Sidewall Metrology Protocol for Superconducting Qubit Devices

AI summary

Overview

Research area: Quantum hardware metrology — specifically, the quantitative characterization of sidewall geometry in superconducting qubit devices using scanning probe microscopy (AFM).

Technical level: Intermediate. The physics motivation (dielectric loss, participation ratios, transmon coherence) is explained at an accessible level, but the descriptor mathematics, filter cutoffs, and uncertainty budget require familiarity with surface metrology and roughness analysis. Advanced readers will find the uncertainty table and probe-deconvolution discussion most useful.

Scope (one sentence): The paper proposes and experimentally demonstrates a standardized AFM-based measurement protocol, with two morphological descriptors and an uncertainty budget, for quantifying the sidewall roughness, curvature, and angle of superconducting qubit components.

What This Paper Is About

Sidewall profiles and angles are suspected to influence superconducting qubit coherence, but there is no standardized way to measure them quantitatively — cross-sectional FIB-SEM and HR-TEM give high-resolution images but struggle to deliver statistically representative numbers across a device. The authors build a scanning-probe-metrology framework that turns tilted-sample AFM data into reproducible numbers: a lateral roughness descriptor, a new curvature-based descriptor for the vertical profile, and procedures for locating the sidewall region and extracting sidewall angles. The goal is a protocol that lets different laboratories compare fabrication processes on equal terms, as a prerequisite for eventually correlating sidewall geometry with qubit performance.

Key Contributions

1. A refined RMS lateral sidewall roughness descriptor, $R_q^{\mathrm{LSP}}$. Defined as the average, over $N$ measured profiles, of the band-pass-filtered RMS amplitude of the lateral sidewall profile $X_{\mathrm{LSP},i}(y)$, i.e. the sidewall position projected onto a plane parallel to the substrate.

2. A new curvature-equivalent radius, $R_\rho$, for vertical sidewall morphology. Because a residual-amplitude descriptor has no meaningful reference profile in the vertical (cross-sectional) direction, the authors instead define $R_\rho$ as the inverse of the profile-averaged RMS curvature. Small $R_\rho$ means many sharp features with small local radii of curvature; large $R_\rho$ means smoother sidewalls. An auxiliary minimum radius of curvature, $\rho_{\min}$, is also defined.

3. A sidewall region detection framework and an angle-distribution analysis. Three algorithms — height-based, gradient-based, and cumulative-summed-gradient-based detection — are compared for defining the foot, midpoint, and top of the sidewall. Local sidewall angles are then binned into normalized probability density functions (PDFs) on either side of the sidewall midpoint, characterized by median $\bar{\alpha}$ and interquartile range (IQR).

4. A complete measurement protocol with a quantified uncertainty budget. Table 1 collects the descriptors, probe specification requirements, measurement parameters, and data-processing steps; Table 2 lists sources of variability, mitigation strategies, and residual variation after mitigation, yielding an estimated combined measurement uncertainty ($k=1$) of 15–30%.

Main Findings

• Lateral roughness spans roughly an order of magnitude across components. $R_q^{\mathrm{LSP}}$ is approximately 2–5 nm for Josephson junction structures and approximately 20 nm for readout resonators; a sidewall containing a visible local pit reaches approximately 30 nm. Profile-to-profile variation within the same image area, $\sigma(R_{q,i}^{\mathrm{LSP}})$, is approximately 0.3–0.6 nm for qubit components and up to approximately 5 nm for readout resonators.

• Vertical morphology separates electrodes from capacitor pads. The curvature-equivalent radius $R_\rho$ is 35–82 nm for Josephson junction electrodes and 51–151 nm for capacitor pads across devices. Variation between profiles taken at the same spot is 2–10 nm and is generally larger for larger $R_\rho$.

• A single profile is not representative. Most profiles on the same chip are qualitatively similar, but one capacitor-pad profile (device 4) is significantly different, leading the authors to state that one edge profile is insufficient to characterize the angle and shape of a chip's sidewall.

• Sharp protrusions reach the probe resolution limit. In the single capacitor profile of device 2, a distinct sharp protrusion near the top of the aluminum pad has an observed local minimum radius $\rho_{\mathrm{OBS}} = -21.8$ nm (negative for protrusions), equal to the conservative resolution-limit estimate of the probes. After correcting for probe shape, $\rho = \rho_{\mathrm{OBS}} - \rho_{\mathrm{probe}}^{\mathrm{eff}} = -21.8\ \mathrm{nm} + 10.9\ \mathrm{nm} = -10.9$ nm, with an associated probe-to-probe uncertainty of 6.3 nm.

