored set when the batch is dropped, so the two effects very nearly cancel. The framework thus surfaces the consequential drop-versus-keep decision while remaining invisible to the imputation value. The level means read against design intent, not as raw grades. The batch mean of 72.9 is low by design, dragged by the Quantity score that encodes small N as a measurable property; the wafer mean of about 95 is the canonical “is the dataset usable” figure; and the linkage score of about 98 is high because L1 and L2 are both perfect, with only L3 below 100. Figure 5 presents the full matrix as three heatmap panels, the baseline and Arm B panels being identical by the W1 rule. That L3 score is where linkage earns its place. L3 scores about 89 in both arms; the per-wafer die-count distribution is so concentrated at its modal value that the MAD is zero, the 3 MAD term in Eq. (2) vanishes, and the five-percent-of-median floor becomes the active band. Without that floor the band would collapse to a single value and score every routine variation as anomalous. What L3 surfaces is not one anomaly but three structurally distinct patterns, all in batches 5, 8 and 11: a deliberate doublemeasurement protocol at exactly twice the modal count; extra measurement-letter scans 14–23% above modal; and a small set of extreme overcounts several times modal. These have three different causes (a measurement artefact, an intentional protocol and a genuine overcount), yet a single-level wafer scorer would have absorbed all of them without comment, since each wafer’s own gain summary is internally complete and consistent. Only the cross-level check, comparing each die count against the corpus envelope, flags them and separates them into meaningful groups. Linkage earns its place not by lowering the score but by surfacing structure the per-level dimensions cannot see. Online Resource 1 plots each wafer’s die count against its batch with the band overlaid, showing the three tiers as distinct clusters above the dense in-band population. 20

Multi-level DQ scoring matrix: baseline | Arm A | Arm B (post-clean corpus) Post-clean baseline N(batch)=13, N(wafer)=285

Arm A (primary) N(batch)=12, N(wafer)=260

Arm B (sensitivity) N(batch)=13, N(wafer)=285

BATCH WAFER

DIE

LINKAGE

BATCH WAFER

DIE

LINKAGE

BATCH WAFER

DIE

LINKAGE

Traceability

100.0

100.0

INH

n/a

Traceability

100.0

100.0

INH

n/a

Traceability

100.0

100.0

INH

n/a

Completeness

74.9

100.0

100.0

n/a

Completeness

75.2

100.0

100.0

n/a

Completeness

74.9

100.0

100.0

n/a

Accuracy

99.2

100.0

99.9

n/a

Accuracy

100.0

100.0

99.9

n/a

Accuracy

99.2

100.0

99.9

n/a

Consistency

80.0

80.0

X

n/a

Consistency

80.0

80.0

X

n/a

Consistency

80.0

80.0

X

n/a

Redundancy

X

99.3

X

n/a

Redundancy

X

99.3

X

n/a

Redundancy

X

99.3

X

n/a

Balance

X

79.1

69.9

n/a

Balance

X

76.4

71.3

n/a

Balance

X

79.1

69.9

n/a

Uniqueness

INH

100.0

98.3

n/a

Uniqueness

INH

100.0

98.1

n/a

Uniqueness

INH

100.0

98.3

n/a

Usefulness

X

100.0

X

n/a

Usefulness

X

100.0

X

n/a

Usefulness

X

100.0

X

n/a

Quantity

10.2

100.0

X

n/a

Quantity

9.4

100.0

X

n/a

Quantity

10.2

100.0

X

n/a

97.9

97.7

LINKAGE (cross-level) (cross-level) (cross-level) L1=100 L2=100 L3=89

LINKAGE (cross-level) (cross-level) (cross-level) L1=100 L2=100 L3=88

97.9

LINKAGE (cross-level) (cross-level) (cross-level) L1=100 L2=100 L3=89

≡ Arm B at every dimension (W1 inheritance) SCORED low (<50)

SCORED mid

SCORED high (>=90)

INH (inherited; not in mean)

X (excluded with reason)

Fig. 5 MLDQ scoring matrix under three corpus scopes: baseline, Arm A and Arm B. Each panel gives the per-dimension scores at the batch, wafer and die levels together with the linkage sub-scores. The baseline and Arm B panels are identical by the inheritance rule, and the Arm A differences are at most half a point at every level mean

5 Models and findings This section presents the modelling in two parts. The first (Sections 5.1–5.3) is the scientific finding: device gain is governed as much by between-run variance as by the recipe, and the within-run position effect changes sign from run to run, a structure invisible to any single pooled model. The second (Sections 5.4–5.7) is the prediction capability: forward gain prediction, inverse recipe search and a relative uncertainty signal. Throughout, the strict arm (Arm A, which imputes nothing) is the primary reference and the inclusive arm (Arm B) is reported as a sensitivity check; the two agree on every qualitative conclusion, and where they differ numerically the difference is small and stated.

