🔬 applied physics

Fabrication Optimization of Suspended Stencil Mask Lithography for Multi-Terminal Josephson Junctions

This study systematically optimizes the fabrication of multi-terminal Josephson junctions using suspended stencil mask lithography by evaluating 270 mask designs to establish reliable yield, minimal dimensions down to 40 nm, and dependencies on terminal count.

Original authors: Justus Teller, Abdur Rehman Jalil, Florian Lentz, Detlev Grützmacher, Thomas Schäpers

Published 2026-03-26

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CC BY 4.0

Original authors: Justus Teller, Abdur Rehman Jalil, Florian Lentz, Detlev Grützmacher, Thomas Schäpers

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

Imagine you are a master chef trying to build a very delicate, multi-layered sandwich. But there's a catch: you can't touch the ingredients with your hands, and you can't let any air or dust get between the layers, or the sandwich will spoil instantly.

This is exactly the challenge scientists face when building Josephson Junctions. These are tiny, super-conducting bridges used in quantum computers. To make them work perfectly, the layers of metal and semiconductor must be built in a vacuum so clean that not even a single atom of dust can land on them.

The Problem: The "Shadow" Technique

Traditionally, to build these bridges, scientists use a technique called lithography, which is like using a stencil to spray paint a design. But in the quantum world, you can't just lift the stencil off and put it back down; the vacuum chamber is too sensitive.

So, the researchers use a Suspended Stencil Mask.

The Analogy: Imagine holding a piece of paper with a tiny hole cut in it, suspended in mid-air just above a table.
The Process: You spray a layer of "paint" (superconducting metal) from above. The paper blocks the paint everywhere except where the hole is. When you lift the paper, you are left with a perfect, clean bridge of paint on the table.
The Goal: They want to make these bridges incredibly narrow (to make the quantum connection strong) and they want to connect three or four different "wires" (terminals) to a single central point, like a starfish shape.

The Experiment: The "Paper Bridge" Test

The problem is that if the hole in your paper stencil is too narrow, the paper itself might snap under its own weight or the pressure of the spray. If the stencil breaks, the whole experiment fails.

The team at Jülich Research Center in Germany decided to play a massive game of "How small can we go before it breaks?"

The Setup: They designed 270 different stencils.
- Some had 2 arms (like a dumbbell).
- Some had 3 arms (like a peace sign).
- Some had 4 arms (like a plus sign).
The Variables: They changed the width of the "arms" and the width of the "center" (the hole).
The Test: They built all of them and looked at them under a super-powerful microscope (SEM) to see which ones survived and which ones snapped.

The Findings: The "Goldilocks" Zone

Here is what they discovered, translated into everyday terms:

The More Arms, The Stronger the Bridge: Surprisingly, stencils with 4 arms could be made with a smaller center hole than stencils with just 2 arms.
- Why? Think of a 4-armed starfish. The arms support each other like a tripod. A 2-armed dumbbell is like a tightrope walker; if the rope is too thin, it snaps. The 4-arm design distributes the stress better, allowing the center to be incredibly tiny (down to 30 nanometers—that's 30 billionths of a meter!).
The "Real" Size vs. The "Planned" Size: When they tried to make the tiniest holes, the actual holes ended up slightly larger than they planned.
- The Analogy: It's like trying to draw a perfect circle with a thick marker. No matter how carefully you aim, the ink spreads a little. The "ink" here is the chemical etching process, which behaves differently depending on the angles of the corners.
The Sweet Spot: They found that for most designs, they could reliably make a hole as small as 40 nanometers without the stencil breaking. This is small enough to create the super-strong quantum connections needed for future computers.

Why Does This Matter?

Think of a quantum computer as a symphony orchestra.

Old way: You have pairs of musicians (2 terminals) playing together.
New way: This research allows us to have a whole section of musicians (3 or 4 terminals) all playing from the exact same sheet of music at the exact same time, meeting in the center.

This creates a "central scattering" effect, which is like a magical meeting point where new, exotic states of matter (like topological states) can be born. These states are the building blocks for fault-tolerant quantum computers—machines that won't crash when the temperature changes or a tiny vibration occurs.

The Takeaway

The scientists successfully mapped out the "blueprint" for building these microscopic stencils. They proved that by designing the mask with the right number of arms and the right angles, we can build the tiniest, cleanest, and most complex quantum bridges ever made.

It's like they just handed the world a new, ultra-precise cookie cutter that can cut shapes so small you need a microscope to see them, paving the way for the next generation of super-computers.

1. Problem Statement

Multi-terminal Josephson junctions (MTJJs) are critical for advanced quantum devices, including topological quantum computing and artificial topological states (e.g., Weyl singularities). A key requirement for these devices is the realization of a central scattering region where multiple superconducting electrodes converge closely at a single weak link.

