Rapid subsurface analysis of frequency-domain thermoreflectance images with K-means clustering

As microelectronics devices continue to scale down, heat flux per unit area continues to increase,¹ in turn increasing the need for better thermal dissipation. Though defects in devices can lead to increased heat generation via hot spots,² the pain point for thermal transport is often interfaces. Interfaces of two different material types create the potential for impeded phonon and electron transport.³ The thermal effect of interfaces is crucial in applications such as power electronics^4–6 and other heterogeneously integrated microelectronics.^7–10 In these systems, uniform bond quality at material interfaces is critical to optimal heat dissipation as well as mechanical performance. However, bond quality can vary substantially over an interface, even between similar materials. Therefore, rapid and accurate defect screening for interfaces is needed to characterize device performance. Thermal transport across an interface can serve as an effective measurement of bond quality,^11,12 often expressed as thermal boundary conductance (TBC).

Through the use of thermoreflectance techniques, effective and repeatable measurements of TBC can be obtained. However, bond quality is often not uniform across the entirety of the bonded area.^28,29 To quantify spatial nonuniformities in bulk material properties, many studies have used a moving stage and performed thermoreflectance measurements at many locations on the sample surface.^{28,29,32–36} This mapping capability has been extended in recent years to show variations in TBC of interfaces in both TDTR³⁷ and FDTR.^{28,29,38–40} Combining mapping capability with high sensing depths in FDTR measurements opens the possibility of FDTR maps of bond quality. In order to achieve the best possible spatial resolution, small distances between measurements are desirable but create a large amount of data to be processed. Additionally, information about bond quality across device-scales is desired putting an even larger demand on computational resources to process the vast amount of data generated from such experiments. Despite these challenges, evaluation of methods to speed up and simplify data processing of such large data sets remains largely unexplored.

The existing analytical solution⁴¹ only applies to radially symmetric geometries, which may not be an accurate physical representation of more geometrically complex samples. Finite element analysis (FEA) representations can accommodate arbitrary geometries and have recently been used for FDTR data analysis⁴² and inverse fitting of bond quality.²⁹ However, repeated finite element solves can become quite computationally expensive, especially for high thermal conductivity materials where $L p$ is relatively large. Machine learning (ML) offers a computationally inexpensive alternative to analytical and FEA methods. ML approaches have been recently used as an alternative to TDTR⁴³ data analyses and even to reconstruct depth varying thermal conductivity.⁴⁴ 

In this work, we do not seek to replace the established physical thermal models, analytical nor numerical. Rather, we present an unsupervised machine learning supplement to the analytical and FEA methods of previous studies, specifically taking advantage of analytical solutions to the heat diffusion equation,^21,41,45 however, the approach does not exclude the FEA and ML solutions, described above.^29,42–44 Here, K-means clustering is used to group the unique spectra at every pixel across an FDTR map of a Si–Si bonded material stack into three clusters. It should be noted that this method is not limited to three components and the optimal number of components for a given data set will be discussed. Each cluster is then associated with one of three component spectra extracted from K-means clustering. The thermal properties of these representative component spectra are fit using a standard thermal model.^21,41,45 Assigning each pixel with thermal properties of the cluster they group with, however, generates non-physical and abrupt transitions between regions with different thermal character. To address this, each pixel is expressed as a linear combination of the three component spectra with coefficients regressed against the experimental phase spectra. The coefficient weighting is then applied across the component thermal properties to generate thermal property maps. The accuracy of thermal property projection using this approach is then compared to explicit fits at select pixels. We show good agreement between the explicit fits and the K-means cluster assisted projections.

While the projections are not expected to achieve as accurate of thermal property values as explicit fits to each pixel, the orders of magnitude decrease in computational demand enabled by the K-means projection establishes the utility of this approach as a high-throughput interfacial screening method. This screening can be used across die-scale maps to down-select for relevant regions to be explicitly fit to the thermal model for the most accurate thermal properties. To the authors’ knowledge, this is the first application of data dimension reduction techniques to FDTR maps for rapid evaluation of the spatial distribution of thermal properties across bonded interfaces. This method is not limited to analysis of interfaces but is applicable to any varying thermal property, such as grain boundary variations on thermal conductivity,³⁹ within a system that is sensitive to the FDTR/TDTR technique.

