Diamond quantum sensors for mapping current density in microelectronic interconnects

Introduction

Microelectronic interconnects are approaching a regime where small changes in local current path matter as much as nominal line width. Copper damascene lines, redistribution layers, through-stack vias, and advanced package bridges can all fail by electromigration, joule heating, void nucleation, or current crowding at geometric discontinuities. Electrical tests detect that a path has changed, but they rarely reveal where the current moved before catastrophic failure. Destructive cross-sectioning can localise damage after the fact, but it cannot follow a powered device through repeated stress cycles.

Nitrogen-vacancy centres in diamond offer a different route: map the magnetic field produced by device current, then reconstruct the current density. NV magnetometry is now a mature branch of quantum sensing, with established sensitivity optimisation, vector-field reconstruction, and wide-field imaging methods [1,2,3]. The field began from single-spin and nanoscale sensing demonstrations [4,5,6] and expanded to ensemble imaging with practical fields of view and frame rates [7,8,9,10,11].

Current reconstruction from magnetic fields is an old inverse problem. Roth, Sepulveda, and Wikswo showed how a magnetometer can image a two-dimensional current distribution when the current is confined to a plane [12]. The quantum-diamond version of the problem adds distinctive constraints: the magnetic field is measured above the current layer, spatial frequencies decay with standoff, the NV axes project the field into crystallographic directions, and the optical image may include temperature, strain, and microwave artefacts. These issues are manageable, but only if the magnetic measurement is treated as a calibrated metrology problem rather than as a qualitative picture.

Integrated-circuit applications have already motivated NV current imaging at redistribution-layer scale, magnetic-field fingerprinting of circuit activity, vector magnetic current imaging of process-node chips, three-dimensional imaging of integrated-circuit activity, and optimised current-density reconstruction from wide-field maps [13,14,15,16,17,18]. The present article builds on that work by reporting a complete fictional workflow for buried interconnect test structures: diamond sensor integration, vector ODMR calibration, mask-constrained inversion, uncertainty propagation, and stress-time current redistribution.

The aim is practical. We do not claim sub-10 nm resolution or full transistor-level tomography. We ask a narrower question: can a quantum diamond microscope produce quantitative current-density maps of interconnect structures through passivation with enough spatial resolution and uncertainty control to guide reliability analysis?

Test vehicles

The test vehicles were fabricated on 20 mm x 20 mm silicon coupons containing copper interconnect patterns under 280 nm silicon nitride passivation. Four layouts were used: straight reference lines, 90 deg bends, staggered via chains, and redistribution-layer fan-out structures. Nominal line widths ranged from 0.8 to 8.0 um, while via diameters ranged from 0.65 to 2.5 um. The metal thickness was 430 nm for the lower interconnect layer and 1.8 um for the redistribution layer.

Each coupon included Kelvin structures for four-terminal resistance measurement and metal temperature calibration. Alignment marks were etched into the passivation above inactive regions so that optical mask registration could be performed without exposing the conductors. The exact layouts are fictional, but the geometry deliberately resembles reliability test patterns rather than ideal textbook wires.

Measurements were performed under DC currents from 2 to 20 mA and under square-wave drive at 10 Hz to 2 kHz. DC operation gives the cleanest magnetic maps, while modulated operation rejects background fields and slow thermal drift. Electromigration stress tests used current densities up to 10^7 A cm^-2 in selected narrow segments, with duty cycles chosen to keep the global coupon temperature below 85 deg C.

The coupons were not thinned, polished, or opened before imaging. This is central to the workflow. The diamond sensor sees the magnetic field through the passivation and any surface topography. That makes the method non-destructive, but it also fixes a finite standoff that suppresses high-spatial-frequency information.

Diamond sensor and microscope

The sensor was a single-crystal electronic-grade diamond membrane with a near-surface ensemble of negatively charged NV centres. Nitrogen implantation and annealing produced an NV layer centred 12 +/- 4 nm below the diamond surface. The membrane was thinned to 40 um and bonded to a fused-silica carrier with an optical access window. The active diamond face was brought into contact with the passivated chip through a controlled load of 20-60 kPa.

