Exploring effective charge in electromigration using machine learning

Benjamin Afflerbach; Dane Morgan; Ryan Jacobs; Shih-kang Lin; Yu-Chen Liu

arxiv: 1907.01480 · v1 · pith:WVUARQSOnew · submitted 2019-07-02 · ❄️ cond-mat.mtrl-sci · physics.comp-ph

Exploring effective charge in electromigration using machine learning

Yu-Chen Liu , Benjamin Afflerbach , Ryan Jacobs , Shih-kang Lin , Dane Morgan This is my paper

Pith reviewed 2026-05-25 11:02 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci physics.comp-ph

keywords electromigrationeffective chargemachine learningalloysmaterials predictioninterconnection reliability

0 comments

The pith

Machine learning models effective charge z* in electromigration as a linear function of elemental properties

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper employs machine learning to model the effective charge z* that characterizes electromigration effects determining interconnection reliability in electronics. It expresses z* as a linear function of physically meaningful elemental properties and validates the fit through 5-fold leave-out-alloy-group cross-validation. The resulting model reaches an average RMSE divided by standard deviation of 0.37 and R squared of 0.86. It also tests extrapolation to entirely new alloys and applies the model to predict z* for relevant host-impurity pairs. A sympathetic reader would care because such predictions could reduce reliance on direct measurements when designing more reliable electronic materials.

Core claim

The effective charge z* can be modeled as a linear function of elemental properties using machine learning approaches, with 5-fold leave-out-alloy-group cross-validation yielding RMSE/σ of 0.37 ± 0.01 and R² of 0.86, plus limited but potentially useful predictive ability when extrapolating to z* of totally new alloys.

What carries the argument

Linear regression model relating z* to elemental properties, fitted via machine learning

Load-bearing premise

The effective charge for new alloys follows the same linear relationships with elemental properties observed in the training set of measured alloys

What would settle it

A measured z* value for a new alloy that lies well outside the model's predicted range would disprove the extrapolation claim

Figures

Figures reproduced from arXiv: 1907.01480 by Benjamin Afflerbach, Dane Morgan, Ryan Jacobs, Shih-kang Lin, Yu-Chen Liu.

**Figure 3.** Figure 3: The RMSE/σ and R-squared (R 2 ) value for the prediction with and without randomized test. One concern is that given the small data set with limited sampling of the alloy space, the use of SFS and/or adjustments made during model construction (e.g., choice of LR vs. other methods) might have created apparently physical correlations that are not real. To test if this occurred, we performed what we call a ra… view at source ↗

**Figure 4.** Figure 4: The contribution of each descriptor to the effective charge value [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗

**Figure 5.** Figure 5: The exploration of effective charge of impurities in Al, Co, Cu, Ag, Sn, or Au host. It is worth noting that Co is proposed to be a new alternative to Cu in the backend-of-line (BEOL) interconnection material for the latest technology node, which is essentially the physical size of a transistor made in a particular technology, so obtaining an improved understanding of EM in this host is particularly impor… view at source ↗

read the original abstract

The effective charge of an element is a parameter characterizing the electromgration effect, which can determine the reliability of interconnection in electronic technologies. In this work, machine learning approaches were employed to model the effective charge (z*) as a linear function of physically meaningful elemental properties. Average 5-fold (leave-out-alloy-group) cross-validation yielded root-mean-square-error divided by whole data set standard deviation (RMSE/$\sigma$) values of 0.37 $\pm$ 0.01 (0.22 $\pm$ 0.18), respectively, and $R^2$ values of 0.86. Extrapolation to z* of totally new alloys showed limited but potentially useful predictive ability. The model was used in predicting z* for technologically relevant host-impurity pairs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Linear regression on elemental features gives decent CV scores for z* but the leave-out-group protocol may allow feature leakage.

read the letter

The main takeaway is that this paper fits a linear model to effective charge z* using elemental properties and reports solid 5-fold leave-out-alloy-group cross-validation numbers (RMSE/σ 0.37, R² 0.86), plus some extrapolation tests on new alloys. It is a straightforward application of existing regression techniques to an established materials property rather than a new method or derivation. The work does a reasonable job of framing the problem around interconnect reliability and shows the model can be used to screen host-impurity pairs. The choice of leave-out-group CV is a clear step above random splits and indicates they tried to test generalization across alloys. The reported metrics are concrete and the extrapolation section, while limited, is at least attempted. That is the useful part for anyone who needs quick estimates of z* without new experiments. The soft spot is the potential for leakage in the CV. If alloy groups share hosts or impurities, the elemental feature vectors are not fully independent between folds, so the good scores could partly reflect that overlap rather than true out-of-distribution performance. The abstract does not spell out the exact grouping rules, so this remains an open question. The model itself is a post-hoc linear fit with no first-principles grounding, which limits how far the predictions can be trusted for truly novel alloys. This is the kind of paper that belongs in a materials informatics or reliability engineering reading group. A reader who works on electromigration screening or wants an example of applied linear modeling on small materials datasets will get practical value from the numbers and the protocol. It is not for people seeking new theory or large advances in ML methodology. I would send it to peer review because the CV results are specific enough that referees can check the data handling and grouping details, and the application target is relevant even if the underlying technique is routine.