As mentioned previously, FDTR is a pump–probe laser technique capable of extracting the thermal properties of an underlying material system.^21–27,29 In our FDTR setup, a 488 nm pump laser whose intensity is periodically modulated at $f$ is used to drive heat into the sample, while a 532 nm continuous-wave probe laser monitors the surface temperature fluctuations through the thermoreflectance effect.²⁶ Both lasers are focused through a 5 $×$ objective resulting in $1/ e 2$ radii of $w pump=7.03±0.19 μ m$ and $w probe=5.38±0.05 μ m$ for the pump and probe, respectively.^28,46 As shown in Fig. 1(a), we perform FDTR experiments on a sample consisting of thinned silicon (Si) that is directly bonded to a Si substrate. Here, two identical coupons of doped-Si undergo a benchtop solvent clean, followed by an Ar plasma treatment. The treated surfaces of the Si coupons are brought into contact under ambient fab conditions and are squeezed under tweezer force to initiate Si–Si bonding. Afterward, one side of the doped-Si stack is thinned and polished resulting in an overlayer thickness of 6–9  $μ m$⁠. The sample is then blanket coated in gold (Au) with a 5 nm titanium adhesion layer, which acts as the transducer layer for FDTR measurements. The measured frequency-dependent phase lag, $θ(f)$⁠, between temperature fluctuations and pump laser heating is extracted after calibration of the system phase and probe noise,²⁶ which is often fit with a thermal model to extract thermophysical properties of each layer:⁴⁵ volumetric heat capacity (⁠ $ρ c p$⁠), thermal conductivity (⁠ $k$⁠), thickness (⁠ $d$⁠), and TBC (⁠ $G$⁠).

FDTR is implemented as an imaging modality by raster scanning the sample and continuously monitoring $θ(f)$ for each pixel. In our FDTR setup, the probe signal is analyzed with a multi-channel lock-in amplifier (UHFLI, Zurich Instruments) and enables simultaneously signal demodulation at up to 8 frequencies. Thus, we can acquire an FDTR hyperspectral data-cube with 16 logarithmically spaced values of $f$ (⁠ $1 kHz$– $60 MHz$⁠) by scanning the sample twice over the same spatial domain (see Fig. S1 in the supplementary material). In this work, FDTR images have a field of view of 200  $μ m$ by 200  $μ m$ with a maximum pixel size of 2  $μ m$⁠. The full hyperspectral data series took 50 min to acquire (i.e., 800  $μ m 2/ min$⁠). Figure 1(b) provides a portion of the hyperspectral data-cube at five characteristic frequencies (2.0 kHz, 9.0 kHz, 82 kHz, 740 kHz, and 29 MHz) in terms of delta phase: $Δθ(f)=θ(f)− avg[θ(f)]$⁠. The pump frequency can be related to the sensing depth, $D s$⁠, using formulations from Ref. 27 and illustrates that frequencies above 740 kHz (⁠ $D s=4.0μ m$⁠) only sample the thinned Si overlayer and Au transducer, while frequencies below 9.0 kHz (⁠ $D s=34μ m$⁠) sense deep into the Si substrate. Intermediate frequencies, e.g., 82 kHz (⁠ $D s=14μ m$⁠), sample near the interface between the thinned Si and Si substrate and show significant variation in $Δθ(f)$⁠. As we showed in Ref. 29, this behavior in the FDTR data-cube is often attributed to bond quality at a subsurface interface. However, this can only be determined after thermal analysis.

Figure 1(c) provides an FDTR $θ$ map at $f=82 kHz$⁠, where three distinct thermal regions are identified. While not immediately intuitive, $θ$ can be used to interpret the ability of a region to conduct heat: high phase (closer to $θ=0$⁠) regions correspond to increased thermal conductance, while low phase (more negative $θ$⁠) regions correspond to reduced thermal conductance.²⁹ Using this method, pixels P $1$⁠, P $2$⁠, and P $3$ in Fig. 1(c) correspond to regions of increasing thermal conductance, respectively. Quantitative interpretation is only obtained after fitting the $θ(f)$ spectra with a radially-symmetric, frequency-domain heat diffusion calculation as shown in Fig. 1(d).^21,26,41 The inset provides the model setup, where each layer has associated $ρ c p$⁠, $k$⁠, $d$⁠, and $G$ values. It should be noted, however, that the substrate is thick enough to ignore the backside $G$⁠. In order to constrain the problem, the Au thermal properties (⁠ $ρ c p | 1$ and $k 1$⁠) were determined by FDTR performed on an SiO $2$ calibration sample [see Fig. S2(a) in the supplementary material], while its thickness (⁠ $d 1$⁠) was obtained via profilometry. The Au properties were assumed constant across the FDTR field of view. Similarly, $ρ c p | 2$ for both pieces of Si were assumed constant at literature values,^47,48 while $k 3$ was initialized at a typical doped-Si value [see Fig. S2(b) in the supplementary material]. Subsequent analyses fix $k 3$ at $k 2$ obtained from the P $1$ value.

