Device variability of Josephson junctions induced by interface roughness

Technical Summary: Device Variability of Josephson Junctions Induced by Interface Roughness

Problem Statement
As superconducting quantum processors scale toward the thousands of qubits required for fault-tolerant quantum computation (e.g., Shor's algorithm), device-to-device variability has emerged as a critical bottleneck. Superconducting qubits, particularly transmons, rely on Al/AlOx/Al Josephson junctions operating in the tunneling regime. In this regime, the critical current $I_c$—and consequently the Josephson energy $E_J$—depends exponentially on microscopic barrier details. Even minor variations in the Al/AlOx interface geometry, such as local thickness fluctuations or contamination, can induce substantial shifts in qubit transition frequencies and gate fidelities. While extensive experimental efforts have focused on improving junction reproducibility through process optimization and microscopy, a quantitative theoretical framework connecting microstructural disorder (specifically interface roughness) to junction-level energy statistics has been lacking.

Methodology
The authors develop a quantitative modeling framework that bridges atomic-scale interface disorder with macroscopic junction transport properties. The approach integrates three primary components:

1. Interface Roughness Modeling: The Al/AlOx interface is modeled as a Gaussian random field $h(x,y)$ characterized by two parameters: the root-mean-square (RMS) roughness amplitude $\sigma$ and the transverse correlation length $\xi$. These parameters are derived from existing experimental literature (e.g., TEM studies of grain sizes and thickness histograms) and new molecular dynamics (MD) simulations. The MD simulations, utilizing the CHGNet machine learning potential, model the oxidation of an Al surface to estimate $\sigma$ under idealized conditions.
2. Quantum Transport Approximation: To address the computational intractability of solving the Bogoliubov–de Gennes (BdG) equation for realistic junction sizes ($\sim 100 \text{ nm} \times 100 \text{ nm}$), the authors employ two key approximations:
   • Ambegaokar–Baratoff (AB) Relation: The critical current is calculated via the AB relation, linking $I_c$ to the normal resistance $R_N$. This reduces the computational complexity by avoiding integration over the full energy spectrum and focusing on the transmission coefficient at the Fermi energy.
   • Local Thickness Approximation: Given that the Fermi wavelength ($\lambda_F \approx 3.6$ Å) and barrier decay length ($\lambda_D \approx 1.9$ Å) are much smaller than the correlation length of the roughness ($\xi \sim 10$ nm), the junction is treated as an ensemble of independent tunneling channels. The total conductance is approximated as the integral of local conductance densities, where the local barrier thickness varies according to the roughness profile.
3. Numerical Simulation: The model is applied to an ensemble of 5,000 Josephson junctions with varying tunnel barrier profiles generated from the Gaussian random field model.

Key Results
The numerical simulations yield several quantitative insights into the statistical behavior of $E_J$:

• Log-Normal Distribution: The distribution of Josephson energy $E_J$ across the ensemble is not Gaussian but follows a log-normal distribution. This arises because $E_J$ depends exponentially on the barrier thickness, which is normally distributed. Consequently, the probability density function is skewed toward larger values.
• Impact of RMS Roughness ($\sigma$): Increasing the RMS roughness amplitude $\sigma$ leads to a significant increase in both the mean value $\langle E_J \rangle$ and the variance $\text{Var}(E_J)$. The increase in the mean is attributed to the exponential sensitivity of tunneling, where rare, anomalously thin regions dominate the total conductance.
• Impact of Correlation Length ($\xi$): Increasing the transverse correlation length $\xi$ causes a slight decrease in the mean $\langle E_J \rangle$ but a marked increase in the variance $\text{Var}(E_J)$. The variance increases because a larger $\xi$ reduces the number of statistically independent tunneling channels (self-averaging), thereby enhancing sample-to-sample fluctuations. The mean decreases slightly due to finite-size effects; as $\xi$ approaches the junction dimensions, the probability of realizing the rare, extremely thin barrier regions that dominate tunneling is reduced.
• Qubit Frequency Variability: Using the derived statistics for $E_J$ and assuming a typical charging energy $E_C$, the model predicts a standard deviation in the qubit transition frequency $\omega_{01}/2\pi$ of approximately 0.364 GHz around a mean of 6.145 GHz for the specific parameters tested.

Significance and Claims
The paper claims to provide a "quantitative and intuitive picture" of how surface roughness induces variability in Josephson energy. By connecting microstructural parameters ($\sigma$ and $\xi$) directly to junction-level statistics, the work offers a framework for understanding yield and uniformity challenges in large-scale quantum processors. The authors emphasize that their results have "direct relevance for junction design," suggesting that controlling interface roughness parameters is crucial for minimizing device variability.

The authors maintain a modest stance regarding the scope of their work. They note that their model assumes $\sigma$ and $\xi$ are independent and identical for both interfaces in the junction, an assumption that requires experimental verification under diverse growth conditions. Furthermore, they identify future directions as connecting this model to experimental characterization of fabricated junctions and integrating their superconductivity solvers with microwave engineering software to enable the co-design of junctions and circuit elements. The paper does not propose new experimental protocols or claim to solve the variability problem entirely, but rather establishes a theoretical baseline for quantifying the impact of interface roughness.