5.1 Variance hierarchy: why a recipe-only model cannot succeed Before fitting any predictive model we decompose the variance of wafer-level gain into between-run and within-run components, because the result determines which model classes can work at all. A one-way decomposition with the process run as the grouping factor gives an intraclass correlation coefficient (ICC) (Nakagawa et al., 2017), the share of total variance attributable to the run, ICC =

2 σrun , 2 + σ2 σrun resid

(3)

2 2 where σrun is the between-run variance and σresid the within-run (residual) variance. In our corpus this ICC is 0.537 in Arm A and 0.501 in Arm B; that is, roughly

21

half of the total variance in gain lies between runs rather than within them at the point estimate. Figure 6 shows the decomposition for both arms, and Figure 7 the residual within-run variability that recipe-only models cannot access. The between-run and residual variance components are comparable in magnitude (Arm A: between-run 7987, residual 6893; Arm B: 7336 and 7319 in the same standardized units), and an independent estimate from the null mixed-effects model agrees (ICC 0.516 and 0.481 respectively). The corresponding one-way analysis of variance rejects equality of run means in both arms (F = 26.4, p ≈ 10−38 in Arm A; F = 22.9, p ≈ 10−36 in Arm B). A cluster bootstrap over whole process runs (B = 2000) places the 95% interval at [0.03, 0.74] (Arm A) and [0.03, 0.73] (Arm B), with 65% and 63% of resamples respectively retaining an ICC above 0.40.3 The interval is wide, an expected consequence of estimating a between-group variance from only 13 to 14 groups, and we therefore do not claim a precise value; we report the interval rather than a bootstrap median, since the median of a non-negative variance ratio is an unstable summary when a share of resamples sits near the boundary. What the data support is that a substantial between-run component is present in both arms: the point estimates place roughly half the variance between runs, and the bootstrap intervals are consistent with them. Critically, the modelling consequence that follows does not depend on the precise figure. Even at the lower end of the plausible range, enough variance lies between runs that a recipe-only model, whose inputs are constant within a run, is bounded well below what within-run information could achieve, which is confirmed directly in Section 5.3: the recipe-only model fails to transfer across runs, its pooled leave-one-batch-out R2 only marginally positive. This single number has a strong modelling consequence. A model whose only inputs are recipe parameters, which are constant within a run, can by construction explain only the between-run share of variance, and only to the extent the recipe actually drives it; it has no access to the within-run half at all. With about half the variance sitting between runs and the recipe explaining only part of that, a recipe-only model is inadequate not because it is poorly fit but because the variance structure forbids it. The decomposition therefore motivates the modelling choice that follows: a mixedeffects formulation that represents the run explicitly as a random effect, rather than a single-level model that must pretend the runs are exchangeable.

5.2 Mixed-effects forward model and the within-run sign flip We model wafer-level gain yij for wafer j in run i as yij = x⊤ ij β + ui + vi pij + εij ,

(4)

3 Variance-component estimates are sensitive to the fitting method at this sample size. A single-start LBFGS REML fit converges to the zero-variance boundary in 41% (Arm A) and 49% (Arm B) of resamples, which depresses the lower confidence bound to zero; in simulation with the same group structure and a known between-run variance matching the fitted value, that fit returned exactly zero in 49% of replicates. The bootstrap reported here therefore uses an optimizer-free one-way ANOVA moment estimator, which admits the boundary and reaches it in about 0.5% of resamples. The interval is close to estimator-invariant: the lower bound lies near 0.03 and the upper near 0.74 under this estimator and under a multi-start REML alternative alike. The point estimates quoted above are REML fits, with which the moment estimator agrees closely (Arm A bootstrap median 0.54 against a REML point estimate of 0.54).

22

1.0

proportion of GAIN variance

0.8

Between-run vs within-run variance; ICC between-run within-run (residual) within 0.463

within 0.499

2 resid = 6,893.4

2 resid = 7,318.7

0.6

0.4

0.2

0.0

between ICC = 0.537 2 = 7,987.1 run

between ICC = 0.501 2 run = 7,335.7

Arm A

Arm B

Fig. 6 Variance decomposition of wafer-level gain into between-run and within-run (residual) components for Arm A and Arm B, with the intraclass correlation annotated on each panel. Roughly half the variance lies between runs in both arms

where xij are the standardized recipe features (emitter dose, screen-oxide thickness, emitter sheet resistivity and its within-batch spread) with fixed effects β, pij is the within-run position, ui ∼ N (0, σu2 ) is the per-run random intercept, vi ∼ N (0, σv2 ) the per-run random slope on position, and εij ∼ N (0, σε2 ) the residual. Comparing nested specifications by likelihood-ratio test selects the model with both a random intercept and a random slope on within-run position (per-run intercept and slope, unstructured covariance). The random slope is strongly warranted. A likelihood-ratio test of the random-slope model against the random-intercept-only model gives χ2 = 60.2 (Arm A) and 52.1 (Arm B). The two models differ by two parameters: the random-slope variance, which is constrained to be non-negative and so takes its null value on the boundary of the parameter space, and the intercept–slope covariance, which is unrestricted. The usual χ22 reference therefore does not apply; under these boundary conditions the asymptotic null distribution is a 50:50 mixture, 21 χ21 + 12 χ22 (Self & Liang, 1987, Case 6), which is stochastically smaller than χ22 . The naive χ22 tail probabilities (≈ 10−14 for Arm A and 10−12 for Arm B) are thus conservative upper bounds on the true p-value, which is smaller still. The random slope is not a marginal addition. Among the fixed effects, emitter dose and screen-oxide thickness carry the largest positive standardized coefficients (dose +46.3, thickness +37.8 in Arm A; +43.0 and +30.6 in Arm B, all with p < 0.005 in both arms), and the resistivity spread carries a significant negative coefficient (−32.7, p = 0.037 in Arm A; −37.6, p = 0.009 in Arm B). The mean resistivity is not individually significant in either arm. That dose and thickness dominate is consistent with the device physics: both act on the emitter, which sets the gain. 23

