⚛️ quantum physics

Josephson energy of superconducting junctions: amorphous versus crystalline tunnel barriers

Using first-principles NEGF-DFT modeling, this study demonstrates that the Josephson energy in superconducting junctions is exponentially sensitive to the atomic structure of the tunnel barrier, revealing that amorphous Al $_2$ O $_3$ barriers exhibit significantly higher and more variable energy values than crystalline ones due to stoichiometric inhomogeneity creating percolation-like tunneling pathways.

Original authors: Wanting Zhang, Aldilene Saraiva-Souza, Félix Beaudoin, Xianghua Kong, Hong Guo, Yu Zhu

Published 2026-09-10

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CC BY 4.0

Original authors: Wanting Zhang, Aldilene Saraiva-Souza, Félix Beaudoin, Xianghua Kong, Hong Guo, Yu Zhu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet world of superconducting circuits, where electricity flows without resistance, tiny components called Josephson junctions act as the essential switches and valves. These junctions are the heart of the most advanced quantum computers, specifically a type of qubit known as the transmon. The behavior of these qubits is governed by a specific energy value, which determines how fast the computer can think and how clearly it can distinguish between different states of information. This energy depends entirely on how easily electrons can tunnel through an incredibly thin barrier of aluminum oxide, a layer only one or two nanometers thick. Because this barrier is so thin, its atomic arrangement—whether the atoms are lined up in a perfect grid or jumbled in a random, glass-like structure—has a massive impact on how the electrons move. Understanding exactly how this microscopic disorder affects the energy of the circuit is crucial for building reliable quantum machines, yet the connection between the chaotic atomic structure of the barrier and the macroscopic energy of the device has remained difficult to pin down.

A team of researchers has now mapped this connection by building detailed computer models of these junctions, comparing a perfectly ordered, crystalline barrier against ten different versions of a disordered, amorphous barrier. Using a powerful method that combines the laws of quantum mechanics with the specific arrangement of every atom, they simulated how electrons travel through these structures. The study focused on aluminum oxide barriers of the same thickness but with different internal architectures. In the crystalline model, the atoms form a neat, repeating pattern, while in the amorphous models, the atoms were generated by a process that mimics melting and rapid cooling, resulting in a random, frozen-in disorder. The researchers then calculated the transmission of electrons through these barriers to determine the resulting Josephson energy, a key parameter that sets the operating frequency of the qubit.

The results revealed a striking difference between the two types of barriers. For the single crystalline junction, the calculated energy was a steady 0.73 gigahertz. However, the ten amorphous junctions showed a wild spread of values. While the average energy for the amorphous group was 2.78 gigahertz, the individual results varied dramatically, ranging from as low as 0.18 gigahertz to as high as 15.1 gigahertz. This means that the disordered samples did not just produce a slightly different result; they produced outcomes that spanned nearly two orders of magnitude. The distribution was heavily skewed, with most samples clustering near the crystalline value, but a few outliers with extremely high energy pulling the average up. This variability suggests that the random arrangement of atoms in the amorphous barrier creates a landscape where the path for electrons is not uniform, but instead depends on the specific, accidental arrangement of atoms in each sample.

To understand why this happens, the researchers looked closely at how the electrons moved through the barriers. They found that the transport mechanism is quantum tunneling, where electrons pass through the barrier even though they lack the energy to climb over it. In the amorphous samples, the chemical composition of the oxide is not perfectly uniform; some regions are richer in aluminum, while others are richer in oxygen. The aluminum-rich areas act as lower, easier barriers for the electrons to cross, while the oxygen-rich areas are higher and harder to cross. In some of the amorphous samples, these low-barrier regions happened to line up, creating a continuous, percolation-like path that allowed electrons to flow much more easily than they would through a uniform barrier. In other samples, these easy paths did not connect, forcing the electrons to struggle through higher barriers. This explains why some amorphous junctions had energy values far higher than the crystalline reference, while others were lower.

The study also addressed how these microscopic findings relate to the real-world devices used in laboratories. The computer models represented tiny slices of material, roughly 1.44 by 1.25 nanometers in size. A real Josephson junction used in a quantum computer is much larger, typically measuring 200 by 200 nanometers. This means a real device contains more than 20,000 of these microscopic regions. Because the real junction is so large, it effectively averages out the extreme variations seen in the tiny models. The final energy of a real device is likely a blend of many different microscopic pathways, smoothing out the wild swings seen in the individual simulations. This suggests that while the atomic disorder creates significant variability at the smallest scale, the macroscopic device benefits from a self-averaging effect that stabilizes its performance.

The researchers concluded that the variability in the Josephson energy is a direct consequence of the stoichiometric inhomogeneity—the uneven mix of aluminum and oxygen—within the amorphous barrier. The presence of aluminum-rich, low-barrier regions that can connect across the oxide creates pathways that strongly enhance conductance, leading to the high-energy outliers. Conversely, samples without these connected paths can be even less transparent than the perfect crystal. While the study relied on simulations and the specific mathematical methods used tend to slightly underestimate the energy gap of the oxide, the trends observed provide a clear, quantitative route from the microscopic structure of the oxide to the energy scale of the superconducting circuit. This work establishes that the random atomic structure of the barrier is not just a minor detail, but a fundamental factor that dictates the energy landscape of the quantum computer, offering a new way to think about how these devices are built and how their performance might be optimized.