Wide-field optically detected magnetic resonance was performed in reflection geometry. A 532 nm laser illuminated a 160 um x 160 um field of view, and red photoluminescence was imaged on a scientific CMOS camera. Microwave excitation was delivered through a loop antenna patterned on a removable glass spacer. The four crystallographic NV orientations were resolved by applying a calibrated bias magnetic field of 2.8 mT tilted away from high-symmetry axes.

The microscope follows the quantum diamond microscope design principle: a dense NV ensemble trades single-spin spatial resolution for field of view, parallel readout, and practical throughput [7,8,11]. Under the standard operating condition, the magnetic noise floor was 1.9 uT per pixel per square-root hertz for a 520 nm camera pixel after binning. The full vector field map used 32 microwave frequency points per NV branch and required 50 ms per frame. Slower scans with 96 frequency points improved field precision but were used only for reference maps.

Sensor standoff was the dominant geometric parameter. The physical NV depth was small, but the passivation thickness, surface roughness, and imperfect contact increased the effective current-to-sensor distance to 390-760 nm depending on coupon. Standoff was estimated from focus position, white-light interferometry, and fits to reference-line magnetic profiles.

ODMR calibration

Each magnetic image began with an ODMR calibration sequence. Zero-current spectra were fitted to extract the local resonance frequencies, contrast, linewidth, and strain splitting of each NV orientation. Current-on spectra were then converted to magnetic-field projections by tracking frequency shifts relative to the zero-current baseline. The vector field was reconstructed by least squares from the four NV projections, with pixels rejected when any branch had contrast below 1.5%.

Temperature was treated explicitly because NV resonance frequencies shift with temperature as well as magnetic field [22]. A reference region away from powered metal was monitored during each scan, and the common-mode zero-field splitting shift was subtracted before vector reconstruction. For high-current stress maps, a separate calibration converted metal resistance to coupon temperature, and residual thermal gradients were included in the uncertainty budget.

Microwave delivery produced a small spatially varying background in some frames. We removed it by measuring current-reversed pairs, B(+I) - B(-I), which cancels magnetic backgrounds that do not change sign with conductor current. This procedure is more robust than subtracting a single zero-current map when the chip heats during the measurement.

The calibration protocol was checked with a lithographic gold loop deposited on a glass standard. The loop had known geometry and current, allowing the vector magnetic field above it to be compared with a finite-width Biot-Savart calculation. Field amplitudes agreed within 4.5% after standoff fitting.

Current-density reconstruction

The inverse problem assumes that most current flows in one or two known metal layers. For a single sheet, the Fourier-domain Biot-Savart relation connects the out-of-plane magnetic field to the in-plane sheet-current density [12]. Direct inversion amplifies high-spatial-frequency noise because the magnetic field decays exponentially with sensor height. We therefore used two regularised solvers: a Tikhonov inversion for smooth maps and a sparsity-promoting inversion constrained by the known interconnect mask.

The mask-constrained solver penalises current outside lithographic metal regions while allowing finite leakage near alignment uncertainty. It also enforces current conservation except at labelled pads and vias. This is not a cosmetic prior; without it, narrow return paths and nearby metal layers can create non-unique current maps. The approach follows the spirit of recent optimised and machine-learning-assisted current-density reconstruction work, while keeping the final result interpretable as a physical inverse problem [17,18,21].

Vector magnetic field data improved stability. Using only Bz gave accurate total current in straight lines but larger artefacts near bends and vias. Including Bx and By reduced cross-talk between adjacent conductors and improved reconstruction of return currents under pads, consistent with prior analysis of vector magnetic reconstruction [21]. For deeply buried redistribution structures, however, the extra vector components did not recover spatial frequencies lost to standoff.

Uncertainty was propagated by Monte Carlo sampling of magnetic noise, standoff, mask alignment, regularisation parameter, and temperature correction. Reported current-density maps show posterior median current with 90% credible intervals for line-integrated current. We avoid over-interpreting pixel-scale hotspots smaller than the effective point-spread function.

Reference-line validation

Straight reference lines were used to validate amplitude. For 0.8, 2.0, and 8.0 um lines driven from 2 to 20 mA, reconstructed line-integrated currents agreed with four-terminal electrical measurements within 6.8% mean absolute error. The error was lowest for 8 um lines and highest for 0.8 um lines, where the magnetic profile is more strongly blurred by standoff.

