Josephson Diode Efficiency: Mechanisms & Metrics

• Josephson diode efficiency is defined as the normalized difference between forward and reverse critical supercurrents, reflecting the degree of nonreciprocity.
• Key insights include the role of finite Cooper pair momentum and screening currents in breaking inversion and time-reversal symmetries.
• Device performance is optimized by tuning junction parameters, achieving up to 40% efficiency without relying on spin–orbit coupling.

A Josephson diode is a nonreciprocal superconducting circuit element that supports a larger dissipationless supercurrent in one direction than in the other—an effect quantified as the Josephson diode effect. The efficiency of a Josephson diode is a central figure of merit, directly measuring the asymmetry between the maximum critical supercurrents achievable in opposite directions. Efficiency in this context not only benchmarks the effectiveness of diode-based rectification in superconducting circuits but also reflects the underlying microscopic mechanisms—such as symmetry breaking, the interplay between bound state spectra, and the contributions from continuum states—that facilitate nonreciprocal transport. Recent research has established a theoretical and practical basis for realizing significant diode efficiencies in a wide variety of short Josephson junction geometries, including those that do not require spin–orbit coupling or exotic materials platforms.

1. Definition and Mathematical Quantification of Josephson Diode Efficiency

The Josephson diode efficiency, denoted $\eta$, characterizes the degree of nonreciprocity in a Josephson junction's critical current. For a junction supporting maximal dissipationless (supercurrent) flows $I_{c+}$ (forward direction) and $I_{c-}$ (reverse direction), the efficiency is defined as:

$\eta \equiv \frac{|\Delta I_c|}{I_{c+} + I_{c-}}$

where $\Delta I_c = I_{c+} - I_{c-}$ is the difference between forward and reverse critical currents. In the physical regime of interest, $I_{c+} > 0$, $I_{c-} > 0$ (both are defined as magnitudes), and $\eta \in [0,1]$, with $\eta = 0$ for a perfectly reciprocal device and $\eta \to 1$ for an ideal diode.

In the context of (Davydova et al., 2022), diode efficiency values up to approximately $40\%$ are predicted, corresponding to a ratio $I_{c+}/I_{c-} \approx 230\%$—meaning the forward critical current can be more than twice the reverse one. This formalism allows systematic benchmarking across different junction mechanisms and device architectures.

2. Microscopic Mechanism: Finite Cooper Pair Momentum and Symmetry Breaking

The essential microscopic origin of Josephson diode efficiency in the universal scenario is the generation of finite Cooper pair momentum $q$ in the superconducting leads. Application of a small in-plane magnetic field $B_y < B_{c1}$ induces screening currents near the superconductor surfaces, leading the order parameter in each lead to acquire a spatial phase:

$$\Delta_1(x) = \Delta e^{2iqx}, \quad \Delta_2(x) = \Delta e^{i\varphi + 2iqx}$$

Here, $x$ is the spatial coordinate along the junction, $\Delta$ is the superconducting gap, and $\varphi$ is the macroscopic phase difference across the junction. The resulting phase winding explicitly breaks spatial inversion symmetry (distinguishes $+x$ from $-x$), and the field-induced screening currents break time-reversal symmetry. This simultaneous symmetry breaking is a necessary condition for producing a nonreciprocal Josephson effect and achieving nonzero diode efficiency.

Importantly, this mechanism operates independent of spin–orbit or Zeeman interaction, thus extending the relevant materials platform for the diode effect to essentially all conventional superconductors.

3. Spectral Contributions: Doppler-Shifted Andreev Bound States and Continuum Modes

Finite-momentum pairing directly impacts the Andreev bound state (ABS) spectrum in the weak-link region. Upon linearization near $\pm k_F$, the ABS energies for a short, clean junction of length $d$ are given by:

$$E_1 = -\Delta \cos\left(\frac{\tilde{\varphi}}{2}\right) + q v_F \qquad (\text{right-movers})$$

$$E_2 = \Delta \cos\left(\frac{\tilde{\varphi}}{2}\right) - q v_F \qquad (\text{left-movers})$$

$\tilde{\varphi} = \varphi + 2qd$

with $v_F$ as the Fermi velocity. The $\pm q v_F$ term represents a Doppler-like shift: right- and left-moving branches shift in energy oppositely, tilting the current–phase relation (CPR).

Additionally, the continuum of states (energies outside the superconducting gap) contributes a phase-independent current:

$I_{\text{cont}} = \frac{2e q v_F}{\pi \hbar}$

In a symmetric configuration ($q = 0$), this continuum contribution cancels; for $q \ne 0$, the cancellation is incomplete and the net continuum current further amplifies the CPR asymmetry.