• Sidewall angle distributions are wide, asymmetric, and sometimes bimodal. For the Josephson junction element (device 5): $\bar{\alpha}{\mathrm{JJ,foot\to mid}} = 51^\circ$ with $\mathrm{IQR} = 53^\circ$, and $\bar{\alpha}{\mathrm{JJ,mid\to top}} = 79^\circ$ with $\mathrm{IQR} = 27^\circ$. For the coupling element (device 6): $\bar{\alpha}{\mathrm{Coupl,foot\to mid}} = 47^\circ$ with $\mathrm{IQR} = 28^\circ$, and $\bar{\alpha}{\mathrm{Coupl,mid\to top}} = 21^\circ$ with $\mathrm{IQR} = 13^\circ$. The authors argue that such shallow, broad, non-unimodal distributions problematize describing sidewall geometry by a single angle.

• Probe choice dominates cross-laboratory comparability. Table 2 reports standard uncertainties before mitigation of 10–20% (probe-to-probe, $R_q^{\mathrm{LSP}}$), 5–8 nm (probe-to-probe, $R_\rho$), 5–15% (probe degradation and instability), up to roughly 50% (particles and contamination), up to roughly 20% (additional experimental parameters such as applied force), and up to roughly 10% (data processing). With the listed mitigation strategies — single pre-characterized probe, probe stability monitoring, particle removal, more detailed parameter specification, standardized processing — residual variations fall to ≤4% or 3–5 nm in most cases, while spatial inhomogeneity of the sample "varies" and cannot be mitigated to a fixed number.

• Probe geometry is treated as part of the method, not an error term. The effective probe radius of the batch used, including wear, is $\rho_{\mathrm{probe}}^{\mathrm{eff}} = 10.9$ nm with $u(\rho_{\mathrm{probe}}^{\mathrm{eff}}) = 6.3$ nm. Because traceable reference standards for validating reconstructed probe geometries below 10 nm are, to the authors' knowledge, unavailable, the recommendation is to report descriptors together with the probe geometry and data processing — analogous to specifying stylus geometry in conventional roughness measurement.

• Total uncertainty and calibration. The microscope-calibration contribution is estimated below 3% based on certified reference standards; the combined measurement uncertainty ($k=1$) for the morphological descriptors is estimated at 15–30%.

• Filtering sets a hard lower limit on observable curvature. Short-wavelength cutoffs of $\lambda_S = 8$ nm for capacitor profiles and $\lambda_S = 25$ nm for electrode profiles were applied, which strongly attenuates features well below these spatial wavelengths and therefore limits the possible observation of radii of curvature well below those values.

• Roughness is framed as a proxy, not a quality metric. The paper explicitly states that no universal monotonic relation between surface roughness and internal quality factors should be expected across materials and fabrication processes, citing that electronic disorder in disordered or granular superconductors can increase $T_c$ or kinetic inductance while preserving low losses.

Methodology in Plain English

The team measures sidewalls by physically tilting the sample so the atomic force microscope's tip can scan down the wall rather than only seeing the top surface — a configuration that also reaches undercut structures. Because the recorded profile is the real surface smeared by the tip's own rounded shape, they first estimate an effective tip radius from sharp protrusions on a probe-characterization sample, then subtract that radius from observed protrusion curvatures.

To turn raw height traces into comparable numbers, they cut each profile into two families: lateral profiles, which give the horizontal position of the wall at various heights (used for roughness), and vertical profiles, which give the cross-section of the wall (used for curvature). Both are band-pass filtered to remove long-wavelength drift and short-wavelength noise. Locating the foot, midpoint, and top of the wall is done by three competing rules — a height threshold (2% of the step height at each end), a gradient threshold, or a threshold on the cumulative squared gradient — and the authors compare which behaves most robustly on real device edges. Sidewall angles are then histogrammed into probability densities above and below the midpoint, giving a shape fingerprint instead of a single number.

For the sample set, they study seven wet-etched transmon devices (devices 1–7) with 200 nm aluminum on high-resistivity silicon, fabricated with e-beam lithography, Transene Type D wet etching, and Manhattan-style double-angle Josephson junction evaporation, plus five additional readout resonators from an alternative process (devices 8–12) on C-plane sapphire with 100 nm aluminum, wet-etched using ma-D 332. Measurements were made on a Park NX20 AFM in intermittent-contact mode. The final protocol is packaged as a table of descriptors, probe specifications, measurement parameters, and analysis steps, with a second table assigning an uncertainty and a mitigation strategy to each source of variability.