Challenge: Fabricating these junctions with extremely small electrode separations (nanometer scale) while maintaining mechanical stability is difficult.
Limitation of Current Methods: Traditional resist-based lithography often requires breaking the ultra-high vacuum (UHV) for oxidation or contamination, degrading interface quality. While Stencil Mask Lithography allows for in-situ fabrication (preserving clean interfaces via MBE or evaporation), the suspended masks are mechanically fragile.
Specific Gap: There was a lack of systematic data on the design limits of suspended stencil masks. Specifically, it was unknown how narrow the mask features (center width) could be made before mechanical failure (breaking) occurred, and how the number of terminals (2, 3, or 4) influenced these limits.

2. Methodology

The authors conducted a systematic study involving the fabrication and characterization of 270 unique stencil mask designs.

Fabrication Process:
- Substrate: Si(111) with a 300 nm SiO₂ layer and a 100 nm Si₃N₄ layer.
- Patterning: Electron Beam Lithography (EBL) using a 100 kV Raith EBPG 5200 tool. Fine and coarse structures were written with specific beam currents and proximity effect corrections.
- Etching: Inductively Coupled Plasma Reactive-Ion Etching (ICP-RIE) for the Si₃N₄, followed by wet etching (Buffered Oxide Etch and HF) to undercut the SiO₂ pillars, creating the suspended Si₃N₄ mask.
- Design Parameters: Masks were designed for 2, 3, and 4 terminals. Key variables included:
  - Arm Width ($aw$): 100 nm to 500 nm (in 50 nm steps).
  - Center Width ($cw$): 10 nm to 100 nm (in 10 nm steps).
  - Center Length ($cl$): Fixed at 1 µm.
  - Angle of Sharpness ( $\alpha$ ): Defined by the geometry, determining how much the coupling is concentrated in the center.
Characterization:
- All masks were inspected via Scanning Electron Microscopy (SEM) to determine if they were "intact" or "broken."
- Geometrical dimensions ($cw$ and $aw$) of the intact masks were measured using line scans to compare designed vs. realized dimensions.

3. Key Contributions

Systematic Yield Analysis: The study provides the first comprehensive dataset correlating mask geometry (specifically center width and terminal count) with fabrication yield (intact vs. broken).
Design Guidelines for MTJJs: It establishes specific "safe zones" for mask design, allowing researchers to fabricate multi-terminal junctions with minimal weak-link lengths without risking mask failure.
Quantification of Fabrication Artifacts: The paper identifies and explains systematic deviations between designed and realized dimensions caused by EBL proximity effects and angle-dependent wet etching rates.
Optimization of Central Scattering Models: It demonstrates how to maximize the "angle of sharpness" ( $\alpha$ ) to ensure the superconducting coupling is localized in the central region, which is essential for the central scattering transport model.

4. Key Results

A. Mask Yield and Stability Limits

Reliability: Masks can be fabricated with close to 100% yield if the center width ($cw$) exceeds a certain threshold.
Terminal Dependence: The more terminals a mask has, the smaller the center width can be designed before failure occurs:
- 2-Terminal Masks: Reliable down to 50 nm (designed).
- 3-Terminal Masks: Reliable down to 40 nm (designed).
- 4-Terminal Masks: Reliable down to 30 nm (designed).
Realized Dimensions:
- 2-Terminal: The smallest realized center width was 40 nm.
- 3-Terminal: The smallest realized width exceeded the design (approx. 60–70 nm for a 40 nm design).
- 4-Terminal: The realized width scattered around a constant 90 nm, regardless of the design (even for designs as small as 20–30 nm).
- Reasoning: This deviation is attributed to the 90° angles in 4-terminal masks, which limit reactant transport during wet etching and exacerbate EBL proximity effects compared to the 180° (2-terminal) or 120° (3-terminal) geometries.

B. Geometrical Accuracy

Arm Width ($aw$): Realized arm widths were consistently ~10 nm smaller than designed across all mask types.
Angle of Sharpness ( $\alpha$ ): The study successfully realized angles greater than 10°, which is advantageous for isolating the central scattering region. The minimum center width ($cw$) showed a slight increase with higher $\alpha$ due to EBL proximity effects.

C. Trade-offs

There is a direct trade-off between mechanical stability (wider masks) and superconducting coupling strength (narrower masks). The study identifies the optimal balance where masks are narrow enough for strong coupling but wide enough to survive the process.

5. Significance and Impact

Enabling Advanced Quantum Devices: By proving that suspended stencil masks can reliably achieve lateral dimensions down to 40 nm (and potentially smaller with optimized designs), this work enables the fabrication of high-quality, multi-terminal Josephson junctions with clean interfaces.
Facilitating Topological Research: The ability to create central scattering regions with multiple electrodes is a prerequisite for creating artificial topological states and Weyl singularities, which are vital for topological quantum computing.
Standardization: The authors propose using their data to create a library of reliable mask designs (via PHIDL software) for the community, reducing trial-and-error in future experiments.
Process Insight: The findings highlight the importance of accounting for angle-dependent etching and EBL proximity effects when designing complex nanofabrication masks, particularly for high-terminal-count devices.

In conclusion, this paper provides the essential "rulebook" for fabricating suspended stencil masks for multi-terminal Josephson junctions, bridging the gap between theoretical device design and practical, high-yield nanofabrication.