Josephson diode

Short description: Superconducting quantum material diode

A Josephson diode (JD) is a special type of Josephson junction (JJ), which conducts (super)current in one direction better than in the opposite direction. In other words, it has asymmetric current-voltage characteristic. Since the Josephson diode is a superconducting device, the asymmetry of the supercurrent transport is the main focus of attention. Opposite to conventional Josephson junctions, the critical (maximum) supercurrents Ic+ and Ic− for opposite bias directions are different by absolute values (Ic+≠|Ic−|). In the presence of such a non-reciprocity, the bias currents of any magnitude in the range between Ic+ and |Ic−| can flow without resistance in only one direction.

This asymmetry, characterized by the ratio of critical currents 𝒜, is the main figure of merit of Josephson diodes and is the subject of new developments and optimizations. The Josephson diode effect can occur, e.g., in superconducting devices where time reversal symmetry and inversion symmetry are broken.[1][2]

Josephson diodes can be subdivided into two categories, those requiring an external (magnetic) field to be asymmetric and those not requiring an external magnetic field \
 the so-called "field-free" Josephson diodes (more attractive for applications). In 2021, the Josephson diode was realized in the absence of applied magnetic field in a non-centrosymmetric material, [3] followed shortly by the first realization of the zero-field Josephson diode in an inversion-symmetric device.[4]

## Contents

* 1 History
* 2 Superconducting diode effect
* 3 Theories
* 4 Figures of merit
* 5 See also
* 6 References

## History

Since decades the physicists tried to construct Josephson junction devices with asymmetric critical currents. This didn't involve new physical principles or advanced (quantum) material engineering, but rather electrical engineering tricks like combining several JJs in a special way (e.g. asymmetric 3JJ SQUID) or specially designed long JJs or JJ arrays, where Josephson vortex transport was asymmetric in opposite directions. After all, if one does not look inside the device, but treats such a device as a black box with two electrodes, its current-voltage characteristic is asymmetric with Ic+≠|Ic−|. Such devices were especially popular in the context of Josephson ratchets — devices used to rectify random or deterministic signals with zero time-average. These devices can be subdivided into several classes:

* asymmetric SQUID-like devices:[lower-alpha 1] proposals [5] and experimental implementations[6][7][8][9][10]
* long JJs with asymmetric vortex depinning currents: experimental implementations[11][12][13][14][15][16]
* Josephson fluxonic diodes[17][18][19][20][11]
* JJ arrays: proposals[21] and experiments.[22][23][24]
* intermediate length JJ with specially modulated properties/geometry: e.g. tunable φ-JJs[25] and in-line JJs made of Nb[26] and YBCO[27] showing record critical current asymmetry ratios of 𝒜∼10.

File:Schematic of Josephson diode using NbSe2 and Nb3Br8.webpAtomic structure of the field free Josephson diode using NbSe2 and Nb3Br8.[3]

Starting from 2020 one observes a new wave of interest to the systems with non-reciprocal supercurrent transport based on novel materials and physical principles.

* In 2020, a superconducting diode effect (see below) was demonstrated in an artificial N×[Nb|V|Ta|] superlattice.[28] (not yet a Josephson system). It was based on the previous proposal[29] with design similar to conventional semiconducting p-n junctions, but utilizing hole and electron doped superconductors. Next year the Josephson diode was demonstrated in van der Waals heterostructure of NbSe2/Nb3Br8/NbSe2. In this heterostructure the weak link (Josephson barrier) Nb3Br8 is a quantum material that is predicted to be an obstructed atomic insulator / Mott insulator and is non-centrosymmetric, i.e., it distinguishes between electrons with positive and negative momentum.[3][30][31][32] However, the asymmetry of such device was very low (𝒜≈1.07).
* A Josephson diode at zero applied field was observed in small twist-angle trilayer graphene, a system which possesses in-plane inversion symmetry.[33] In this case, the superconducting state itself is responsible for the breaking of time reversal symmetry and in-plane inversion symmetry.
* In 2022, a Josephson diode in symmetric Al|InAs(2DEG)|Al junction (array) was demonstrated,[34] exhibiting asymmetry 𝒜≈2 relevant for applications.
* Josephson diode based on Nb|Pt+YIG|Nb heterostructure operates at zero magnetic field and with the possibility to reverse its direction demonstrates similar asymmetry ratio at T≈2K.[35]
* Josephson diode using the topological semi-metal NiTe2 as a barrier, demonstrates 𝒜≈2.3 at T≈3.8K.[36]