Referee Report

2 major / 2 minor

Summary. The manuscript applies linear regression to model the effective charge z* in electromigration as a linear function of physically meaningful elemental properties for host-impurity pairs. It reports average 5-fold leave-out-alloy-group cross-validation results of RMSE/σ = 0.37 ± 0.01 and R² = 0.86, claims limited but potentially useful extrapolation to new alloys, and applies the model to technologically relevant pairs.

Significance. If the reported cross-validation performance holds under a leakage-free protocol, the work offers a practical, feature-based route to estimate z* without new experiments, which could aid reliability predictions in microelectronics. The choice of physically interpretable inputs and the leave-out-group CV protocol are strengths that reduce some overfitting risk compared to random splits.

major comments (2)

[Abstract] Abstract (cross-validation and extrapolation paragraphs): The leave-out-alloy-group 5-fold CV is presented as testing extrapolation to 'totally new alloys,' yet the manuscript does not specify whether the alloy groups are constructed to ensure that no host or impurity element appears in both training and test folds. Shared elemental features across groups would allow the linear model to see the same property vectors during training, undermining the independence required for the reported R² = 0.86 and the extrapolation claim.
[Abstract] Abstract (model description): The central claim that z* is modeled 'as a linear function of physically meaningful elemental properties' is load-bearing for the extrapolation results, but the manuscript provides no explicit list of the selected features, their physical justification, or evidence that the linear form was not chosen after inspecting the data; this makes it difficult to assess whether the RMSE/σ = 0.37 reflects genuine predictive power or post-hoc fitting.

minor comments (2)

[Abstract] The abstract reports two RMSE/σ values (0.37 ± 0.01 and 0.22 ± 0.18) without clarifying what the second value represents; this notation should be defined in the methods or results section.
No table or figure is referenced in the provided abstract that shows the actual feature coefficients or the list of host-impurity pairs used; adding such a table would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. We respond to each major comment below and will revise the manuscript to improve clarity and address the concerns.

read point-by-point responses

Referee: [Abstract] Abstract (cross-validation and extrapolation paragraphs): The leave-out-alloy-group 5-fold CV is presented as testing extrapolation to 'totally new alloys,' yet the manuscript does not specify whether the alloy groups are constructed to ensure that no host or impurity element appears in both training and test folds. Shared elemental features across groups would allow the linear model to see the same property vectors during training, undermining the independence required for the reported R² = 0.86 and the extrapolation claim.

Authors: The leave-out-alloy-group CV partitioned the data by unique host-impurity pairs, holding out all entries for each group in a given fold. We agree the manuscript does not explicitly describe whether this construction eliminates overlap of individual elements (hosts or impurities) between folds. In our dataset, some elements do appear in multiple pairs, allowing partial feature overlap. This is a valid point that weakens the 'totally new alloys' phrasing. We will revise the methods and abstract to specify the grouping procedure, report the extent of element overlap, and moderate the extrapolation language to match what the protocol actually tests. revision: yes
Referee: [Abstract] Abstract (model description): The central claim that z* is modeled 'as a linear function of physically meaningful elemental properties' is load-bearing for the extrapolation results, but the manuscript provides no explicit list of the selected features, their physical justification, or evidence that the linear form was not chosen after inspecting the data; this makes it difficult to assess whether the RMSE/σ = 0.37 reflects genuine predictive power or post-hoc fitting.

Authors: Feature selection was performed prior to any model fitting or CV, drawing on established physical factors known to influence electromigration (atomic size mismatch, electronegativity difference, valence electron count, and melting point). The linear form was chosen a priori for interpretability with the limited data available. We will add to the revised manuscript an explicit table listing all features with their physical motivations and a statement confirming the pre-specified nature of the model and features. revision: yes

Circularity Check

0 steps flagged

No significant circularity in empirical ML fit with group CV

full rationale

The paper fits a linear model of z* to elemental properties and reports performance via explicit 5-fold leave-out-alloy-group cross-validation plus limited extrapolation tests. This is a standard supervised learning workflow in which held-out groups supply the validation metric; the reported RMSE/σ and R² are therefore not equivalent to the training inputs by construction. No self-definitional equations, fitted-input-renamed-as-prediction, or load-bearing self-citation chains appear in the provided text. The central claim remains an empirical correlation whose validity rests on the independence of the CV splits rather than on any internal reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The model rests on a fitted linear relationship whose coefficients are determined from experimental data and on the domain assumption that elemental properties suffice to capture z* variation across alloys.