The thermal analysis begins with P $1$⁠, which appears to be in the most poorly conductive region of the three pixels. This enables fitting for local values of $G 1$⁠, $k 2$⁠, $d 2$⁠, and $G 2$ [see Fig. S3(a) in the supplementary material]. As seen in the plot, a good fit is obtained for P $1$ given the thermophysical properties in Table I. In particular, we find $k 2=86.9±7.3 W / m K$ which is inline with the thermal conductivity of doped-Si at the extracted thickness of $d 2=8600±350 nm$ and an unbonded value for $G 2=0.4±0.03 MW / m 2 K$⁠. Moving to P $2$⁠, while fixing $k 3$ and $d 2$⁠, we find a similar value for $k 2=88.4±7.2 W / m K$ and a partially bonded value for $G 2=9.9±2.7 MW / m 2 K$⁠. Finally, P $3$ demonstrates a well bonded value for $G 2>100 MW / m 2 K$ [see Figs. S3(c) and S4 in the supplementary material], enabling extraction of $k 2=92.1±7.5 W / m K$⁠. Here, the uncertainty analysis follows Ref. 25. It should be noted that the value of $G 1$ for each point also agrees well with literature values of the Au–Si thermal boundary conductance.⁴⁸ By comparing $G 2$ values for P $1$⁠, P $2$⁠, and P $3$ as well as their representative colorbar values, P $3$ (bonded) transitions to P $1$ (unbonded) through a gradient region typified by P $2$⁠.

K-means clustering is an unsupervised machine learning algorithm that seeks to iteratively partition $N$ observations into $K≤N$ clusters.^49,50 This is achieved by minimizing the within-cluster-variance, or distance of each point assigned to a cluster from the cluster centroid. Applying this approach to FDTR maps, each hyperspectral pixel in the map represents one of the set of $N$ observations, where $N$ is the total number of pixels. To maintain high spatial resolution, maps may contain $> 10 6$ pixels, straining computational resources for explicit thermal fitting analysis as described in Sec. III for every pixel of an image. By partitioning the total number of pixels, $N$⁠, into $K$ components, the data set, and thus the computational expense, can be dramatically reduced. This strategy of data reduction is applied to FDTR maps to rapidly assess the spatial dependence of subsurface thermal properties.

K-means clustering is applied using the built-in functionality in Matlab 2022b. Specifically, the K-means++ algorithm⁵¹ is implemented using a squared-euclidean distance metric. We applied this algorithm to the unaltered hyperspectral FDTR data-cube with $K=3$ and 50 replicates. It should be noted, $K=4$ and $K=5$ were compared to $K=3$ by computing the mean silhouette values. The mean silhouette of $K=3$ was found to be 0.77 compared to 0.40 and 0.37 for $K=4$ and $K=5$⁠, respectively, confirming that $K=3$ is the optimal choice for the number of clusters (see Fig. S5 of the supplementary material). K-means clustering can be applied to more complex, heterogeneous samples, where the number of clusters likely needs to be increased from the simple material system presented here.

K-means analysis of the FDTR hyperspectral data-cube produces $K=3$ spectral components. Each pixel in the map is correspondingly assigned to one of these $K=3$ clusters by the algorithm described above. Figure 2(a) shows the grouping of pixels into clusters K $1$⁠, K $2$⁠, and K $3$⁠, expressed for three frequency selections. The frequencies chosen to express the grouping behavior map to different sensing depths allowing for a simultaneous comparison across multiple regions of the material stack. In Fig. 2(a), the z axis shows the phase of every pixel at $D s=1.6μ m$ (⁠ $f=29 MHz$⁠). This frequency shows the behavior within the top Si layer. The x axis shows the phase of every pixel at $D s=14μ m$ (⁠ $f=82 kHz$⁠). This frequency shows the behavior close to the Si–Si interface and captures information about the quality of the interface through spatially resolved features. This is immediately apparent from the phase map at this frequency, shown in Fig. 1(c), where phase variations contour a featured landscape across the map, highlighting the spatial dependence of bond quality at a material interface. Finally, the y axis shows the phase of every pixel at $D s=52μ m$ (⁠ $f=2 kHz$⁠). This represents the material stack beyond the interface, into the underlying Si substrate.