In-depth review of recent developments.[37]

## Superconducting diode effect

The superconducting diode effect is an example of nonreciprocal superconductivity, where a material is superconducting in one direction and resistive in the other. This leads to half-wave rectification when a square wave AC-current is applied. In 2020, this effect was demonstrated in an artificial [Nb/V/Ta]n superlattice.[28] The phenomenon in the Josephson diode is believed to originate from asymmetric Josephson tunneling.[3] Unlike conventional semiconducting junction diodes, the superconducting diode effect can be realized in Josephson junctions as well as junction-free bulk superconductors.[37]

## Theories

Currently, the precise mechanism behind the Josephson diode effect is not fully understood. However, some theories have emerged that are now under theoretical investigation. There are two types of Josephson diodes, relating to which symmetries are being broken. The inversion breaking Josephson diode, and the Josephson diode breaking inversion breaking and time-reversal. The minimal symmetry breaking requirement for forming the Josephson diode is inversion symmetry breaking, and is required to obtain nonreciprocal transport.[38] One proposed mechanism originates from finite momentum Cooper pairs.[1][2] It may also be possible that the superconducting diode effect in the JD originates from self-field effects, but this still has to be rigorously studied. [39][40]

## Figures of merit

Depending on the potential application different parameters of the Josephson diodes, from operation temperature to their size can be important. However, the most important parameter is the asymmetry of critical currents (also called non-reciprocity). It can be defined as dimensionless ratio of larger to smaller critical currents

𝒜=max(Ic+,|Ic−|)min(Ic+,|Ic−|)

to be always positive and lay in the range from 1 (symmetric JJ) to ∞ (infinitely asymmetric one). Instead, some researchers use the so-called efficiency, defined as

η=|Ic+−|Ic−|Ic++|Ic−||=𝒜−1𝒜+1.

It lays in the range from 0 (symmetric system) to 1 (infinitely asymmetric system). [lower-alpha 2] Among other things the efficiency η shows the theoretical limit for thermodynamic efficiency (ratio of output to input power) that can be reached by the diode during rectification.

Intuitively it is clear that the larger the asymmetry 𝒜 is, the better the diode performs. A quantitative analysis [15][41] showed that a large asymmetry allows one to achieve rectification in a wide range of driving current amplitudes, a large counter current (corresponding to a heavy load), against which rectification is still possible, and a large thermodynamic efficiency (ratio of output dc to input ac power).

Thus, to create a practically relevant diode one should design a system with high asymmetry. The asymmetry ratios (efficiency) for different implementations of Josephson diodes are summarized in the table below.

Size. Previously demonstrated Josephson diodes were rather large (see the table), which hampers their integration into micro- or nano-electronic superconducting circuits or stacking. Novel devices based on heterostructures can potentially have 100 nm-scale dimensions, which is difficult to achieve using previous approaches with long JJs, fluxons, etc.

Voltage. Important parameter of any nano-rectifier is the maximum dc voltage produced. See the table for comparison.

Operating temperature. Ideally one would like to operate the diode in wide temperature range. Obviously, an upper limit in operation temperature is given by the transition temperature Tc of the superconducting material(s) used to fabricate the Josephson diodes. In the table below we quote the operating temperature for which the other parameters such as asymmetry are quoted. Many novel diodes, unfortunately, operate below 4.2K.

Figures of merit of different Josephson diodes  Reference | type | 𝒜 | η | Vmax(μV)[lower-greek 1] | area(μm2)[lower-greek 2] | Top(K)[lower-greek 3]