Why This Matters

The paper addresses a gap that sits between fabrication and quantum performance: many groups suspect sidewall geometry contributes to loss and device-to-device variability, but without a shared descriptor set, one laboratory's "rough sidewall" cannot be compared to another's. By defining measurands together with probe specifications and an uncertainty budget, the protocol makes fabrication comparisons potentially traceable and repeatable across institutions — a prerequisite for any statistically meaningful correlation with coherence times.

• Fabrication process benchmarking: comparing wet- versus dry-etched, or different chemistries and resists, using a common roughness and curvature vocabulary rather than SEM images.
• Inter-laboratory metrology comparisons: the uncertainty table is explicitly structured to separate short-timescale repeatability (same probe) from cross-laboratory comparability (different probes, different times).
• Josephson junction process development: junction electrodes show the smallest $R_q^{\mathrm{LSP}}$ (2–5 nm) and tightest $R_\rho$ range (35–82 nm), giving a baseline for evaluating junction fabrication changes where the electric field concentrates.
• Resonator loss studies: readout resonators show the largest lateral roughness (approximately 20 nm, up to approximately 30 nm at a pit), making them the natural first target for roughness-versus-quality-factor studies.

Industry relevance: superconducting qubit foundries and process-integration teams need yield and consistency metrics that are cheaper and more statistically tractable than cross-sectional TEM. A tilted-AFM protocol with a stated 15–30% combined uncertainty is a practical candidate for incoming inspection, etch-tool qualification, and supplier comparison — provided the probe-specification requirement is respected, since probe-to-probe variation (10–20% before mitigation) is identified as a leading limit on comparability.

Future Directions

• A comprehensive uncertainty analysis, which the conclusion explicitly names as future work beyond the current estimate of 15–30% ($k=1$).
• Additional morphological descriptors extending beyond $R_q^{\mathrm{LSP}}$, $R_\rho$, and $\rho_{\min}$.
• Statistical correlations between sidewall characteristics and device performance — the stated long-term motivation, which the paper does not yet attempt, and which the authors caution requires first understanding the relative contributions of the various decoherence channels.
• Cross-laboratory validation of the protocol, implicit in the goal of comparing fabrication processes across laboratories; the residual variations in Table 2 for $s(R_\rho)$ are described only as approximate estimates based on experimental observations and laboratory experience, so independent replication would test them.

An open question the paper raises but does not resolve is how sidewall granularity and topography interact with superconducting electrodynamics at finite penetration depth — for a 200 nm Al film the authors cite a penetration depth around $\lambda = 60$ nm and a coherence length of roughly $\xi \approx 100$–250 nm depending on mean free path, scales comparable to the 10–100 nm curvature range being measured.

Target Audience

Metrologists and instrumentation scientists working on nanoscale surface characterization; superconducting qubit process engineers and fabrication teams who need quantitative rather than anecdotal sidewall metrics; and quantum computing experimentalists who want to test whether sidewall geometry correlates with $T_1$ and resonator quality factors. Beginners will get the physics motivation from Section II but will need background in roughness parameters ($R_q$, filter cutoffs, stylus/AFM convolution) to use the protocol. Readers should note that the text supplied here is truncated mid-sentence in Section VI.3 (instrumentation and measurement), so the remaining measurement details and several appendices — including the full curvature definitions in Appendix B, the probe-radius estimation in Appendix E, validity criteria in Appendix D, and mitigation strategies in Appendix G — are not available for this summary.

Authors’ abstract

Sidewall morphology is a potentially important factor influencing superconducting qubit performance, yet its quantitative characterization remains challenging due to the absence of standardized measurement protocols. In this work, we develop a scanning probe microscopy-based measurement framework for the quantitative characterization of superconducting qubit sidewalls. The framework introduces a refined root mean square lateral sidewall roughness descriptor and a new curvature-equivalent radius for quantifying vertical sidewall morphology. The descriptors and measurement procedures are applied to different components of qubit devices, forming the basis of a proposed sidewall characterization protocol. This provides a foundation for quantitative comparisons of fabrication processes across laboratories, enabling future investigations of correlations between sidewall geometry and qubit performance.