The total Josephson current is thus:

$$I(\varphi) = \frac{e\Delta}{\hbar} \sin\left(\frac{\tilde{\varphi}}{2}\right) \cdot \text{sgn}\left[\Delta \cos\left(\frac{\tilde{\varphi}}{2}\right) - q v_F\right] + \frac{2e q v_F}{\pi \hbar}$$

Since the CPR is now non-antisymmetric ($I(\varphi) \ne -I(-\varphi)$), the maximal positive and negative critical currents differ, giving rise to finite $\eta$.

4. Device Architecture, Analytical Formulas, and Experimental Implementation

The standard device geometry comprises two massive superconducting slabs connected by a narrow metallic or semiconducting bridge. The critical step is to apply an in-plane field $B_y$ (satisfying $B_y < B_{c1}$ to remain in the Meissner state) to generate screening currents and thus a phase gradient:

$q \approx B_y \lambda_L$

where $\lambda_L$ is the London penetration depth. Via the proximity effect, the bridge inherits the modulated pairing potential from the leads: $\Delta(x) \sim e^{2iqx}$. The approach is compatible with a wide range of materials, does not demand special spin–orbit engineered systems, and is controlled via the applied magnetic field and device geometry.

The diode efficiency in this scheme can be engineered by tuning $q$ (via $B_y$), the bridge length $d$, and the interface transparency. Maximum efficiency, reaching $\approx 40\%$ ($I_{c+}/I_{c-} \approx 2.3$), occurs when the Doppler shift $q v_F$ becomes comparable to $\Delta$.

Summary of key relations:

Quantity	Formula
Lead pair potentials	$\Delta_1(x) = \Delta e^{2iqx}$, $\Delta_2(x) = \Delta e^{i\varphi + 2iqx}$
Doppler-shifted ABS energies	$$E_{1,2} = \mp \Delta \cos(\tilde{\varphi}/2) \pm qv_F$$
Continuum current	$I_{\text{cont}} = \dfrac{2e q v_F}{\pi\hbar}$
Total Josephson current	$$I(\varphi) = \dfrac{e\Delta}{\hbar} \sin(\tilde{\varphi}/2) \, \text{sgn}[\,\cdots\,] + I_{\text{cont}}$$
Diode efficiency	$\eta = \dfrac{|I_{c+} - I_{c-}|}{I_{c+} + I_{c-}}$

5. Performance Regimes, Scaling, and Optimal Conditions

Performance metrics from (Davydova et al., 2022) indicate that the supercurrent rectification effect can be strong, with diode efficiencies $\eta$ up to $0.4$ ($40\%$) in the optimal parameter regime. The asymmetry ratio $I_{c+}/I_{c-} \approx 2.3$ is a direct function of the ratio $q v_F/\Delta$ and increases as the phase gradient and Doppler shift rise relative to the superconducting gap. The upper bound for the efficiency is set by material properties (maximum sustainable screening current before vortex entry), device dimensions, and the transparency of the junction interfaces.

Operation at maximal efficiency requires careful tuning: if $q$ is too small, asymmetry vanishes; if $q$ is too large (exceeding the critical field), superconductivity is destroyed. Conventionally, fields on the order of several mT (below $B_{c1}$) and junction lengths at or below the superconducting coherence length are appropriate.

6. Implications for Platform Engineering and Broader Device Applicability

A central implication of this universal mechanism is the elimination of the need for materials with strong spin–orbit coupling or exotic internal structures to realize significant Josephson diode efficiency. The proposal extends the platform for nonreciprocal superconducting devices to any system capable of sustaining a Meissner state and proximity-induced phase gradients. This opens pathways for incorporating diode functionality into a broad range of device architectures (e.g., SNS, ScS, and S–semiconductor–S) and materials, including niobium, aluminum-based, or any conventional superconductors.

The mechanism's independence from spin–orbit effects and Zeeman splitting also circumvents potential pair-breaking effects associated with high magnetic fields in spintronic-based platforms. This robustness may provide decisive advantages for scaling, device integration, and the realization of dissipationless rectification elements in superconducting digital circuits and quantum information networks.

7. Summary and Outlook

The Josephson diode efficiency, as developed in (Davydova et al., 2022), is set by the degree of CPR asymmetry arising from finite-momentum Cooper pairing—a direct consequence of simultaneous inversion and time-reversal symmetry breaking in the superconducting leads. The mechanism is analytically transparent and amenable to engineering in a broad class of materials and device geometries. Doppler-shifted Andreev bound state spectra and incomplete cancellation of the continuum current are the microscopic origins of the rectification, with quantitative efficiency metrics anchored in the device's key parameters. Peak efficiencies of $\approx 40\%$, as well as pronounced current asymmetry ($I_{c+}/I_{c-} \approx 2.3$), are achievable without recourse to spin–orbit coupling, Zeeman fields, or nonstandard materials. This universality and compatibility with established superconducting platforms position the described scheme as a foundational mechanism for engineering efficient, scalable, and robust Josephson diodes.