free parameters (1)

linear regression coefficients
Weights multiplying each elemental property in the expression for z* are obtained by fitting to the collected dataset.

axioms (1)

domain assumption Effective charge z* is adequately described by a linear combination of elemental properties
This modeling choice is stated directly in the abstract as the basis for the machine-learning approach.

pith-pipeline@v0.9.0 · 5674 in / 1282 out tokens · 42313 ms · 2026-05-25T11:02:05.160370+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Original data sets: The complete databases used in the study, including all effective charges and all descriptors for all the alloy and pure metal system s is titled “Dataset(used)”. The complete initially developed database is titled “Dataset(whole)”. The text file titled “Reference_dataset” on Figshare lists the references used to obtain the database of...

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Gilder and D

H.M. Gilder and D. Lazarus: Effect of High Electronic Current Density on the Motion of Au195 and Sb125 in Gold. Phys. Rev. 145, 507 (1966)

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Bekiaris, Z

N. Bekiaris, Z. Wu, H. Ren, M. Naik, J.H. Park, M. Lee, T.H. Ha, W. Hou, J.R. Bakke, M. Gage, Y. Wang and J. Tang: Cobalt fill for advanced interconnects, in 2017 IEEE International Interconnect Technology Conference (IITC) (2017), pp. 1

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[1] [1]

Figures Data: Fig X.csv and Fig SX.csv contain all the data used to make Figure X and Figure SX in the manuscript and the SI, respectively

work page

[2] [2]

Dataset(used)

Original data sets: The complete databases used in the study, including all effective charges and all descriptors for all the alloy and pure metal system s is titled “Dataset(used)”. The complete initially developed database is titled “Dataset(whole)”. The text file titled “Reference_dataset” on Figshare lists the references used to obtain the database of...

work page

[3] [3]

K.N. Tu, Y. Liu and M. Li: Effect of Joule heating and current crowding on electromigration in mobile technology. Applied Physics Reviews 4, 011101 (2017)

work page 2017

[4] [4]

Huntington and A.R

H.B. Huntington and A.R. Grone: Current-induced marker motion in gold wires. Journal of Physics and Chemistry of Solids 20, 76 (1961)

work page 1961

[5] [5]

Bosvieux and J

C. Bosvieux and J. Friedel: Sur l'electrolyse des alliages metalliques. Journal of Physics and Chemistry of Solids 23, 123 (1962)

work page 1962

[6] [6]

Blech: Electromigration in thin aluminum films on titanium nitride

I.A. Blech: Electromigration in thin aluminum films on titanium nitride. Journal of Applied Physics 47, 1203 (1976)

work page 1976

[7] [7]

Lin, Y.-c

S.-k. Lin, Y.-c. Liu, S.-J. Chiu, Y.-T. Liu and H.-Y. Lee: The electromigration effect revisited: non-uniform local tensile stress-driven diffusion. Scientific Reports 7, 3082 (2017)

work page 2017

[8] [8]

Sorbello: Theory of electromigration

R.S. Sorbello: Theory of electromigration. Solid State Physics 51, 159 (1997). 30

work page 1997

[9] [9]

Ho and T

P.S. Ho and T. Kwok: Electromigration in metals. Reports on Progress in Physics 52, 301 (1989)

work page 1989

[10] [10]

Shi and H.B

J. Shi and H.B. Huntington: Electromigration of gold and silver in single crystal tin. J. Phys. Chem. Solids 48, 693 (1987)

work page 1987

[11] [11]

van Ek, J.P

J. van Ek, J.P. Dekker and A. Lodder: Electromigration of substitutional impurities in metals: Theory and application in Al and Cu. Phys. Rev. B: Condens. Matter 52, 8794 (1995)

work page 1995

[12] [12]

Dekker, A

J.P. Dekker, A. Lodder and J. van Ek: Theory for the electromigration wind force in dilute alloys. Phys. Rev. B: Condens. Matter 56, 12167 (1997)

work page 1997

[13] [13]

Dekker and A

J.P. Dekker and A. Lodder: Calculated electromigration wind force in face- centered-cubic and body-centered-cubic metals. Journal of Applied Physics 84, 1958 (1998)

work page 1958

[14] [14]

Dekker, P

J.P. Dekker, P. Gumbsch, E. Arzt and A. Lodder: Calculation of the electromigration wind force in Al alloys. Phys. Rev. B: Condens. Matter 59, 7451 (1999)

work page 1999

[15] [15]

