Overlapped Junctions in Quantum Devices

• Overlapped junctions are defined by the spatial superposition of electrodes, layers, or gates that set the active transport region with high lithographic and electrostatic precision.
• They are implemented in diverse platforms such as superconducting circuits, graphene devices, long Josephson junctions, and gate-defined semiconductor systems, each leveraging unique fabrication techniques.
• This overlap strategy offers significant advantages over shadow evaporation and split-gate methods by enhancing control over tunneling, phase evolution, and valley transport for improved device performance.

An overlapped junction is a junction in which the active transport region is defined by the physical overlap of separately defined electrodes, layers, contacts, or gates. In the superconducting literature, an overlap Josephson junction consists of two superconducting electrodes deposited in separate lithography steps such that the top electrode physically overlaps the bottom electrode over a well-defined area $A$, with an $\mathrm{AlO_x}$ tunnel barrier formed by oxidation of the bottom electrode (Wu et al., 2017). In graphene transport, the overlap is a finite AB-stacked bilayer segment between monolayer leads, so that all electronic transport between source and drain must traverse the overlapped region (Tamura, 2023). In gate-defined semiconductor devices, overlapping top gates use stacked dielectric layers to define adjacent carrier-density regions without a lithographic split gap (Fuchs et al., 2024). Across these settings, the overlap geometry is used to control tunneling, phase evolution, electrostatics, or wave-function matching with a degree of lithographic and electrostatic precision that differs materially from shadow-evaporation, split-gate, and side-contact alternatives.

1. Geometric archetypes and defining features

The defining geometric feature of an overlapped junction is that the junction area is set by spatial superposition rather than by a suspended shadow mask, a lateral vacuum gap alone, or a purely coplanar split. In the superconducting sub-micron implementation, the overlap area is $A \simeq 0.01$–$0.02\,\mu\mathrm{m}^2$, with bottom-electrode thickness $t_1 \approx 20\,\mathrm{nm}$ and top-electrode thickness $t_2 \approx 40\,\mathrm{nm}$; the tunnel barrier is created by oxidation of the cleaned bottom aluminum electrode (Wu et al., 2017). In the long Josephson tunnel junction geometry, the tunnel barrier overlaps a doubly connected bottom loop for a length $L \gtrsim \pi \lambda_J/2$ and width $W \lesssim \lambda_J$, so the overlap couples the Josephson phase to loop inductance and fluxoid quantization (Monaco et al., 2011).

In partially overlapped graphene, the overlap is the central finite AB-stacked bilayer region, with monolayer graphene on the left and right. Two canonical configurations are used: “low-bi-up” and “low-bi-low,” or equivalently “up” and “down,” depending on whether the monolayer leads connect to different layers or to the same layer of the bilayer (Tamura, 2023). In a side-contact graphene junction, two separate graphene sheets overlap at their edges, and the overlap area is $A=N\cdot S_0$, where $$S_0=|a_1\times a_2|=(\sqrt{3}/2)a^2 \simeq 5.24\,\text{\AA}^2$$ (Li et al., 2015). In HgTe Hall bars, overlapping top gates are separated vertically by low-temperature ALD $\mathrm{AlO_x}$0 layers, and Gate II and Gate III overlap Gate I laterally by $\mathrm{AlO_x}$1–$\mathrm{AlO_x}$2 (Fuchs et al., 2024).

A recurrent distinction is with techniques whose critical dimension is defined by a lateral gap or a deposition angle. The superconducting overlap junction explicitly contrasts with shadow-evaporation using a single, suspended, bi-layer resist mask and angled evaporation of Al; the overlapped top-gate architecture contrasts with split gates, for which the minimum lithographic gap directly controls the depleted neck region (Wu et al., 2017, Fuchs et al., 2024). This suggests that “overlap” is not merely a descriptive geometric label, but a fabrication and field-control strategy.

2. Superconducting overlap Josephson junctions

In high-coherence superconducting circuits, overlap Josephson junctions were developed as a two-step, normal-angle process that eliminates the angle-dependent shadow masks typically used for small junctions. The bottom electrode is evaporated Al at normal incidence and defined by a first e-beam pattern; after air exposure, a second e-beam lithography step opens the top-electrode region, the bottom-electrode surface is cleaned in situ with Ar RF plasma at $\mathrm{AlO_x}$3 and $\mathrm{AlO_x}$4 for $\mathrm{AlO_x}$5–$\mathrm{AlO_x}$6, oxidation at $\mathrm{AlO_x}$7 and room temperature for $\mathrm{AlO_x}$8–$\mathrm{AlO_x}$9 forms the tunnel barrier, and a second normal-angle Al evaporation completes the top electrode (Wu et al., 2017). The barrier thickness is estimated as $A \simeq 0.01$0–$A \simeq 0.01$1.