In Fig. 2(a), projections onto x–y and x–z planes are shown to highlight the correlation of behaviors at these sensing depths. Projections along y–z are largely overlapped and do not show any additional trends. The overlap of the clusters on the y–z plane is physically interpreted as the independence of top-layer Si from the bottom-layer Si substrate; the top and bottom materials are identical and do not strongly influence the resultant thermal transport.

When projecting onto the x–y plane, we are examining the correlation between phase behavior in the bottom Si substrate (y axis) and the Si–Si interface (x axis). Along this projection, clusters K $2$ and K $3$ are constant in the low-frequency Si substrate region and only show variation at the interface (i.e., phase variation along the x axis). K $1$ is distinguished from K $2$ and K $3$ by its stronger phase dependence on the low-frequency y axis. Additionally, the phase along the x axis is simultaneously decreased, albeit to a lesser extent. The rapid downturn to lower phase along the low-frequency y axis captures the strong sensitivity of the underlying Si substrate to small changes at the interface. Physically, this shows that the interface quality is the limiting factor for heat transfer. Small changes in the interface equate to large changes in the underlying substrate heat extraction.

The x–z plane projections show a more straightforward trend, where independent of the phase behavior along the z axis (high-frequency, top-Si layer), the clustering captures the changes at the x axis, interfacial region. The spread in the clusters is similar along the z axis, while the clustering distinguishes pixels along the interfacial x axis. The physical interpretation is that the thermal wave proceeds unimpeded in the top Si layer until reaching the interface. Thus, the interfacial quality has a minimal impact on the phase character of the high-frequency z axis, since its phase character is solely determined by the top-layer Si through which it traverses, neglecting thermal reflection at the interface.

Figure 2(b) shows the frequency-dependent $θ$ spectra associated with each cluster: K $1$⁠, K $2$⁠, and K $3$⁠. These spectra are analogous to the manually chosen pixel spectra, P $1$⁠, P $2$⁠, and P $3$⁠, in Fig. 1(d) in that the three spectra express the varying $θ(f)$ across the FDTR map. The difference is that the three spectra extracted by K-means clustering analysis find spectral bounds in K $1$ and K $3$ where maxima and minima are observed, respectively, in the 82 kHz interfacial region. K $2$ falls in between these bounds in phase and, when exploring $K>3$ in the K-means clustering analysis, additional components simply fill in the gaps between the phase extremes set by K $1$ and K $3$ in the 1 kHz–1 MHz region. For $K>3$⁠, phase values outside of the bounds set by K $1$ and K $3$ are not observed. Thus, the three components extracted by K-means using $K=3$ likely characterize the bounding set of possible phase profiles across the FDTR map, allowing for projection of every pixel onto a weighted linear combination of these three basic components. In other words, every pixel in the FDTR map can be expressed as a linear combination of these basis spectra.

Given this deduction, we fit the three spectral components shown in Fig. 2(b) to the same thermal model as in Fig. 1(d). This produces $K=3$ unique $G 1$⁠, $G 2$⁠, and $k$ parameters, $G 1 , { 1 , 2 , 3 }$⁠, $G 2 , { 1 , 2 , 3 }$⁠, $k { 1 , 2 , 3 }$⁠. These thermal parameters can now be applied across the FDTR maps according to established weighting of the component contribution per pixel, as in Eqs. (3)–(5). However, we must first establish the fit model parameters and the per-pixel coefficient weighting, $c K( i x y)$⁠.

As similarly described in the pixel analysis, the cluster thermal analysis begins with $K 1$⁠, which appears to be in the most poorly conductive component spectra. A good fit is obtained for $K 1$ given the thermophysical properties in Table II, where $k 2=88.7±7.5 W / m K$ and $d 2=8400±350 nm$⁠, and an unbonded value for $G 2=1.05±0.09 MW / m 2 K$⁠. Moving to $K 2$⁠, while fixing $d 2$ and $k 3$⁠, we find $k 2=88.9±7.3 W / m K$ and a partially bonded value for $G 2=8.0±2.0 MW / m 2 K$⁠. Finally, $K 3$ demonstrates a well bonded value for $G 2>100 MW / m 2 K$ (see Fig. S4 in the supplementary Material), $k 2=93.0±7.6 W / m K$⁠. The exact thermal properties from the component spectra (Table II) are consistent with those from the manually selected pixels in Table I, suggesting that the K-means algorithm is correctly representing these regions of interest. Moreover, by comparing $G 2$ values for K $1$⁠, K $2$⁠, and K $3$ as well as their representative colorbar values, K $3$ (bonded) transitions to K $1$ (unbonded) through a gradient region typified by K $2$⁠.