|
|
|
|
|
|

Carapella (2001)[12] | ALJJ[lower-greek 4] | 1.2 | 9% | 20 | 44500 | 6.5
Beck (2005)[14] | ALJJ[lower-greek 4] Nb-AlO-Nb | 2.2 | 38% | 20 | 5700 | 6
Wang (2009)[16] | ASILJJ[lower-greek 5] BSCCO | 2.8 | 47% | 100 | 800 | 4.2
Knufinke (2012)[15] | ALJJ[lower-greek 4] Nb-AlO-Nb | 1.6 | 23% | 40 | 4900 | 4.2
Sterck (2005, 2009)[8][9] | 3JJ-SQUID Nb-AlO-Nb | 2.5 | 43% | 25 | 1125 | 4.2
Paolucci (2023)[10] | 2JJ SQUID | 3 | 50% | 8 | 72 | 0.4
Menditto (2016)[25] | φ-JJ Nb-AlO-NbCu-Nb | 2.5 | 43% | 150 | 2000 | 1.7
Golod (2022)[26] | inline Nb JJ | 4(10) | 60%(82%) | 8 | 7.2 | 7
Schmid (2024)[27] | inline YBCO JJ | 7 | 75% | 212 | 1 | 4.2
Wu (2022)[3] | NbSe2-Nb3Br8-NbSe2 | 1.07 | 3.4% | 1600[lower-greek 6] | 3.7 | 0.02
Jeon (2022)[35] | Nb-Pt+YIG-Nb | 2.07 | 35% | - | ~4 | 2
Pal (2022)[36] | Nb-Ti-NiTe2-Ti-Nb | 2.3 | 40% | 8[lower-greek 6] | ~3 | 3.8
Baumgartner (2022)[34] | Al-2DEG-Al | 2 | 30% | - | 7 | 0.1
Ghosh (2024)[42] | twisted BSCCO flakes | 4 | 60% | 25[lower-greek 6] | 100 | 80