The junction design is described by the standard relations

$A \simeq 0.01$2

$A \simeq 0.01$3

$A \simeq 0.01$4

and

$A \simeq 0.01$5

Experimentally, the room-temperature normal resistance scales empirically as $A \simeq 0.01$6, with $A \simeq 0.01$7 giving $A \simeq 0.01$8–$A \simeq 0.01$9 for $0.02\,\mu\mathrm{m}^2$0–$0.02\,\mu\mathrm{m}^2$1, and typical $0.02\,\mu\mathrm{m}^2$2 values of $0.02\,\mu\mathrm{m}^2$3–$0.02\,\mu\mathrm{m}^2$4 per junction from $0.02\,\mu\mathrm{m}^2$5 and the Ambegaokar–Baratoff relation (Wu et al., 2017). In a concentric transmon, the reported parameters were $0.02\,\mu\mathrm{m}^2$6, $0.02\,\mu\mathrm{m}^2$7, $0.02\,\mu\mathrm{m}^2$8, $0.02\,\mu\mathrm{m}^2$9, $t_1 \approx 20\,\mathrm{nm}$0, and Hahn-echo $t_1 \approx 20\,\mathrm{nm}$1; the measured $t_1 \approx 20\,\mathrm{nm}$2 approached the Purcell limit of $t_1 \approx 20\,\mathrm{nm}$3, indicating low intrinsic junction loss (Wu et al., 2017). A common misconception is that the extra lithography step or the in-situ Ar cleaning necessarily degrades coherence; the reported result was that the overlap geometry with in-situ Ar cleaning did not degrade $t_1 \approx 20\,\mathrm{nm}$4 or $t_1 \approx 20\,\mathrm{nm}$5 compared to SE-based junctions.

The same overlap concept was later extended to micrometer-scale junctions for superconducting quantum electronics and amplifiers. On a $t_1 \approx 20\,\mathrm{nm}$6 intrinsic Si wafer, the bottom electrode was formed from $t_1 \approx 20\,\mathrm{nm}$7 Al, patterned by stepper lithography and wet etch; after native $t_1 \approx 20\,\mathrm{nm}$8 regrowth, a $t_1 \approx 20\,\mathrm{nm}$9 PMMA protective layer and an LOR/imaging-resist bilayer were used for top-electrode definition, followed by an $t_2 \approx 40\,\mathrm{nm}$0 plasma ash at $t_2 \approx 40\,\mathrm{nm}$1 and $t_2 \approx 40\,\mathrm{nm}$2 for $t_2 \approx 40\,\mathrm{nm}$3, an Ar RF-plasma clean at $t_2 \approx 40\,\mathrm{nm}$4 and $t_2 \approx 40\,\mathrm{nm}$5 Ar for $t_2 \approx 40\,\mathrm{nm}$6, and oxidation with ultra-high-purity $t_2 \approx 40\,\mathrm{nm}$7 using dose A ($t_2 \approx 40\,\mathrm{nm}$8, $t_2 \approx 40\,\mathrm{nm}$9), dose B ($L \gtrsim \pi \lambda_J/2$0, $L \gtrsim \pi \lambda_J/2$1), or dose C ($L \gtrsim \pi \lambda_J/2$2, $L \gtrsim \pi \lambda_J/2$3) (Bal et al., 2020). For example junctions with $L \gtrsim \pi \lambda_J/2$4, the measured pairs were $L \gtrsim \pi \lambda_J/2$5, $L \gtrsim \pi \lambda_J/2$6, and $L \gtrsim \pi \lambda_J/2$7 (Bal et al., 2020).