Lodder: Direct Force Controversy in Electromigration Exit

A. Lodder: Direct Force Controversy in Electromigration Exit. Defect and Diffusion Forum 261-262, 77 (2007). 31

work page 2007

[16] [16]

fourth paradigm

A. Agrawal and A. Choudhary: Perspective: Materials informatics and big data: Realization of the “fourth paradigm” of science in materials science. APL Materials 4, 053208 (2016)

work page 2016

[17] [17]

Ramprasad, R

R. Ramprasad, R. Batra, G. Pilania, A. Mannodi-Kanakkithodi and C. Kim: Machine learning in materials informatics: recent applications and prospects. npj Comput. Mater. 3, 54 (2017)

work page 2017

[18] [18]

L. Ward, A. Agrawal, A. Choudhary and C. Wolverton: A general-purpose machine learning framework for predicting properties of inorganic materials. npj Comput. Mater. 2, 16028 (2016)

work page 2016

[19] [19]

W. Li, R. Jacobs and D. Morgan: Predicting the thermodynamic stability of perovskite oxides using machine learning models. Comput. Mater. Sci. 150, 454 (2018)

work page 2018

[20] [20]

Dimiduk, E.A

D.M. Dimiduk, E.A. Holm and S.R. Niezgoda: Perspectives on the Impact of Machine Learning, Deep Learning, and Artificial Intelligence on Materials, Processes, and Structures Engineering. Integrating Materials and Manufacturing Innovation 7, 157 (2018)

work page 2018

[21] [21]

H. Wu, A. Lorenson, B. Anderson, L. Witteman, H. Wu, B. Meredig and D. Morgan: Robust FCC solute diffusion predictions from ab-initio machine learning methods. Comput. Mater. Sci. 134, 160 (2017). 32

work page 2017

[22] [22]

Tanaka, K

I. Tanaka, K. Rajan and C. Wolverton: Data-centric science for materials innovation. MRS Bulletin 43, 659 (2018)

work page 2018

[23] [23]

De Jong, W

M. De Jong, W. Chen, R. Notestine, K. Persson, G. Ceder, A. Jain, M. Asta and A.J.S.r. Gamst: A statistical learning framework for materials science: application to elastic moduli of k-nary inorganic polycrystalline compounds. Sci. Rep. 6, 34256 (2016)

work page 2016

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Mueller, A.G

T. Mueller, A.G. Kusne and R. Ramprasad: Machine learning in materials science: Recent progress and emerging applications. Rev. Comput. Chem. 29, 186 (2016)

work page 2016

[25] [25]

Pedregosa, Ga, #235, l

F. Pedregosa, Ga, #235, l. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, #201 and d. Duchesnay: Scikit-learn: Machine Learning in Python. J. Mach. Learn. Res. 12, 2825 (2011)

work page 2011

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Morgan, B

D. Morgan, B. Afflerbach, R. Jacobs, T. Mayeshiba and H. Wu: MAterials Simulation Toolkit – Machine Learning (MAST-ML), (GitHub, GitHub repository, 2017). 33

work page 2017

[27] [27]

Raschka: MLxtend: Providing machine learning and data science utilities and extensions to Python's scientific computing stack. J. Open Source Softw. 3 (2018)

work page 2018

[28] [28]

DiGiacomo, P

G. DiGiacomo, P. Peressini and R. Rutledge: Diffusion coefficient and electromigration velocity of copper in thin silver films. J. Appl. Phys. 45, 1626 (1974)

work page 1974

[29] [29]

Park and R.W

C.W. Park and R.W. Vook: Electromigration-resistant Cu-Pd alloy films. Thin Solid Films 226, 238 (1993)

work page 1993

[30] [30]

Lee, C.K

K.L. Lee, C.K. Hu and K.N. Tu: In situ scanning electron microscope comparison studies on electromigration of Cu and Cu(Sn) alloys for advanced chip interconnects. J. Appl. Phys. 78, 4428 (1995)

work page 1995

[31] [31]

Gilder and D

H.M. Gilder and D. Lazarus: Effect of High Electronic Current Density on the Motion of Au195 and Sb125 in Gold. Phys. Rev. 145, 507 (1966)

work page 1966

[32] [32]

Bekiaris, Z

N. Bekiaris, Z. Wu, H. Ren, M. Naik, J.H. Park, M. Lee, T.H. Ha, W. Hou, J.R. Bakke, M. Gage, Y. Wang and J. Tang: Cobalt fill for advanced interconnects, in 2017 IEEE International Interconnect Technology Conference (IITC) (2017), pp. 1

work page 2017