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The headline finding lives in the position term. For variance explained we use the marginal and conditional R2 of Nakagawa and Schielzeth (2013) in the random-slope form of Johnson (2014); because the response is Gaussian with an identity link, the distribution-specific variance is zero and the two reduce to 2 Rmarginal =

σf2 , 2 σf + σα2 + σε2

2 Rconditional =

σf2 + σα2 , 2 σf + σα2 + σε2

(5)

where σf2 is the variance of the fixed-effect predictions and σε2 the residual variance. Here σα2 is not a single random-intercept variance but the mean random-effect variance of the fitted intercept-and-slope structure, the covariate-averaged term Tr(ZΣZ⊤ )/n of Johnson (2014), which is what makes these R2 values well defined once the model carries a random slope on position. The marginal R2 of the selected model, the variance explained by the fixed effects alone, is 0.468 (Arm A) and 0.466 (Arm B), while the conditional R2 , fixed plus random effects, is 0.678 and 0.640. The gap between them carries the finding in two numbers: a substantial share of what the model captures, between a quarter and a third of the explained variance, is run-specific structure the fixed effects cannot see, which is exactly the reading Nakagawa and Schielzeth (2013) give to the marginal–conditional difference, namely the share of variability residing in the random effects. Adding the random slope moves the two measures in opposite directions: marginal R2 falls relative to the random-intercept model (0.509 → 0.468 in Arm A) while conditional R2 rises (0.564 → 0.678 in Arm A; both fits are shown in Fig. 8). The drop in marginal R2 is expected rather than anomalous: Johnson (2014) notes that the fixed-slope estimate, and with it σf2 , can differ between a random-intercept and a random-slope fit, with the difference small when the design is balanced but potentially 24

considerable when the random-intercept slope is biased toward an unrepresentative subset of runs, so marginal R2 need not be preserved when the slope is freed. In our data the direction of that change is the quantitative counterpart of a position effect that is real within runs but not shared in sign across them: pooling it into one fixed slope averages the per-run effects into a single coefficient that is statistically significant yet matches no individual run. We do not assume that sign inconsistency here; it is established directly by the per-run analysis below, and the opposing movement of the two R2 values is its summary in one pair of numbers. Figure 9 plots wafer-level gain against within-run position and motivates the perrun analysis that follows. We confirm the sign flip through two genuinely independent statistical views, with two further corroborating observations, all four collected in Fig. 10. First, the pooled fixed-effect position coefficient, the single slope a recipe-plusposition model would report, is significant and negative (−3.68 raw gain units per position step, p ≈ 10−5 , Arm A; −3.96, p ≈ 10−6 , Arm B; about −23 to −26 in standardized units, the fourth largest of the five fixed effects). Taken alone it suggests a uniform “later wafers have lower gain” effect, spanning roughly 90 gain units across a 25-wafer run. Second, the per-run rank correlations tell a different story. Fitting position against gain within each run separately, four runs reach Bonferroni-corrected significance in both arms, and they do not agree in sign: run 1 is strongly positive (Spearman ρ = +0.89) while runs 6, 9 and 11 are strongly negative (ρ = −0.83, −0.72, −0.81). A single pooled slope cannot represent one run rising and three falling; the pooled coefficient is a weighted average that mostly cancels. Third, the model’s own per-run random slopes (best linear unbiased predictors, BLUPs) span both signs and recover exactly this: run 1’s slope is about +80 gain units while run 6’s is about −80, in both arms. The per-run slopes from the mixed-effects model and the independent per-run Spearman correlations agree closely (Pearson r ≈ 0.88–0.90 between the two), so the random-slope structure is not an artefact of the model; it tracks a pattern visible in the raw within-run data. Fourth, the batch-3 cassette split (Section 4.3) is the same phenomenon caught at a batch boundary: re-indexing position at the boundary flips the within-segment correlation from a non-significant +0.25 to −0.47 (p = 0.001), with a coherent negative within-half slope (ρ = −0.50, p = 0.025). This re-indexing signature does not depend on the boundary being the maximal cut, so it is unaffected by the boundary having been selected from a scan. The two independent views (per-run rank correlations and mixed-model random slopes, agreeing at Pearson r of approximately 0.89), together with the corroborating pooled coefficient and the change-point re-indexing, are mutually consistent: the within-run position effect is real and substantial within individual runs, but its sign is not shared across runs, so any model that forces a single position coefficient will report a slope that fails to represent any individual run and understates how much position matters within the runs where the effect is strong. We do not interpret position as a furnace-slot coordinate; as noted in Section 4.3 the physical load position is not recorded, so position is treated as within-run sequence order.