The micrometer-scale process enabled an overlap-junction-based Josephson parametric amplifier using only 2 layers. The nonlinear inductive element was an 8-SQUID array with junction area $L \gtrsim \pi \lambda_J/2$8, shunt capacitance $L \gtrsim \pi \lambda_J/2$9, and coupling capacitance $W \lesssim \lambda_J$0; the resonance was tunable over $W \lesssim \lambda_J$1–$W \lesssim \lambda_J$2, with center frequency $W \lesssim \lambda_J$3 at zero flux detuning, peak gain $W \lesssim \lambda_J$4–$W \lesssim \lambda_J$5, $W \lesssim \lambda_J$6 bandwidth $W \lesssim \lambda_J$7, negligible insertion loss to the cold load, and added noise lower bound $W \lesssim \lambda_J$8 photons at $W \lesssim \lambda_J$9 (Bal et al., 2020). The process was contrasted with trilayer fabrication requiring $A=N\cdot S_0$0 photolithography and etch/deposition steps and with shadow evaporation requiring suspended resist bridges and multi-angle e-beam deposition.

3. Long Josephson overlap junctions with doubly connected electrodes

A distinct meaning of overlap junction appears in long Josephson tunnel junctions with doubly connected electrodes. Here the device is a planar long Josephson tunnel junction in the “in-line” configuration in which one or both superconducting electrodes is a doubly-connected thin-film loop, and the tunnel barrier overlaps the bottom loop (Monaco et al., 2011). The essential feature is that a circulating current

$A=N\cdot S_0$1

flows in the loop and generates a radial field felt by the junction, while fluxoid quantization constrains the overall phase drop across the junction.

Inside the barrier the gauge-invariant phase $A=N\cdot S_0$2 obeys the static sine–Gordon equation

$A=N\cdot S_0$3

or, in normalized units $A=N\cdot S_0$4,

$A=N\cdot S_0$5

The self-field boundary conditions are

$A=N\cdot S_0$6

$A=N\cdot S_0$7

with $A=N\cdot S_0$8, $A=N\cdot S_0$9, and $$S_0=|a_1\times a_2|=(\sqrt{3}/2)a^2 \simeq 5.24\,\text{\AA}^2$$0 (Monaco et al., 2011). Fluxoid quantization imposes an additional global constraint on

$$S_0=|a_1\times a_2|=(\sqrt{3}/2)a^2 \simeq 5.24\,\text{\AA}^2$$1

The central physical consequence is phase frustration. Because $$S_0=|a_1\times a_2|=(\sqrt{3}/2)a^2 \simeq 5.24\,\text{\AA}^2$$2 is fixed by flux quantization, the phase profile cannot simultaneously satisfy the ideal boundary fields needed for the maximum supercurrent. The resulting magnetic diffraction patterns are piecewise linear and asymmetric, and trapping one extra flux quantum shifts the critical current by

$$S_0=|a_1\times a_2|=(\sqrt{3}/2)a^2 \simeq 5.24\,\text{\AA}^2$$3

The total energy combines Josephson energy and magnetic energy in the loop arms, and the stable states are obtained by a self-consistent application of the principle of minimum energy (Monaco et al., 2011). Experimental Nb/Al–AlOx/Nb trilayer devices with $$S_0=|a_1\times a_2|=(\sqrt{3}/2)a^2 \simeq 5.24\,\text{\AA}^2$$4, $$S_0=|a_1\times a_2|=(\sqrt{3}/2)a^2 \simeq 5.24\,\text{\AA}^2$$5, $$S_0=|a_1\times a_2|=(\sqrt{3}/2)a^2 \simeq 5.24\,\text{\AA}^2$$6, $$S_0=|a_1\times a_2|=(\sqrt{3}/2)a^2 \simeq 5.24\,\text{\AA}^2$$7, and $$S_0=|a_1\times a_2|=(\sqrt{3}/2)a^2 \simeq 5.24\,\text{\AA}^2$$8–$$S_0=|a_1\times a_2|=(\sqrt{3}/2)a^2 \simeq 5.24\,\text{\AA}^2$$9 showed suppression of $\mathrm{AlO_x}$00 by $\mathrm{AlO_x}$01 in the purely asymmetric bias and resolved steps of $\mathrm{AlO_x}$02–$\mathrm{AlO_x}$03 per trapped flux quantum (Monaco et al., 2011).

4. Partially overlapped graphene layers and valley transport

In graphene, a partially overlapped junction is formed by a finite AB-stacked bilayer region between monolayer leads. In the “low-bi-up” or “up” geometry, the left and right monolayer leads connect to different layers, so the total current is forced to flow through the interlayer path; in the “low-bi-low” or “down” geometry, both leads connect to the same layer, so electrons can bypass the interlayer path (Tamura, 2023, Tamura, 22 Jan 2025). The tight-binding description uses intralayer hopping $\mathrm{AlO_x}$04, direct interlayer coupling $\mathrm{AlO_x}$05, skew couplings $\mathrm{AlO_x}$06 and $\mathrm{AlO_x}$07, and layer potentials $\mathrm{AlO_x}$08 induced by a vertical field (Tamura, 2023).