The component contribution per pixel is shown in Fig. 3 where cluster K $1$ is distributed as in Fig. 3(a), cluster K $2$ is distributed as in Fig. 3(b), cluster K $3$ is distributed as in Figs. 3(c), and 3(d) shows the combined contribution across all clusters. The important point to note from this figure is that the weightings, $c K( i x y)$⁠, show the spatial character of each component. For example, K $1$ was described above as being distinguished from K $2$ and K $3$ in the correlation analysis on the x–y plane. In the spatial analysis of the coefficient fit results, clusters K $2$ and K $3$ appear to represent gradient-to-high-phase and high-phase, respectively, in the 1 kHz–1 MHz region. This is contrasted with K $1$⁠, which occurs isolated as a low-phase island, as in the upper-right-hand corner of Fig. 3(a). Also consider the faint circle near $Y=150μ m$⁠, $Y=40μ m$⁠. This faint circle emphasizes the evolution of a gradient following K $2$ toward K $1$ in this region. In sum, the individual component contribution can be digested from the map shown in Fig. 3(d), where color-mixing between the components, weighted according to their spatial contribution, shows that hardly any given pixel is a pure representative of one of the three basis components we have assumed to define the system. This establishes the need to extend the K-means analysis to an interpolation by a linear combination of K-means basis components. With the interpolation procedure established above by Eqs. (2)–(5), we now present the K-means partitioned maps in terms of projected thermal parameters, $G ~ 1$⁠, $k 2 ~$⁠, and $G ~ 2$⁠.

Figures 4(a)–4(c) provide thermophysical property maps for $G ~ 1$⁠, $k 2 ~$⁠, and $G ~ 2$⁠, respectively, as extracted from the K-means clustering analysis. The maps for $G ~ 1$ and $k 2 ~$ show a mostly uniform distribution varying by $123±1.3 MW / m 2 K$ and $92±1.8 W / m K$⁠, respectively. It should be noted the maximum/minimum variation in $G ~ 1$ and $k 2 ~$ are within the fitted error bound of our FDTR method as shown in Table I, which is applied as the colorbar bounds. The slight variation seen in $G ~ 1$ and $k 2 ~$ is a result of overlapping sensitivity in the FDTR fitting process applied to the K-means cluster spectra [see Fig. S6 and its discussion in the supplementary material]. Conversely, significant variation in $G ~ 2$ is retrieved from the K-means clustering method. Here, the blue regions take a value near $1 MW / m 2 K$⁠, while the red regions are bounded by the $>100 MW / m 2 K$ sensitivity limit (see Fig. S4 in the supplementary material). The resulting $G ~ 2$ map aligns very well with the intuition developed by the interpretation of Fig. 1(c). Moreover, we interestingly find that the $G ~ 2$ map captures gradual variation from unbonded to bonded regions. To highlight this effect, a vertical line trace is pulled from Fig. 4(c) and plotted in 4(d) for $X=174.5μ m$⁠. Low values of $Y$ correspond to the most unbonded region of Fig. 4(c) which gradually becomes bonded at $Y=25μ m$⁠. Within approximately $25μ m$⁠, more than two orders of magnitude change in $G ~ 2$ is observed, which again reduces by more than one order of magnitude to the partially bonded value at $Y=175μ m$⁠. This map underscores the importance of buried interface quality in microelectronics as a device sitting over a blue region in Fig. 4(c) will have dramatically different thermal performance than one sitting over a red region even though their proximity to one another may only be 10s of $μ m$⁠.

Table III compares the K-means interpreted thermophysical properties at the same points extracted in Table I. Clearly, the K-means method is able to pull out meaningful variations in the thermophysical properties when compared to performing a manual pixel analysis. In particular, $G 2$ gradually increases from P $1$ (unbonded) to P $3$ (bonded) for both pixel and K-means analysis methods, where the intermediate value, P $2$⁠, is within the fitted uncertainty. We note that the extreme unbonded value deviates from the expected value from the pixel method, likely due to the equal weighting in the non-linear regression applied across the frequency domain. This could be improved by weighting the frequencies that probe the interface more heavily.