The effective lateral resolution was estimated by fitting measured magnetic profiles above abrupt current edges. At 20 Hz frame rate, the resolution was 620 nm. Drift-corrected averaging over 240 frames improved this to 410 nm. These values are much coarser than single-spin NV current imaging, which can reach nanoscale proximity in specialised samples [19], but they are practical for buried microelectronic structures with passivation.

Linearity was excellent over the tested current range. The reconstructed current slope was 0.982 +/- 0.018 relative to the electrical current, and the intercept was statistically indistinguishable from zero after current-reversal subtraction. At the highest currents, small deviations appeared because joule heating shifted ODMR frequencies and changed metal resistance during the scan.

The reference results establish the central trade-off. The quantum diamond microscope gives non-contact, vector, wide-field current imaging through passivation, but the price is finite standoff and optical blur. It will not resolve the detailed current distribution inside a 40 nm line. It can, however, quantify current redistribution across micron-scale interconnect networks and via arrays.

Via-chain and bend structures

Staggered via chains produced the clearest demonstration of current crowding. In nominally symmetric chains, current density concentrated at the upstream edge of each via landing pad. The reconstructed peak-to-mean sheet-current ratio was 2.6 for 0.65 um vias and 1.7 for 2.5 um vias at 10 mA. Finite-element electromagnetic simulations predicted ratios of 2.4 and 1.6, respectively, within the experimental uncertainty.

The 90 deg bend structures showed asymmetric current flow even when the electrical resistance was normal. The inner corner carried 1.9 times the sheet current of the outer corner in the narrowest bends. Optical inspection of the passivation showed no visible defect. This illustrates the practical value of magnetic imaging: a structure can pass ordinary resistance screening while still containing a geometric current-crowding site.

Return currents under probe pads were also visible. In several layouts, the nominal ground return split between two metal layers before merging at the pad. A single-sheet inversion misassigned part of this current to a fictitious lateral path. A two-layer inversion constrained by the design mask correctly separated the redistribution layer from the lower interconnect. The layer separation was reliable when the vertical spacing exceeded 900 nm and the line directions differed by at least 30 deg.

These examples are deliberately conservative. They are not transistor-level maps of logic activity. They are interconnect-level maps of powered test structures, the scale at which present quantum diamond current imaging has a clear metrological advantage over destructive sectioning.

Electromigration stress imaging

A subset of 0.8 um lines and via chains was stressed at current density near 10^7 A cm^-2 for up to 18 h. Magnetic maps were acquired every 12 min during the first hour and every 45 min thereafter. Four-terminal resistance increased smoothly for most devices until a final abrupt jump. The magnetic maps revealed spatial redistribution before that electrical signature became large.

In narrow straight lines, current shifted away from one edge over several hours, consistent with progressive void formation or local resistance increase near that edge. The reconstructed centroid of current moved by 310 +/- 80 nm before the line resistance increased by 5%. After failure, optical inspection showed passivation discoloration close to the predicted high-resistance region. The experiment is not a substitute for cross-section confirmation, but it gives a time-resolved failure hypothesis.

Via chains showed a different pattern. Current crowding at the first two vias decreased while crowding at downstream vias increased, suggesting redistribution through parallel paths as one via resistance rose. The magnetic maps therefore distinguished a distributed degradation process from a single sudden open. This distinction matters for reliability models because a chain can accumulate local damage before the measured resistance crosses a specification limit.

Thermal drift was the main challenge during stress imaging. The ODMR temperature correction removed most common-mode shifts, but local heating near stressed lines produced residual artefacts in the reconstructed current. We therefore report stress-time trends only when they are larger than the propagated thermal uncertainty.

Comparison with other current-imaging approaches

Scanning probe magnetic microscopes, electron-beam probing, infrared thermography, lock-in thermoreflectance, and focused-ion-beam cross-sectioning all have roles in failure analysis. NV diamond current imaging is not a universal replacement. Its advantage is a rare combination: it is non-contact, works under ambient or cryogenic conditions, can image through passivation, and returns a magnetic field that is quantitatively linked to current.

Compared with single-NV scanning magnetometry, the wide-field approach sacrifices ultimate spatial resolution for throughput. Single-spin current-density imaging has demonstrated nanoscale capability in favourable geometries [19], while wide-field imaging has proven effective for larger current distributions such as graphene devices and microelectronic structures [13,14,15,16,20]. For interconnect reliability, field of view and non-destructive operation are often more valuable than nanometre standoff.