One major focus is valley current reversal. For each lateral subband, the reversal metric is

$\mathrm{AlO_x}$09

and the average indicator is

$\mathrm{AlO_x}$10

Without vertical field, both low-bi-up and low-bi-low junctions show modest peaks $\mathrm{AlO_x}$11, attributed to monolayer–bilayer matching. As the vertical field increases, $\mathrm{AlO_x}$12 declines in the low-bi-low junction but rises in the low-bi-up junction to about $\mathrm{AlO_x}$13; the enhancement originates from interlayer matching, especially near the bilayer gap edge $\mathrm{AlO_x}$14 (Tamura, 2023). Analytically, near the gap edge and for suitable overlap length, the peak condition is

$\mathrm{AlO_x}$15

with $\mathrm{AlO_x}$16.

A second line of work examined energy symmetry in partially overlapped graphene under a vertical electric field. In the up junction, only the valley-reversed transmission has even symmetry with respect to energy:

$\mathrm{AlO_x}$17

whereas the other channels generally lack this property (Tamura, 22 Jan 2025). The interlayer wave-function ratio

$\mathrm{AlO_x}$18

is asymmetric in energy, $\mathrm{AlO_x}$19, yet in the up geometry its odd-in-$\mathrm{AlO_x}$20 contribution cancels between entrance and exit because of chiral symmetry, $\mathrm{AlO_x}$21-rotation symmetry, and conservation of probability. The paper attributes the unexpected even symmetry to a self-cancellation effect of $\mathrm{AlO_x}$22 (Tamura, 22 Jan 2025). This is not a purely formal statement: when two identical up junctions are placed in series, the symmetric-in-$\mathrm{AlO_x}$23 channel can be fully suppressed over the full $\mathrm{AlO_x}$24 gap, producing a double-↑ valley valve.

The experimental proposals reflect the transport mechanism. For valley current reversal, a four-terminal nonlocal setup and an optical pump–probe scheme were proposed; changing the vertical gate field flips the sign of the nonlocal resistance $\mathrm{AlO_x}$25, while a second-harmonic probe can read out valley sign on the right monolayer (Tamura, 2023). A common misunderstanding is that overlap alone should produce boundary-induced intervalley scattering. In the 2025 formulation, the interfaces are armchair edges chosen so as to ensure no zigzag edge states or intervalley scattering at the boundary (Tamura, 22 Jan 2025).

5. Overlapping top gates and gate-defined lateral junctions

In semiconducting layers, an overlapped junction can be formed electrostatically by overlapping top-gate electrodes rather than by a back gate/top gate combination or a split-gate configuration. In the HgTe implementation, a mesa-defined Hall bar $\mathrm{AlO_x}$26 wide is first covered by a $\mathrm{AlO_x}$27 layer of low-temperature ALD $\mathrm{AlO_x}$28, then Gate I (Ti $\mathrm{AlO_x}$29/Au $\mathrm{AlO_x}$30) is patterned, a second $\mathrm{AlO_x}$31 $\mathrm{AlO_x}$32 layer is deposited, and Gate II and Gate III are defined so that each overlaps Gate I laterally by $\mathrm{AlO_x}$33–$\mathrm{AlO_x}$34 (Fuchs et al., 2024). The dielectric constant is $\mathrm{AlO_x}$35, the gate-to-semiconductor breakdown field is $\mathrm{AlO_x}$36, and the inter-gate breakdown voltage is $\mathrm{AlO_x}$37 over $\mathrm{AlO_x}$38, corresponding to $\mathrm{AlO_x}$39.

The electrostatics are described by Poisson’s equation,

$\mathrm{AlO_x}$40

with Dirichlet conditions under the gates and continuity of $\mathrm{AlO_x}$41 and $\mathrm{AlO_x}$42 at the semiconductor–dielectric interface. In the simplest parallel-plate model,

$\mathrm{AlO_x}$43

$\mathrm{AlO_x}$44

and

$\mathrm{AlO_x}$45

In the overlap region, the local potential is approximated by

$\mathrm{AlO_x}$46

Because the only ungated spacing is the $\mathrm{AlO_x}$47 dielectric thickness, the lateral field profile stays uniform to within $\mathrm{AlO_x}$48 of the nominal gate efficiency through the entire overlap region (Fuchs et al., 2024).