The use of K-means in this work should not be considered a replacement for pixel-by-pixel or FEA fitting methods, but instead to provide rapid screening of thermal properties. The phase images that typically result from FDTR can be difficult to understand, especially when spatial variation occurs at different frequency ranges. To garner insight, it is therefore critical to be able to present experimental data in terms of physical quantities as is shown in Figs. 4(a)–4(c). However, fitting every pixel of this image to generate maps of thermal properties is time intensive. For example, we find that each fit takes approximately 0.6 s to perform. This does not account for the additional time required to load the measurement data. For the current measurements, the FDTR image has 20 200 pixels suggesting that fitting every pixel will at best take 3.4 h to complete. This does not include processing of fit results and intermittent checks that the fit converges at each pixel. Conversely, the K-means method requires the operation of the clustering algorithm (2.6 s), least squares minimization of coefficients (162s), and fitting of only three spectra (1.8 s). This results in a total time of 166.4 s, or about 3 min, which is a 74-fold speed improvement. Afterward, if regions of particular interest arise, more detailed analysis can be performed in a targeted fashion. When considering K-means to speed up FDTR data analysis on other material systems, it is imperative to consider FDTR’s sensitivity relative to the experimental noise floor. Figure S7 in the supplementary material provides FDTR sensitivity contours for material systems that incorporate a low thermal conductivity material (i.e., SiO $2$⁠) as either the overlayer or substrate, where reduced sensitivity is obtained. Reference 27 provides more details on FDTR’s sensitivity as a function of overlayer/substrate material as well as laser spot size.

To summarize, we demonstrate that the analysis of FDTR hyperspectral image data with K-means clustering methods provides rapid quantification of spatially dependent subsurface thermal properties. Given an exemplar sample consisting of 8.6  $μ$m of doped-Si bonded to a doped-Si substrate, FDTR hyperspectral phase data identify spatial contrast for pump frequencies that probe the Si–Si interface. Thermal analysis performed by selecting a handful of pixels in regions of interest enable quantification of the bond quality as well as thermal properties of the overlayer Si. Similarly, K-means clustering with three components allows pixels to be grouped based on the measured phase at frequencies that probe the overlayer Si, Si–Si interface, and Si substrate. By fitting the component spectra to our FDTR thermal model, representative thermal properties for each cluster are obtained. This allows, via the component weighting applied to each pixel, a reconstruction of thermophysical property maps of the sample. In particular, the Si–Si thermal boundary conductance map demonstrates two orders of magnitude variation over 25  $μ$m spatial domain, underscoring the importance of subsurface bond quality on microelectronic thermal performance. The use of K-means, while not a direct replacement for more rigorous analysis methods, provides physical insight via thermophysical property maps at a 74-fold speed improvement over fitting every pixel.

See the supplementary material for the full FDTR hyperspectral images for frequencies from $1 kHz$ to $60 MHz$⁠; FDTR performed on SiO $2$ and doped-Si calibration samples; FDTR sensitivity plots for the pixel-selective thermal analysis; FDTR sensitivity bounding the upper value of $G 2$ that can be determined with the current analysis methods; K-means silhouette plots justifying the use of $K=3$ in the current work; zoomed in color scale thermophysical property maps; and FDTR buried interface sensitivity for various conditions.

We thank Shane Sickafoose for the review of this manuscript. This work was supported by Laboratory Directed Research and Development program at Sandia National Laboratories, a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under Contract No. DE-NA0003525. The views expressed in the article do not necessarily represent the views of the U.S. DOE or the United States Government.

The authors have no conflicts to disclose.

A.J. and Z.T.P. contributed equally to this work.

Amun Jarzembski: Conceptualization (equal); Data curation (lead); Formal analysis (supporting); Investigation (supporting); Methodology (equal); Resources (lead); Software (equal); Supervision (equal); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Zachary T. Piontkowski: Conceptualization (equal); Data curation (supporting); Formal analysis (lead); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (equal). Wyatt Hodges: Conceptualization (supporting); Formal analysis (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Matthew Bahr: Data curation (supporting); Resources (supporting); Writing – review & editing (supporting). Anthony McDonald: Data curation (supporting); Investigation (lead). William Delmas: Conceptualization (supporting); Methodology (supporting); Visualization (supporting); Writing – review & editing (supporting). Greg W. Pickrell: Funding acquisition (supporting); Project administration (supporting). Luke Yates: Conceptualization (equal); Funding acquisition (lead); Methodology (supporting); Project administration (lead); Supervision (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.