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5.3 Wafer-level ridge cross-check: the marginal value of position To check that these conclusions do not depend on the mixed-effects machinery, we repeat the analysis with a transparent wafer-level ridge regression, validated by leave-one-batch-out cross-validation (LOBO-CV). The wafer-level ridge regression minimizes β̂ = arg min ∥y − Xβ∥22 + λ∥β∥22 , (6) β

with penalty λ selected by leave-one-batch-out cross-validation. This is the honest test for a process with strong between-run variance: a standard random split would leak run-level information between train and test and report optimistic numbers. Table 6 collects the results. 26

Per-run position-GAIN correlation -- arm A pooled FE beta (std) = -0.182

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Within each cassette GAIN trends downward with position (b3a rho=-0.38, b3b rho=-0.50). Aggregating across cassettes inverts the sign (whole-batch rho=+0.25) because b3b is offset upward in GAIN. Treating batch 3 as a single run masks the true position effect.

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Fig. 10 The within-run position effect is real but sign-inconsistent across runs. a Per-run position– gain rank correlations, with one run strongly positive and three strongly negative at Bonferronicorrected significance. b Random slopes (best linear unbiased predictors) from the mixed-effects model, which span both signs. c The batch-3 change-point, at which re-indexing flips the correlation sign. A single pooled coefficient averages these to a small pooled value that represents no single run

Two results matter. First, the recipe-only picture is confirmed from the other direction: cross-run generalization is poor. The recipe-only model’s pooled LOBO-CV R2 is only marginally positive (0.062 in Arm A, 0.108 in Arm B); adding within-run position raises the pooled value slightly but drives the mean-of-folds R2 strongly negative (−1.22 in Arm A, −1.06 in Arm B), the pooled and per-fold measures diverging because the model captures part of the between-run offset while having no skill relative to each run’s own mean. We report these values rather than hiding the negative one, because it is the expected consequence of the variance hierarchy in Section 5.1, not a modelling failure. A recipe-only model simply does not transfer across runs. Second, position carries genuine marginal value. Adding within-run position raises the pooled leave-one-batch-out R2 from 0.062 to 0.108 (Arm A) and from 0.108 to 0.155 (Arm B), a difference of +0.046 and +0.047. This improvement is positive in 27

both arms but small relative to its uncertainty: a cluster bootstrap gives a 95% interval of [−0.018, 0.197] (Arm A) whose lower bound sits just below zero, and a one-sided sign-rank test across the held-out-run folds is only marginally significant (p = 0.06 in Arm A, p = 0.12 in Arm B). We therefore characterize position as adding predictive information on average rather than as a decisively positive effect, consistent with the finding of Section 5.2 that its direction varies across runs: a feature whose sign is runspecific improves prediction unevenly and need not register as uniformly significant under leave-one-run-out evaluation. That the same feature is both small relative to its per-run spread as a pooled coefficient (Section 5.2) and yet positive on average as a marginal predictor is not a contradiction; it reflects that position matters within runs even though its direction varies, so it improves prediction without resolving into a clean global sign. The wafer-level ridge model does not store an interpretable coefficient vector in the released results; the mixed-effects fixed-effects table of Section 5.2 serves as the interpretable coefficient summary for the forward model, and the ridge model is used here specifically as a model-class-independent cross-check on the LOBO-CV generalization and the marginal value of position. Table 6 Wafer-level ridge regression: pooled leave-one-batch-out (LOBO-CV) R2 , the incremental ∆R2 from adding within-run position, and the random-forest comparison. The meanof-folds R2 is reported as obtained and is strongly negative, which is the expected consequence of the between-run variance hierarchy rather than a fitting failure. Ridge forward-model root-meansquare error on held-out runs is approximately 115 gain units in Arm A and 111 in Arm B. No in-sample training R2 is stored, so models are compared on cross-validated performance only Arm A

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5.4 Forward prediction: what the model delivers and which features drive it The first part of this section established the variance structure and the within-run position finding. This second part turns the same models toward prediction: what an engineer can forecast before fabrication, what recipe the models recommend for a target gain, and how much each prediction can be trusted. The deployable forward model is the mixed-effects model of Section 5.2, with the wafer-level ridge regression of Section 5.3 as a transparent cross-check, whose leaveone-batch-out performance is reported there. The value of the forward model is not high point accuracy, which the variance hierarchy of Section 5.1 caps, but an honest forecast that, combined with the uncertainty quantification of Section 5.6, tells an engineer both a predicted gain and how far to trust it. 28

Which features drive that prediction is answered first by the model we actually deploy: the mixed-effects fixed effects of Section 5.2. Two independent importance analyses corroborate the ordering reported there. Both are computed on a randomforest4 fit (Breiman, 2001) and explained in-sample, so we read them as descriptive of the associations present in the data rather than as attributions of the deployed model, a distinction that matters because the random forest itself does not generalize across runs (Section 5.7). With that caveat, they agree with the mixed-effects model and with each other: permutation importance ranks emitter dose and within-run position as the two strongest features in both arms (mean ∆R2 ≈ 0.47 and 0.46 in Arm A; in Arm B the order reverses, position 0.45 and dose 0.33), followed by the two resistivity features and thickness; and SHAP5 attributions (Lundberg & Lee, 2017), summarized in Fig. 11, return the same leading set: emitter dose, within-run position and mean resistivity. That a linear mixed-effects model, a permutation-importance analysis on a random forest, and a game-theoretic attribution method independently elevate the same few features is the useful result: the forward model’s dependence on dose, position and the emitter-defining parameters is not an artefact of one method. We note that within-run position ranks highly here as an association even though, as Section 5.2 showed, its pooled coefficient is significant but much smaller than the per-run slopes of about ±80 because its sign varies across runs; the importance methods detect that position carries information without resolving its direction.