This geometry is explicitly contrasted with split gates. For a split-gate gap of about $\mathrm{AlO_x}$49, the gate efficiency at the quantum-well center falls to about $\mathrm{AlO_x}$50 of the far-away value, whereas the overlapped design is lithography-insensitive and dielectric-limited (Fuchs et al., 2024). The resulting junctions were used to study quantum Hall edge-state interaction in the $\mathrm{AlO_x}$51, $\mathrm{AlO_x}$52, $\mathrm{AlO_x}$53, and $\mathrm{AlO_x}$54 regimes. At $\mathrm{AlO_x}$55, the four-probe longitudinal resistance exhibited exact quantization with $\mathrm{AlO_x}$56 in unipolar $\mathrm{AlO_x}$57 or $\mathrm{AlO_x}$58 configurations and $\mathrm{AlO_x}$59 in bipolar $\mathrm{AlO_x}$60 or $\mathrm{AlO_x}$61 configurations, consistent with the Landauer–Büttiker formulas given in the paper. At $\mathrm{AlO_x}$62, up to forty distinct $\mathrm{AlO_x}$63 plateaus were resolved (Fuchs et al., 2024). The paper states that the overlap gates produce atomically sharp junctions $\mathrm{AlO_x}$64 and yield textbook quantization even in $\mathrm{AlO_x}$65 and $\mathrm{AlO_x}$66 regimes.

6. Computational formulations and side-contact realizations

The study of overlapped junctions has also required specialized computational formulations. In computational electromagnetics, a method based on the method of moments was introduced for metallic or dielectric structures in contact with other metallic or dielectric structures. The method combines a generalization of surface integral equation formulations for bodies in contact with an algorithm for enforcing boundary conditions and Kirchhoff’s Law across a junction edge (Gomez-Sousa et al., 2015). The duplicated-unknown system

$\mathrm{AlO_x}$67

is reduced by constraints $\mathrm{AlO_x}$68 and a mapping $\mathrm{AlO_x}$69 or $\mathrm{AlO_x}$70, yielding the projected system

$\mathrm{AlO_x}$71

A key feature is that the generalization does not require duplicating unknowns on the dielectric separation surfaces, while current continuity and Kirchhoff’s law are imposed at arbitrarily complex junctions (Gomez-Sousa et al., 2015). In the grounded monopole validation example, the merged-unknown solution followed measurement to within $\mathrm{AlO_x}$72 over $\mathrm{AlO_x}$73–$\mathrm{AlO_x}$74, improved radiation-pattern agreement to better than $\mathrm{AlO_x}$75, converged $\mathrm{AlO_x}$76 faster in GMRES iteration count, and used $\mathrm{AlO_x}$77 less RAM when LU-factored (Gomez-Sousa et al., 2015).

At the atomistic end of the spectrum, a plane-wave multiple-scattering method was implemented for side-contact graphene junctions formed by two separate graphene sheets with overlapping edges. The transport obeys the Landauer–Büttiker relation

$\mathrm{AlO_x}$78

with

$\mathrm{AlO_x}$79

For non-overlapping tunneling at fixed vertical separation $\mathrm{AlO_x}$80, the transmission decays exponentially with lateral distance,

$\mathrm{AlO_x}$81

with $\mathrm{AlO_x}$82–$\mathrm{AlO_x}$83; at $\mathrm{AlO_x}$84, $\mathrm{AlO_x}$85 peaks at $\mathrm{AlO_x}$86 (Li et al., 2015). In overlapped junctions at fixed $\mathrm{AlO_x}$87, AA stacking shows a superlinear rise and saturation,

$\mathrm{AlO_x}$88

with $\mathrm{AlO_x}$89 and $\mathrm{AlO_x}$90, whereas AB stacking shows nearly linear scaling,

$\mathrm{AlO_x}$91

(Li et al., 2015). The orbital interpretation is correspondingly different: for AA stacking, the effective potential at $\mathrm{AlO_x}$92 forms connected low-barrier channels between layers, while for AB stacking, interlayer potential pockets block direct paths. A plausible implication is that overlap geometry alone does not determine transport; stacking, interlayer separation, and the available interlayer path are equally decisive.