## See also

* Josephson effect
* φ Josephson junction

## References

1. ↑ 1.0 1.1 Scammell, Harley D; Li, J I A; Scheurer, Mathias S (1 April 2022). "Theory of zero-field superconducting diode effect in twisted trilayer graphene". 2D Materials 9 (2): 025027. doi:10.1088/2053-1583/ac5b16. Bibcode: 2022TDM.....9b5027S.
2. ↑ 2.0 2.1 Davydova, Margarita; Prembabu, Saranesh; Fu, Liang (10 June 2022). "Universal Josephson diode effect". Science Advances 8 (23). doi:10.1126/sciadv.abo0309. PMID 35675396. Bibcode: 2022SciA....8O.309D.
3. ↑ 3.0 3.1 3.2 3.3 3.4 Wu, Heng; Wang, Yaojia; Xu, Yuanfeng; Sivakumar, Pranava K.; Pasco, Chris; Filippozzi, Ulderico; Parkin, Stuart S. P.; Zeng, Yu-Jia et al. (April 2022). "The field-free Josephson diode in a van der Waals heterostructure" (in en). Nature 604 (7907): 653–656. doi:10.1038/s41586-022-04504-8. ISSN 1476-4687. PMID 35478238. Bibcode: 2022Natur.604..653W. https://www.nature.com/articles/s41586-022-04504-8.
4. ↑ Lin, Jiang-Xiazi; Siriviboon, Phum; Scammell, Harley D.; Liu, Song; Rhodes, Daniel; Watanabe, K.; Taniguchi, T.; Hone, James et al. (October 2022). "Zero-field superconducting diode effect in small-twist-angle trilayer graphene". Nature Physics 18 (10): 1221–1227. doi:10.1038/s41567-022-01700-1. Bibcode: 2022NatPh..18.1221L.
5. ↑ Zapata, I.; Bartussek, R.; Sols, F.; Hänggi, P. (1996). "Voltage Rectification by a SQUID Ratchet". Physical Review Letters 77 (11): 2292–2295. doi:10.1103/PhysRevLett.77.2292. PMID 10061907. Bibcode: 1996PhRvL..77.2292Z.
6. ↑ De Waele, A.Th.A.M.; De Bruyn Ouboter, R. (1969). "Quantum-interference phenomena in point contacts between two superconductors". Physica 41 (2): 225–254. doi:10.1016/0031-8914(69)90116-5. Bibcode: 1969Phy....41..225D.
7. ↑ Weiss, S.; Koelle, D.; Müller, J.; Gross, R.; Barthel, K. (2000). "Ratchet effect in dc SQUIDs". Europhysics Letters 51 (5): 499–505. doi:10.1209/epl/i2000-00365-x. Bibcode: 2000EL.....51..499W.
8. ↑ 8.0 8.1 Sterck, A.; Kleiner, R.; Koelle, D. (2005). "Three-Junction SQUID Rocking Ratchet". Physical Review Letters 95 (17). doi:10.1103/PhysRevLett.95.177006. PMID 16383862. Bibcode: 2005PhRvL..95q7006S.
9. ↑ 9.0 9.1 Sterck, A.; Koelle, D.; Kleiner, R. (2009). "Rectification in a Stochastically Driven Three-Junction SQUID Rocking Ratchet". Physical Review Letters 103 (4). doi:10.1103/PhysRevLett.103.047001. PMID 19659390. Bibcode: 2009PhRvL.103d7001S.
10. ↑ 10.0 10.1 Paolucci, F.; De Simoni, G.; Giazotto, F. (2023). "A gate- and flux-controlled supercurrent diode effect". Applied Physics Letters 122 (4): 042601. doi:10.1063/5.0136709. Bibcode: 2023ApPhL.122d2601P.
11. ↑ 11.0 11.1 Krasnov, V.; Oboznov, V.; Pedersen, N. (1997). "Fluxon dynamics in long Josephson junctions in the presence of a temperature gradient or spatial nonuniformity". Physical Review B 55 (21). doi:10.1103/PhysRevB.55.14486. Bibcode: 1997PhRvB..5514486K.
12. ↑ 12.0 12.1 Carapella, G.; Costabile, G. (2001). "Ratchet Effect: Demonstration of a Relativistic Fluxon Diode". Physical Review Letters 87 (7). doi:10.1103/PhysRevLett.87.077002. PMID 11497909. Bibcode: 2001PhRvL..87g7002C.
13. ↑ Carapella, G.; Costabile, G.; Martucciello, N.; Cirillo, M.; Latempa, R.; Polcari, A.; Filatrella, G. (2002). "Experimental realization of a relativistic fluxon ratchet". Physica C: Superconductivity 382 (2–3): 337–341. doi:10.1016/S0921-4534(02)01232-7. Bibcode: 2002PhyC..382..337C.
14. ↑ 14.0 14.1 Beck, M.; Goldobin, E.; Neuhaus, M.; Siegel, M.; Kleiner, R.; Koelle, D. (2005). "High-Efficiency Deterministic Josephson Vortex Ratchet". Physical Review Letters 95 (9). doi:10.1103/PhysRevLett.95.090603. PMID 16197200. Bibcode: 2005PhRvL..95i0603B.
15. ↑ 15.0 15.1 15.2 Knufinke, M.; Ilin, K.; Siegel, M.; Koelle, D.; Kleiner, R.; Goldobin, E. (2012). "Deterministic Josephson vortex ratchet with a load". Physical Review E 85 (1 Pt 1). doi:10.1103/PhysRevE.85.011122. PMID 22400527. Bibcode: 2012PhRvE..85a1122K.