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4 A random forest is an ensemble machine-learning model that, for regression, averages the predictions of many decision trees. Each tree is grown on a bootstrap resample of the data, and at every split a fresh random subset of the features is considered as split candidates; this per-node feature sampling decorrelates the trees, and averaging over them yields a flexible non-linear regressor. 5 SHAP (SHapley Additive exPlanations) is a feature-attribution framework that uses the game-theoretic Shapley value to assign each feature a contribution to a given prediction, quantifying how much each feature moves the prediction above or below its baseline. It can be estimated model-agnostically (Kernel SHAP) or with faster model-specific approximations such as Linear, Max and Deep SHAP.

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5.5 Inverse recipe search: recommending a recipe for a target gain The forward model is invertible: given a target gain, we search the recipe space for the settings predicted to achieve it. We do this directly, by evaluating the mixedeffects forward model across a grid of candidate recipes (3,750 points spanning the three normalized emitter-dose set-points, fifty resistivity values and the twenty-five within-run positions, with thickness and resistivity spread held at their central values) and retaining the recipes whose marginal prediction band intersects the target band. The prediction band is the fixed-effects prediction plus or minus one standard error, a roughly 68% band, not a 95% interval. The grid is evaluated as a vectorized batch prediction, so a full sweep across target gains is immediate; it does not require a separately trained surrogate model. We report the procedure and its admissible regions; the released results do not record an execution-time benchmark, so we make no quantitative speed claim. Run across the full target range with a ±5% band, the search recovers a coherent and physically sensible picture of which gains are achievable, shown in Fig. 12. The number of admissible recipes is unimodal in the target: zero at 600 and 700, rising sharply to a peak near a target of 1000 (about 1,970–1,985 admissible recipes), and falling back to zero by 1400–1500. In other words, the model reports that mid-range gains are broadly achievable across many recipes while the extremes of the deviceacceptance window are reachable by few recipes or none, exactly the kind of feasibility map a process engineer needs before committing a run. Position here is treated as within-run sequence order, as in Section 5.2; this is the working interpretation of the position index and is not independently confirmed in the process records (Section 4.3). Because the inverse search uses the populationaverage (fixed-effect) position coefficient rather than any single run’s random slope, a reported position reflects the average across runs and is not a guarantee for one future run: as Section 5.2 showed, the realized position effect varies in sign from run to run. We therefore report position as a recommendable setting only at the population level, while making explicit that its run-specific realization cannot be predicted in advance. Within an achievable target, the search exposes a three-tier identifiability structure in the emitter dose. At a representative target of 1100 (±5%), the low dose set-point (normalized 0.929) admits no recipe in either arm; the mid set-point (1.000) admits an intermediate fraction (662 of 1,250 grid recipes in Arm A, 682 in Arm B); and the high set-point (1.071) admits all 1,250. The three tiers have distinct operational meaning. The low dose is eliminative: it cannot reach the target, so observing this target rules it out. The mid dose is identifying: only some combinations of the other parameters reach the target, so the target meaningfully constrains the recipe. The high dose is saturated : it reaches the target across the whole sub-grid, so the target tells you little about the remaining parameters. This structure is itself useful guidance, since it tells an engineer that at the high dose the target is easy but under-determined, while at the mid dose hitting the target pins down the recipe more tightly. That the emitter implant dominates the recipe in this way is consistent with independent device-physics evidence: in polysilicon-emitter bipolar transistors, the emitter–base interface oxide, whose thickness varies with furnace boat position (top versus bottom of the tube), is 30

reported to be a controlling factor on current gain (Adukkadukkam, Chow, & Hiew, 2025), an emitter-side sensitivity our model recovers from data alone. Matching recipe counts per sweep target (stacked by emitter_dose)

