---
title: Overlap Junction Arrays
url: https://www.emergentmind.com/topics/overlap-junction-arrays
type: topic
---

# Overlap Junction Arrays

An overlap junction array is a dense assembly of single Al/AlOx/Al Josephson tunnel junctions fabricated using overlap geometry, wherein the bottom and top superconducting aluminum electrodes overlap across a nano-oxide tunnel barrier. These arrays, which do not use SQUID loops, are engineered so that a perpendicular magnetic field is focused into the junction barrier by demagnetization effects in the wider electrodes, enabling efficient and tunable modulation of the Josephson inductance via Fraunhofer-type supercurrent interference. Overlap junction arrays provide high-density, high-inductance, and field-tunable transmission line media, with experimentally demonstrated impedance tuning that spans the superconducting resistance quantum without the additional capacitance and fabrication overhead typical of SQUID-based approaches [2210.12119].

## 1. Junction Geometry and Fabrication

Overlap junction arrays utilize planar Al/AlOx/Al tunnel junctions with specific geometry to maximize field tunability and packing density. The fabrication involves the following sequence:
- **Bottom electrode:** Aluminum (∼30 nm), e-beam evaporated.
- **Oxide barrier:** Native AlOx, grown in situ by static oxidation (O₂ at ∼5 mbar for ∼5–10 min), thickness $d_\mathrm{ox} \approx 1$–$2$ nm.
- **Top electrode:** Aluminum (∼60 nm).
- **Patterning:** Standard Dolan-bridge e-beam lithography on an MMA/PMMA bilayer resist, followed by double-angle evaporation and liftoff.

The geometry is defined by overlapping two aluminum films in a $3.0\,\mu$m (width) × $0.4\,\mu$m (length) region, yielding a uniform tunnel barrier area $A = 1.2\,\mu\textrm{m}^2$. Adjacent junctions are separated by a $0.2\,\mu$m gap. A typical array comprises two parallel series chains, each with $N = 3{,}300$ junctions, a unit cell pitch $a = 0.6\,\mu$m, and total length per branch $\sim2.0\,\textrm{mm}$. This design achieves a junction density of approximately $1.7\,\mu\textrm{m}^{-1}$ [2210.12119].

## 2. Array Configuration and Electrodynamics

The physical and electrical configuration of overlap junction arrays is critical for their high impedance and inductance:
- The two parallel series chains form a coplanar stripline in the long-wavelength limit.
- The inductance per unit cell is $L_\mathrm{cell} = 2 L_J$, where $L_J$ is the Josephson inductance.
- The per-cell capacitance, $C_\mathrm{cell}$, arises from capacitive coupling between the two rails.
- Wave velocity and characteristic impedance follow
  \[
  v = \frac{1}{\sqrt{L_\mathrm{cell} C_\mathrm{cell}}}, \qquad
  Z = \sqrt{\frac{L_\mathrm{cell}}{C_\mathrm{cell}}}
  \]
At zero field, measured parameters include $v = 2.33\times10^6\,\textrm{m/s}$ and $Z = 4.2\,\textrm{k}\Omega$; at $B_\perp = 1.3$ mT, $v$ is halved and $Z$ doubles to $8.7\,\textrm{k}\Omega$ [2210.12119].

## 3. Demagnetization and Field-Focusing Effects

A pivotal aspect of overlap junction arrays is demagnetization-driven field focusing:
- With Al film thickness $t\approx50$ nm and width $W=3\,\mu$m, the demagnetization factor is $N_d \approx t/W = 0.02$, leading to a field-focusing factor $\beta \approx 1/N_d = 50$.
- A perpendicular magnetic field $B_\perp$ is concentrated into the junction’s tunnel barrier: $B_\mathrm{eff} = \beta B_\perp$.
- The magnetic flux through the barrier area for $B_\perp = 1$ mT is $\Phi = B_\mathrm{eff} \cdot A = 6 \times 10^{-2}\,\Phi_0$, where $\Phi_0 = h/2e$ is the flux quantum.
This field focusing enables strong tuning of the junction critical current $I_c$ and thus the Josephson inductance with modest applied fields [2210.12119].