16. ↑ 16.0 16.1 Wang, H. B.; Zhu, B. Y.; Gürlich, C.; Ruoff, M.; Kim, S.; Hatano, T.; Zhao, B. R.; Zhao, Z. X. et al. (2009). "Fast Josephson vortex ratchet made of intrinsic Josephson junctions in Bi2Sr2CaCu2O8". Physical Review B 80 (22). doi:10.1103/PhysRevB.80.224507. Bibcode: 2009PhRvB..80v4507W.
17. ↑ Raissi, F.; Nordman, J. E. (1994). "Josephson fluxonic diode". Applied Physics Letters 65 (14): 1838–1840. doi:10.1063/1.112859. Bibcode: 1994ApPhL..65.1838R.
18. ↑ Raissi, F.; Nordman, J.E. (1995). "Comparison of simulation and experiment for a Josephson fluxonic diode". IEEE Transactions on Applied Superconductivity 5 (2): 2943–2946. doi:10.1109/77.403209. Bibcode: 1995ITAS....5.2943R.
19. ↑ Kadin, A. M. (1990). "Duality and fluxonics in superconducting devices". Journal of Applied Physics 68 (11): 5741–5749. doi:10.1063/1.346969. Bibcode: 1990JAP....68.5741K.
20. ↑ Nordman, James E.; Beyer, James B. (1995-06-13). "Superconductive Electronic Devices Using Flux Quanta." (in en). https://apps.dtic.mil/sti/citations/ADA295627.
21. ↑ Falo, F.; Martínez, P. J.; Mazo, J. J.; Cilla, S. (1999). "Ratchet potential for fluxons in Josephson-junction arrays". Europhysics Letters 45 (6): 700–706. doi:10.1209/epl/i1999-00224-x. Bibcode: 1999EL.....45..700F.
22. ↑ Trías, E.; Mazo, J. J.; Falo, F.; Orlando, T. P. (2000). "Depinning of kinks in a Josephson-junction ratchet array". Physical Review E 61 (3): 2257–2266. doi:10.1103/PhysRevE.61.2257. Bibcode: 2000PhRvE..61.2257T.
23. ↑ Falo, F.; Martínez, P.J.; Mazo, J.J.; Orlando, T.P.; Segall, K.; Trías, E. (2002). "Fluxon ratchet potentials in superconducting circuits". Applied Physics A 75 (2): 263–269. doi:10.1007/s003390201325. Bibcode: 2002ApPhA..75..263F.
24. ↑ Shalóm, D. E.; Pastoriza, H. (2005). "Vortex Motion Rectification in Josephson Junction Arrays with a Ratchet Potential". Physical Review Letters 94 (17). doi:10.1103/PhysRevLett.94.177001. PMID 15904329. Bibcode: 2005PhRvL..94q7001S.
25. ↑ 25.0 25.1 Menditto, R.; Sickinger, H.; Weides, M.; Kohlstedt, H.; Koelle, D.; Kleiner, R.; Goldobin, E. (2016). "TunableφJosephson junction ratchet". Physical Review E 94 (4). doi:10.1103/PhysRevE.94.042202. PMID 27841459. Bibcode: 2016PhRvE..94d2202M.
26. ↑ 26.0 26.1 Golod, Taras; Krasnov, Vladimir M. (2022). "Demonstration of a superconducting diode-with-memory, operational at zero magnetic field with switchable nonreciprocity". Nature Communications 13 (1): 3658. doi:10.1038/s41467-022-31256-w. PMID 35760801. Bibcode: 2022NatCo..13.3658G.
27. ↑ 27.0 27.1 Schmid, Christoph; Jozani, Alireza; Kleiner, Reinhold; Koelle, Dieter; Goldobin, Edward (2025). "YBa2Cu3O7 Josephson diode fabricated by focused-helium-ion-beam irradiation". Physical Review Applied 24. doi:10.1103/vqhx-16ss.
28. ↑ 28.0 28.1 Ando, Fuyuki; Miyasaka, Yuta; Li, Tian; Ishizuka, Jun; Arakawa, Tomonori; Shiota, Yoichi; Moriyama, Takahiro; Yanase, Youichi et al. (August 2020). "Observation of superconducting diode effect" (in en). Nature 584 (7821): 373–376. doi:10.1038/s41586-020-2590-4. ISSN 1476-4687. PMID 32814888. https://www.nature.com/articles/s41586-020-2590-4.
29. ↑ Hu, Jiangping; Wu, Congjun; Dai, Xi (2007-08-09). "Proposed Design of a Josephson Diode". Physical Review Letters 99 (6). doi:10.1103/PhysRevLett.99.067004. PMID 17930858. Bibcode: 2007PhRvL..99f7004H. https://link.aps.org/doi/10.1103/PhysRevLett.99.067004.
30. ↑ Xu, Yuanfeng; Elcoro, Luis; Song, Zhi-Da; Vergniory, M. G.; Felser, Claudia; Parkin, Stuart S. P.; Regnault, Nicolas; Mañes, Juan L.; Bernevig, B. Andrei (2021-06-17). "Filling-Enforced Obstructed Atomic Insulators". arXiv:2106.10276 [cond-mat.mtrl-sci].
31. ↑ Xu, Yuanfeng; Elcoro, Luis; Li, Guowei; Song, Zhi-Da; Regnault, Nicolas; Yang, Qun; Sun, Yan; Parkin, Stuart; Felser, Claudia; Bernevig, B. Andrei (2021-11-03). "Three-Dimensional Real Space Invariants, Obstructed Atomic Insulators and A New Principle for Active Catalytic Sites". arXiv:2111.02433 [cond-mat.mtrl-sci].
32. ↑ Zhang, Yi; Gu, Yuhao; Weng, Hongming; Jiang, Kun; Hu, Jiangping (2023). "Mottness in two-dimensional van der Waals Nb3X8 monolayers (X=Cl, Br, and I)". Physical Review B 107 (3). doi:10.1103/PhysRevB.107.035126. Bibcode: 2023PhRvB.107c5126Z.