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5.6 Gaussian-process uncertainty and engineer-facing trust tiers A forward prediction is only actionable if it carries an honest uncertainty. We fit a Gaussian-process regressor (Matérn kernel with ν = 2.5 plus a white-noise term) over the recipe features and use its predictive standard deviation as the basis for engineerfacing trust tiers. The GP excludes the resistivity spread as a feature because its fitted length scale ran to the boundary, so the GP treats it as effectively constant; the GP therefore operates on emitter dose, mean resistivity and thickness. The GP’s behaviour confirms and sharpens the variance-hierarchy story. Under five-fold cross-validation its R2 is 0.501 (Arm A) and 0.447 (Arm B), close to the mixed-effects model’s marginal R2 (0.468 and 0.466) and consistent with the ridge cross-check, so on interpolation within the observed runs the three model families agree on how much variance is explainable; Fig. 13 compares the Gaussian-process and mixed-effects predictions directly. Under leave-one-batch-out validation, however, the GP’s R2 falls to strongly negative (mean-of-folds −1.79 and −1.69). On the same mean-of-folds metric the wafer ridge is also strongly negative (−1.22 and −1.06), so both model families lose predictive skill when asked to generalize to an entirely unseen run; the GP is the worse of the two by about 0.6 in R2 . We compare the two on the mean-of-folds metric because a pooled R2 is recorded for the ridge but not for the GP; the ridge’s pooled value is positive (+0.108 and +0.155), and the contrast between that pooled figure and the strongly negative per-fold figures reflects the difference between the two aggregation schemes rather than a difference between the models. This is not a controlled model comparison in any case, since the GP operates on three recipe features and the ridge on five. This pattern, agreement under cross-validation and divergence under leave-one-batch-out, is not a defect: it is the 31

uncertainty model correctly reporting that predicting an entirely unseen run is far harder than interpolating within seen runs, the signature of the between-run variance established in Section 5.1. That divergence is what makes the GP useful for deployment rather than just for scoring. We translate its predictive standard deviation into three tiers against the typical in-sample spread: a prediction is HIGH trust when the GP standard deviation is within 1.2× the training-region spread (a data-dense region), MEDIUM up to 1.5× (the edge of the training data), and LOW beyond that (an extrapolation zone). The reference spread is about 86–88 gain units. Surfaced alongside a predicted gain, and mapped over the recipe space in Fig. 14, these tiers give an engineer a direct, legible signal: a HIGH-trust prediction sits where the model has seen comparable recipes, while a LOW-trust prediction flags a recipe far from anything fabricated, where the forecast should be treated as a rough indication and verified. Mixed-effects vs GP marginal predictions on recipe grid (Arm A; position_within_run held at median 11.0)

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Fig. 14 Gaussian-process predictive-uncertainty map over the recipe space for Arm A, with trust tiers defined against the training-region spread: HIGH at or below 1.2×, MEDIUM at or below 1.5×, and LOW above 1.5×. LOW-trust regions flag recipes far from anything fabricated

5.7 Model selection and honest limits Our model choices follow from the small-sample, hierarchical setting rather than from a search for the highest in-sample fit. At the wafer level we use ridge regression and a 32

random forest, evaluated by leave-one-batch-out cross-validation. The random forest generalizes worse than ridge on every cross-validated metric: its pooled LOBO-CV R2 is negative (−0.28 in Arm A, −0.22 in Arm B) where ridge is positive (+0.11, +0.16), and its mean-of-folds R2 is more negative than ridge’s (−2.38 versus −1.22 in Arm A; −2.15 versus −1.06 in Arm B). We therefore prefer ridge as the waferlevel cross-check: a flexible, high-capacity learner has no advantage when the binding constraint is between-run variance rather than functional complexity, and its extra capacity simply fits run-specific noise that does not transfer. The released results do not store an in-sample training R2 for these models, so we compare them on crossvalidated performance only, not on a train-versus-test gap; the conclusion rests on the random forest’s worse cross-validated generalization. The honest limits follow directly. The forward model’s cross-run R2 is low by construction, not by under-fitting (Section 5.1). The uncertainty tiers promise a relative signal, denser versus sparser regions of recipe space, not a calibrated coverage guarantee; a HIGH-trust label means that comparable recipes were seen, not that the true gain falls in a stated interval with a stated probability. The inverse search returns recipes the model predicts will hit a target, inheriting all of the forward model’s uncertainty; it is a screening aid, not a guarantee of a fabricated outcome. And every result rests on thirteen to fourteen runs of one device class; the structure we report (variance hierarchy, sign-flipping position effect, dose identifiability tiers) is what the data support, and its generalization to other devices is a hypothesis for future work, not a claim made here. And, considering the unclear impact and meaning of the position of wafers in the furnaces, the systematic recording of how wafers are loaded in each batch will be recommended.

6 Engineer-facing assistant The forward, inverse and uncertainty models of Section 5 are useful to a process engineer only if they can be queried without the statistical tooling that produced them. We therefore provide a natural-language assistant that lets an engineer ask, in plain language, either a forward question (what gain would this recipe give?) or an inverse one (what recipe reaches this gain?) and receive an answer drawn from the validated models. The assistant is a deployment layer over Section 5, not a new model: it adds accessibility, not predictive power.

6.1 Architecture The assistant couples a local, open-weights large language model (Llama 3.1, served through Ollama) to the validated models through a small set of typed tools. Two tools are exposed: a forward tool that takes a recipe and returns the mixed-effects prediction of Section 5.2 together with the Gaussian-process standard error and trust tier of Section 5.6, and an inverse tool that takes a target gain and returns the admissible recipes from the search of Section 5.5, grouped by the dose-identifiability tiers reported there. The language model’s role is deliberately narrow: it interprets the question and decides which tool to call and with what arguments, the tool then executes the validated model in ordinary code, and the language model phrases the returned 33

values back to the engineer. The language model itself performs no arithmetic and fits nothing. A query may be directed to either analysis arm (Section 3.4), so the imputation sensitivity carried through the paper remains visible at the point of use. The whole system runs on a single local machine, with no query or recipe transmitted to an external service, which is a precondition for use on proprietary fabrication data.