Compared with purely electrical tests, the method separates parallel paths. Two structures can have the same total resistance but different current distributions. Compared with thermal imaging, it detects current directly rather than inferring it from heat spreading. Compared with destructive microscopy, it can be repeated during stress. The weaknesses are also clear: buried-layer separation is ill-conditioned, magnetic shielding or ferromagnetic package materials can distort fields, and the inversion requires design information.

The best use case is therefore guided failure analysis. A quantum diamond microscope can identify where current crowding or redistribution is likely, after which cross-sectioning, transmission electron microscopy, or chemical analysis can confirm the physical damage.

Uncertainty budget

The uncertainty budget was assembled from repeated measurements, calibration standards, and inversion sensitivity. For straight lines wider than 2 um, amplitude uncertainty was dominated by magnetic noise and microwave-frequency calibration. For narrow lines and vias, standoff uncertainty dominated because high spatial frequencies decay rapidly with distance. A 100 nm standoff error produced a 6-14% change in peak current density for 0.8 um lines.

Temperature contributed two errors. First, temperature shifts the NV zero-field splitting [22]. Second, it changes metal resistance during current stressing, so the electrical current density inferred from nominal geometry can drift even if the applied current is constant. Current reversal and reference-region subtraction reduce the first error; independent resistance thermometry helps with the second.

Mask alignment contributed 40-110 nm uncertainty depending on optical contrast. This is small compared with the wide-field resolution but important near narrow vias. Regularisation choice contributed little to line-integrated currents but strongly affected pixel-scale peak values. We therefore recommend reporting line-integrated current and peak-to-mean ratios over physically meaningful regions, not isolated pixel maxima.

The total uncertainty for line-integrated current was 5-8% in reference lines, 9-14% in via chains, and 15-24% in two-layer return-path structures. These values are sufficient for reliability screening and model validation, but not for certifying local current density at the scale of individual grains or liners.

Limitations

The method assumes that the relevant current paths are close enough to the diamond sensor for their magnetic fields to survive spatial filtering. Deep package layers can still be measured if currents are large, but fine features are lost. Inverse algorithms cannot restore information that never reaches the sensor plane.

The second limitation is layer ambiguity. A magnetic map above a chip is a projection of all currents. When multiple metal layers have similar orientation and spacing below the standoff length, the layer decomposition becomes non-unique without strong design constraints. This is why mask-constrained inversion is useful, but it also means that the analysis depends on layout metadata.

The third limitation is material compatibility. Ferromagnetic films, magnetic probe fixtures, or large external currents can create backgrounds that reduce dynamic range. Microwave delivery can also perturb sensitive circuits if shielding is inadequate. These engineering details are not glamorous, but they determine whether the method is usable outside a physics laboratory.

Finally, NV magnetometry measures magnetic fields, not damage. A current map can suggest where voiding, local heating, or resistance increase is occurring, but it cannot identify chemistry or microstructure. The method should be paired with conventional failure-analysis tools when root cause is required.

Conclusion

Wide-field NV diamond magnetometry can quantitatively map current density in powered microelectronic interconnect test structures through passivation. In the present workflow, vector ODMR imaging, current-reversal subtraction, standoff calibration, and mask-constrained Biot-Savart inversion produced line-integrated current estimates within 6.8% for reference lines and identified current crowding in bends, vias, and return paths. During electromigration stress, the method tracked current redistribution before large resistance changes appeared.

The practical message is measured optimism. Diamond quantum sensors are not magic microscopes that see arbitrary buried nanoscale currents. They are calibrated magnetic imagers whose strengths are non-destructive access, vector field measurement, and quantitative inversion at micron-scale interconnect features. That is already enough to make them useful for reliability studies, process debugging, and package-level current-path verification.

Data and code availability

The supplementary archive contains raw ODMR image stacks, magnetic-field maps, test-vehicle layouts, inversion scripts, uncertainty-propagation notebooks, finite-element models, and processed current-density maps. Proprietary-like layout labels in the fictional test vehicle are replaced by anonymised net names, but geometry and layer coordinates are preserved for reproducibility.

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