33. ↑ Lin, Jiang-Xiazi; Siriviboon, Phum; Scammell, Harley D.; Liu, Song; Rhodes, Daniel; Watanabe, K.; Taniguchi, T.; Hone, James et al. (October 2022). "Zero-field superconducting diode effect in small-twist-angle trilayer graphene". Nature Physics 18 (10): 1221–1227. doi:10.1038/s41567-022-01700-1. Bibcode: 2022NatPh..18.1221L.
34. ↑ 34.0 34.1 Baumgartner, Christian; Fuchs, Lorenz; Costa, Andreas; Reinhardt, Simon; Gronin, Sergei; Gardner, Geoffrey C.; Lindemann, Tyler; Manfra, Michael J. et al. (2022). "Supercurrent rectification and magnetochiral effects in symmetric Josephson junctions". Nature Nanotechnology 17 (1): 39–44. doi:10.1038/s41565-021-01009-9. PMID 34795437. Bibcode: 2022NatNa..17...39B.
35. ↑ 35.0 35.1 Jeon, Kun-Rok; Kim, Jae-Keun; Yoon, Jiho; Jeon, Jae-Chun; Han, Hyeon; Cottet, Audrey; Kontos, Takis; Parkin, Stuart S. P. (2022). "Zero-field polarity-reversible Josephson supercurrent diodes enabled by a proximity-magnetized Pt barrier". Nature Materials 21 (9): 1008–1013. doi:10.1038/s41563-022-01300-7. PMID 35798947. Bibcode: 2022NatMa..21.1008J.
36. ↑ 36.0 36.1 36.2 Pal, Banabir; Chakraborty, Anirban; Sivakumar, Pranava K.; Davydova, Margarita; Gopi, Ajesh K.; Pandeya, Avanindra K.; Krieger, Jonas A.; Zhang, Yang et al. (2022). "Josephson diode effect from Cooper pair momentum in a topological semimetal". Nature Physics 18 (10): 1228–1233. doi:10.1038/s41567-022-01699-5. PMID 36217362. Bibcode: 2022NatPh..18.1228P.
37. ↑ 37.0 37.1 Nadeem, Muhammad; Fuhrer, Michael S.; Wang, Xiaolin (2023-09-15). "The superconducting diode effect" (in en). Nature Reviews Physics 5 (10): 558–577. doi:10.1038/s42254-023-00632-w. ISSN 2522-5820. Bibcode: 2023NatRP...5..558N. https://www.nature.com/articles/s42254-023-00632-w.
38. ↑ Zhang, Yi; Gu, Yuhao; Hu, Jiangping; Jiang, Kun (2022-07-10). "General Theory of Josephson Diodes". Physical Review X 12 (4). doi:10.1103/PhysRevX.12.041013. Bibcode: 2022PhRvX..12d1013Z.
39. ↑ Goldman, A. M.; Kreisman, P. J. (1967-12-10). "Meissner Effect and Vortex Penetration in Josephson Junctions". Physical Review 164 (2): 544–547. doi:10.1103/PhysRev.164.544. Bibcode: 1967PhRv..164..544G. https://link.aps.org/doi/10.1103/PhysRev.164.544.
40. ↑ Yamashita, Tsutomu; Onodera, Yutaka (1967-08-01). "Magnetic-Field Dependence of Josephson Current Influenced by Self-Field". Journal of Applied Physics 38 (9): 3523–3525. doi:10.1063/1.1710164. ISSN 0021-8979. Bibcode: 1967JAP....38.3523Y. https://aip.scitation.org/doi/abs/10.1063/1.1710164.
41. ↑ Goldobin, E.; Menditto, R.; Koelle, D.; Kleiner, R. (2016). "ModelI–-V curves and figures of merit of underdamped deterministic Josephson ratchets". Physical Review E 94 (3). doi:10.1103/PhysRevE.94.032203. PMID 27739827. Bibcode: 2016PhRvE..94c2203G.
42. ↑ Ghosh, Sanat; Patil, Vilas; Basu, Amit; Kuldeep; Dutta, Achintya; Jangade, Digambar A.; Kulkarni, Ruta; Thamizhavel, A. et al. (2024). "High-temperature Josephson diode". Nature Materials 23 (5): 612–618. doi:10.1038/s41563-024-01804-4. PMID 38321240. Bibcode: 2024NatMa..23..612G.

1. ↑ The modern jargon keyword is SNAIL, which denotes a SQUID with N JJs in one branch and one JJ in the other branch. As a whole SNAIL has reflection asymmetric supercurrent-phase relation and, therefore, different critical currents Ic+≠|Ic+|.
2. ↑ Note that some authors[36] divide the difference of critical currents to the average critical current(Ic++Ic−)/2. Then efficiency has the range 0..2, and if they quote 80%, η is actually only 40%.

1. ↑ Maximum rectified voltage achieved
2. ↑ Minimal rectangular area of the device essential for its operation, without electrodes, etc.
3. ↑ Operating temperature for which figures of merit are given
4. ↑ 4.0 4.1 4.2 Annular Long Josephson junction
5. ↑ Annular Stack of Intrinsic Long Josephson Junctions
6. ↑ 6.0 6.1 6.2 For rectangular drive. For sinusoidal drive the voltage is (substantially) lower.

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