6.2 Provenance and the absence of fabricated numbers The central design property follows from this separation. Because every gain value, every recipe and every trust tier originates from a model call rather than from the language model’s own parameters, the assistant cannot fabricate a numerical result: asked a quantitative question it has no tool for, the language model is constrained to request the missing inputs rather than to invent an answer. Each response carries a provenance record showing which tool was called, the arguments it received and the values it returned (Fig. 15b), so the engineer can confirm that the numbers were produced by the validated models. This makes the honesty the rest of the paper argues for a structural property of the interface rather than a matter of trust in the language model: the natural-language layer may phrase an answer imperfectly, but it cannot manufacture a gain or a recipe. Because the numbers come from the deterministic models, the substantive content of an answer is reproducible even though the language model’s wording is not.

6.3 Demonstration and scope Figure 15 shows the assistant answering an inverse query. For a target gain of 1000 it returns complete recipes spanning the feasible dose tiers, each annotated with predicted gain, standard error and trust tier, reproducing in interactive form the same dose-identifiability structure and trust-tier semantics established in Sections 5.5 and 5.6. The assistant is a demonstrated proof of concept: its purpose is to show that the validated models can be made usable by a non-specialist behind a provenance guarantee, not to claim a contribution in language modelling. Its conversational robustness has not been evaluated in a controlled study, and it inherits every limitation of the models it wraps (Section 5.7).

7 Discussion 7.1 Manufacturing implications The practical motivation for this work is the cost asymmetry involved in customizing recipes and process flows: a phototransistor run commits months of cleanroom time and irreversible material before any device can be measured, so any tool that screens recipes before a run has leverage out of proportion to its accuracy. The forward and inverse models developed here are that tool. Given a candidate recipe they return a predicted gain with a relative trust tier, and given a target gain they return the set of recipes the model judges capable of reaching it, both in the time of a grid evaluation rather than a fabrication cycle. We are deliberate about what this buys. The crossrun predictive accuracy is modest by construction, a pooled leave-one-batch-out R2 of 34

(a)

(b)

Fig. 15 The engineer-facing natural-language assistant. a An inverse query, in which the engineer asks in plain language for a recipe reaching a target gain of 1000; a local language model (Llama 3.1, served through Ollama) interprets the request, invokes the inverse search of Section 5.5, and reports complete recipes with predicted gain, standard error and trust tier. b The accompanying provenance panel, in which every value is produced by the validated models rather than the language model, listing the tool called, the arguments it received and the recipes returned across the feasible dose tiers. Recipe inputs are shown as normalized ratios following the released-dataset convention, and gain is in measured units

roughly 0.11–0.16, and, as Section 5.1 establishes, this is the direct consequence of the variance hierarchy rather than a shortfall to be engineered away. The contribution is therefore an honest screening capability: the models flag, through the inverse search’s admissible-region map and the Gaussian-process trust tiers, which targets are broadly reachable and which predictions sit far from any fabricated recipe. Used this way, as a funnel that removes clearly infeasible recipes and ranks the rest, even a bounded model converts months of blind iteration into a few informed runs. 35

7.2 What the finding means, and what generalizes The scientific finding is that gain in this process is governed as much by between-run structure as by the recipe, and that the within-run position effect is real but signinconsistent across runs. The variance hierarchy, a between-run component estimated near half the total though with wide uncertainty at this sample size, is the load-bearing result: it explains why recipe-only models fail, why a mixed-effects formulation is mandatory, and why the honest predictive ceiling is where it is. The position sign flip, supported by two independent statistical views and corroborated again at the batch3 cassette boundary, shows that a single pooled slope can actively mislead, averaging a real effect to a small, unrepresentative value, and that the structure only becomes visible when the run is modelled explicitly. What transfers is the methodology: the discipline of decomposing variance before selecting a model class, of treating a hierarchical dataset with per-level quality dimensions and an explicit cross-level linkage score, and of replacing silent data omissions with exclusion-with-reason. Any fabrication process with batch, wafer and die strata and small batch-level sample size faces the same structural problems this approach was built for; the multi-level data-quality framework and the variance-decomposition-first modelling sequence are the parts we expect to carry to other industrial hierarchies, even where the physics and the numbers differ entirely.