## 4. Fraunhofer Interference and Inductance Modulation

Each overlap junction exhibits Fraunhofer-pattern critical current oscillations under the modulated effective field:
- The current satisfies the “rectangular-junction” Fraunhofer form:
  \[
  I_c(B) = I_{c0}\,\left|\frac{\sin(\pi \Phi/\Phi_0)}{\pi \Phi/\Phi_0}\right|, \qquad \Phi = B_\mathrm{eff} A
  \]
- This directly modulates the Josephson inductance, which (for small phase bias $\varphi \approx 0$) is
  \[
  L_J(B) \approx \frac{\Phi_0}{2\pi I_c(B)}
  \]
- The array’s total transmission-line impedance is thus field-tunable:
  \[
  Z(\omega,B) \approx \sqrt{\frac{2 L_J(B)}{C_\mathrm{cell}}}
  \]
Experimental tuning demonstrates impedance spanning $4$–$9\,\textrm{k}\Omega$, exceeding the resistance quantum $R_Q = h/(2e)^2 \approx 6.45\,\textrm{k}\Omega$ under $\sim1$ mT perpendicular field [2210.12119].

## 5. Experimental Characterization and Tunability

Overlap junction arrays have been extensively characterized:
- Reflection coefficient $|S_{11}|$ measurements show a doubling of standing-wave resonance density as $B_\perp$ increases, consistent with $v \to v/2$, $L\to4L$ dynamics.
- Dispersion curves $f(k)$ fit
  \[
  2\pi f = \frac{v k}{\sqrt{1 + (v k/\omega_p)^2}}
  \]
where $v$ is the velocity and $\omega_p$ the plasma frequency.
- The array displays uniform impedance tuning and $\sim2\%$ rms fabrication disorder, which does not increase with applied field.
- Qubit-coupled phase shift $\delta(f)$ fits a Lorentzian, with width $\Gamma$ yielding impedance $Z$ in agreement with transmission-line and mode-spacing measurements.

| $B_\perp$ (mT) | $v$ $(10^6\,\textrm{m/s})$ | $Z$ $(\textrm{k}\Omega)$ |
|:---:|:---:|:---:|
|   0   | 2.33 | 4.2 |
| 1.3 | 1.19 | 8.7 |

These results confirm controllable, uniform, in-situ tunability of both Josephson inductance and transmission-line impedance [2210.12119].

## 6. Fabrication Methodology and Design Optimization

Reliable fabrication of dense, reproducible overlap junction arrays is established with:
- MMA/PMMA bilayer resist for undercut and liftoff.
- Dolan-bridge design with $\sim200$ nm bridge width.
- Double-angle evaporation: first Al ($30$ nm), static oxidation, then second Al ($60$ nm).
- Junctions defined as $3\,\mu$m wide strips intersecting $0.4\,\mu$m regions, ensuring control over $I_c$ and capacitance.
- Liftoff in NMP at $80^\circ$C yields clean junctions with reproducible AlOx barriers.
Design choices such as wide overlap ($W/L = 7.5$) are critical for maximizing demagnetization and field focusing effects without introducing additional capacitance or process complexity [2210.12119].

## 7. Implications and Advantages Over SQUID Arrays

The overlap junction approach enables several key advantages:
- Eliminates the excess capacitance, complexity, and area overhead associated with SQUID loops.
- Allows denser packing, achieving higher impedance per length and more compact superconducting metamaterials.
- Achieves impedance tunability across $R_Q$ using modest fields ($\sim1\,\textrm{mT}$), enhancing flexibility for superconducting circuit applications.
- Demonstrates uniform, robust tunability with minimal fabrication disorder.

*A plausible implication is* that overlap junction arrays provide a preferable platform for high-impedance, flux-tunable microwave environments, particularly where compactness, uniformity, and ease of fabrication are paramount [2210.12119].

Source: https://www.emergentmind.com/topics/overlap-junction-arrays