Technical Summary: Josephson Energy of Superconducting Junctions: Amorphous versus Crystalline Tunnel Barriers

Problem Statement
The Josephson energy ( $E_J$ ) is a fundamental parameter determining the frequency and anharmonicity of transmon superconducting qubits. In Al/AlO $_x$ /Al junctions, $E_J$ is governed by electron tunneling through an ultrathin oxide barrier, exhibiting an exponential dependence on the barrier's atomic structure. While macroscopic circuit models treat the junction as a lumped element, the actual barrier is structurally and chemically nonuniform, containing thickness variations, oxygen deficiencies, and stoichiometric inhomogeneities. Existing atomistic models have largely focused on crystalline barriers or specific interface terminations. However, realistic AlO $_x$ barriers are amorphous, introducing stochastic variations in atomic configuration that a single structure cannot represent. The challenge lies in establishing a quantitative, atomistic route from the microstructure of amorphous oxide barriers to the macroscopic energy scale $E_J$ .

Methodology
The authors employ a first-principles device modeling workflow combining Density Functional Theory (DFT) with the Non-Equilibrium Green's Function (NEGF) formalism to simulate quantum transport through Al/Al $_2$ O $_3$ /Al junctions.

Atomic Structure Generation:
- Crystalline Reference: A commensurate supercell of Al(111)– $\alpha$ -Al $_2$ O $_3$ –Al(111) was constructed with 420 atoms. The oxide was terminated asymmetrically (Al on one side, O on the other).
- Amorphous Ensemble: Ten independent amorphous Al $_2$ O $_3$ barriers were generated using a melt–quench molecular dynamics protocol in LAMMPS (using the Matsui force field). These were subsequently inserted between crystalline Al(111) slabs and relaxed using the machine-learning interatomic potential CHGNet to handle the heterogeneous metal–oxide interfaces. Each amorphous junction contains 1080 atoms.
- Geometry: The crystalline and amorphous models possess different lateral periodicities and cross-sectional areas ( $A_0$ ), which are explicitly accounted for in the transport analysis.
Quantum Transport Calculation:
- Transport calculations were performed using the NanoDCAL package with a linear combination of atomic orbitals (LCAO) basis and norm-conserving pseudopotentials (GGA level).
- The energy-dependent transmission coefficient $T(E)$ was computed for the relaxed junctions.
- The normal-state conductance ( $G_N$ ) was derived from the Fermi-level transmission $T(E_F)$ via the Landauer formula, scaled by the simulation cell area $A_0$ and the device area $A$ (set to $200 \times 200$ nm $^2$ for comparison).
- The critical current ( $I_c$ ) and Josephson energy ( $E_J$ ) were extracted using the zero-temperature Ambegaokar–Baratoff relation and the definition $E_J = \hbar I_c / 2e$ , assuming a superconducting gap $\Delta = 0.18$ meV.
Scattering State Analysis:
- To understand the transport mechanism, the authors analyzed the probability of scattering states at the Fermi level, specifically looking for evanescent decay patterns and real-space wavefunction localization.

Key Results

Variability in $E_J$ : The ten amorphous junctions exhibit a wide distribution of Josephson energies. The mean $E_J/h$ is 2.78 GHz with a standard deviation of 4.67 GHz. In contrast, the crystalline reference yields 0.73 GHz.
Skewed Distribution: The distribution is strongly skewed. The median value (0.86 GHz) is close to the crystalline reference, but individual values span nearly two orders of magnitude, ranging from 0.18 GHz (Sample 5) to 15.1 GHz (Sample 6).
Transmission Characteristics: Area-normalized transmission curves ( $T(E)/A_0$ ) show a similar insulating gap structure for all samples. However, the position of the Fermi level relative to the transmission valley varies significantly between samples. Small shifts in the valley position relative to $E_F$ result in large variations in $T(E_F)$ due to the exponential nature of tunneling.
Transport Mechanism: Scattering-state analysis confirms that transport occurs via quantum tunneling. The variability arises from stoichiometric inhomogeneity within the amorphous oxide.
- Percolation Pathways: In high- $E_J$ samples, Al-rich regions (which have a lower effective barrier height) connect across the oxide, forming percolation-like tunneling pathways. This allows electrons to traverse the barrier more easily, significantly enhancing conductance.
- Localization: In low- $E_J$ samples, such connected low-barrier paths are absent, and scattering states remain localized near the incident electrode, resulting in lower transparency than even the crystalline junction.

Significance and Claims
The paper establishes a quantitative link between the microscopic atomic structure of amorphous oxide barriers and the macroscopic Josephson energy of superconducting circuits. The authors claim that the large variance in $E_J$ observed in amorphous junctions is not random noise but a direct consequence of the stochastic formation of percolation-like tunneling channels driven by local stoichiometric fluctuations.

Crucially, the authors note that while individual atomistic models show extreme variability, a realistic macroscopic junction (e.g., $200 \times 200$ nm $^2$ ) encompasses over $2 \times 10^4$ such microscopic regions. Consequently, the device-level $E_J$ is expected to "self-average" over these configurations, reflecting an ensemble average rather than the extreme outliers seen in the small simulation cells.

Limitations and Future Directions
The authors acknowledge that the GGA functional used likely underestimates the oxide band gap, potentially leading to slightly overestimated absolute transmission values, though the trends across samples should remain robust. They suggest that future work should employ GGA+U or hybrid functionals (e.g., HSE) for more quantitative accuracy. Additionally, they propose that computing a larger ensemble of amorphous samples would allow for a proper statistical distribution of $E_J$ to be compared directly with experimental junction-to-junction spreads.

Technical Summary: Josephson Energy of Superconducting Junctions: Amorphous versus Crystalline Tunnel Barriers

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