8 Conclusion and future work We have studied the prediction and inversion of phototransistor gain from fabrication process parameters on a small production corpus, and found that the dominant structure in the data is between-run rather than recipe-driven: roughly half of gain variance lies between process runs, and the within-run position effect changes sign across runs so that a pooled model averages it away. These findings explain why a recipe-only predictor is bounded, and they are recovered only by modelling the run explicitly through hierarchical random effects. On this basis we provide a forward gain predictor with a relative, Gaussian-process-based uncertainty signal and an inverse recipe search, as a screening aid whose accuracy is the expected consequence of the variance structure. Underpinning the modelling, we develop a multi-level data-quality assessment tailored to the nested physical entities of fabrication, with an explicit cross-level linkage score, and we release a normalized dataset and runnable pipeline for full reproducibility. The transferable contribution is methodological: decompose variance before choosing a model, score data quality across physical levels with explicit cross-level linkage, and surface scarcity and exclusions rather than hiding them. New datasets on bipolar phototransistors are expected in the next two years, supporting further validation of the proposed methodology. Furthermore, given the multi-level structure of the device, another exploitation of this work will be in considering different devices and the characterization parameters on which they should be optimized. Beyond the present corpus of data, the validated models are exposed through a local natural-language assistant that makes them usable without machine-learning expertise while guaranteeing that every number it reports is model-derived, and we 36

expect the multi-level, cross-level-linked data-quality approach to be the component most readily carried to other industrial hierarchies with the same nested physical structure and small batch-level samples. Supplementary information. Online Resource 1 (ESM 1.pdf) contains two parts. The first plots the per-wafer die count against batch with the L3 plausibility band of Eq. (2) overlaid, showing the three count-anomaly tiers discussed in Section 4.5 as distinct clusters above the dense in-band population. The second reports the headline results of Section 5 recomputed with physical batch 3 treated as a single process run rather than split, the sensitivity analysis referred to in Section 4.3. Acknowledgements. The phototransistor process data and measurements used in this paper refer to devices fabricated in the Fondazione Bruno Kessler cleanrooms as part of a service provided to OPTOI SRL. The authors thank Alfredo Maglione, OPTOI CEO, for authorizing the use of these data. Use of generative AI and AI-assisted technologies. During the preparation of this work the authors used Claude (Anthropic), ChatGPT (OpenAI) and Grok (xAI) for English language editing and revision of the manuscript, and to assist with analysis code development. The tools were not used to generate scientific content, to produce or analyse data, or to formulate any result, finding or interpretation reported here. All output was reviewed and verified by the authors, who take full responsibility for the content of the publication.

Statements and Declarations Funding This work was carried out using internal institutional resources. No external funding, public or private, was received for this study, and no project or grant supported it. The fabrication service provided to OPTOI SRL, noted below, was a commercial service and did not constitute research funding or support for this study.

Competing interests The devices analysed in this work were fabricated at Fondazione Bruno Kessler as part of a service provided to OPTOI SRL, which authorized the use of the data for this publication. Paolo Conci is the founder of Microfab Solutions; he contributed to this work as an unpaid research collaborator, and no payment or other consideration was made or requested between Microfab Solutions and Fondazione Bruno Kessler in connection with this study. The authors declare no other competing interests.

Ethics approval Not applicable. This study involved no human participants and no animals.

Consent to participate and consent for publication Not applicable. 37

Data availability The normalized wafer-level dataset supporting the findings of this study is openly available in Zenodo under a CC-BY-4.0 licence at https://doi.org/10.5281/zenodo .21379925 (Amirabgir et al., 2026). The raw die-level records and the per-arm normalization constants are not publicly available because they are proprietary process data held under Fondazione Bruno Kessler’s data-sharing policy; they are not required to reproduce any result reported here, since all modelling operates on wafer-level summaries and the normalization leaves every model’s predictions unchanged (the penalized and tree-based models standardize features within the training pipeline).

Code availability The analysis code and trained models are openly available on GitHub at https://github .com/mahshid-amirabgir/phototransistor-virtual-metrology (release v1.1.0), together with the normalized dataset and a dual-mode pipeline that reproduces every figure and number reported here.

Materials availability Not applicable.

Author contributions Conceptualization: Mahshid Amirabgir, Lorenza Ferrario, Paolo Conci, Giancarlo Orengo; Methodology: Mahshid Amirabgir, Lorenza Ferrario, Giancarlo Orengo; Software: Mahshid Amirabgir, Mahdieh Amirabgir; Validation: Mahshid Amirabgir, Lorenza Ferrario, Mahdieh Amirabgir, Paolo Conci, Giancarlo Orengo; Formal analysis: Mahshid Amirabgir, Mahdieh Amirabgir; Investigation: Lorenza Ferrario; Resources: Mahshid Amirabgir, Lorenza Ferrario; Data curation: Mahshid Amirabgir, Lorenza Ferrario; Visualization: Mahshid Amirabgir, Mahdieh Amirabgir; Writing – original draft: Mahshid Amirabgir, Lorenza Ferrario, Mahdieh Amirabgir; Writing – review and editing: Mahshid Amirabgir, Lorenza Ferrario, Mahdieh Amirabgir, Paolo Conci, Giancarlo Orengo; Supervision: Lorenza Ferrario, Paolo Conci, Giancarlo Orengo; Project administration: Lorenza Ferrario, Giancarlo Orengo. All authors read and approved the final manuscript.

ORCID iDs Mahshid Amirabgir 0009-0006-4849-2624; Lorenza Ferrario 0000-0003-2175-4306; Paolo Conci 0000-0003-4011-7695; Mahdieh Amirabgir 0009-0005-0615-1058; Giancarlo Orengo 0000-0003-1044-2999.

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