Phi Josephson junction

Last updated December 16, 2023 • 5 min readFrom Wikipedia, The Free Encyclopedia

A φ Josephson junction (pronounced phi Josephson junction) is a particular type of the Josephson junction, which has a non-zero Josephson phase φ across it in the ground state. A π Josephson junction, which has the minimum energy corresponding to the phase of π, is a specific example of it.

Introduction

The Josephson energy $U$ depends on the superconducting phase difference (Josephson phase) $\phi$ periodically, with the period $2\pi$ . Therefore, let us focus only on one period, e.g. $-\pi <\phi \leq +\pi$ . In the ordinary Josephson junction the dependence $U(\phi )$ has the minimum at $\phi =0$ . The function

U(\phi )={\frac {\Phi _{0}I_{c}}{2\pi }}[1-\cos(\phi )]

,

where $I c$ is the critical current of the junction, and $\Phi _{0}$ is the flux quantum, is a good example of conventional $U(\phi )$ .

Instead, when the Josephson energy $U(\phi )$ has a minimum (or more than one minimum per period) at $\phi \neq 0$ , these minimum (minima) correspond to the lowest energy states (ground states) of the junction and one speaks about "φ Josephson junction". Consider two examples.

First, consider the junction with the Josephson energy $U(\phi )$ having two minima at $\phi =\pm \varphi$ within each period, where $\varphi$ (such that $0<\varphi <\pi$ ) is some number. For example, this is the case for

$U(\phi )={\frac {\Phi _{0}}{2\pi }}\left\{I_{c1}[1-\cos(\phi )]+{\frac {1}{2}}I_{c2}[1-\cos(2\phi )]\right\}$ ,

which corresponds to the current-phase relation

$I_{s}(\phi )=I_{c1}\sin(\phi )+I_{c2}\sin(2\phi )$ .

If $I c1 >0$ and $I c2 <-1/2<0$ , the minima of the Josephson energy occur at $\phi =\pm \varphi$ , where $\varphi =\arccos \left(-2I_{c1}/I_{c2}\right)$ . Note, that the ground state of such a Josephson junction is doubly degenerate because $U(-\varphi )=U(+\varphi )$ .

Another example is the junction with the Josephson energy similar to conventional one, but shifted along $\phi$ -axis, for example $U(\phi )={\frac {\Phi _{0}I_{c}}{2\pi }}[1-\cos(\phi -\varphi _{0})]$ ,

and the corresponding current-phase relation

$I_{s}(\phi )=I_{c}\sin(\phi -\varphi _{0})$ .

In this case the ground state is $\phi =\varphi _{0}$ and it is not degenerate.

The above two examples show that the Josephson energy profile in φ Josephson junction can be rather different, resulting in different physical properties. Often, to distinguish, which particular type of the current-phase relation is meant, the researches are using different names. At the moment there is no well-accepted terminology. However, some researchers use the terminology after A. Buzdin:^[1] the Josephson junction with double degenerate ground state $\phi =\pm \varphi$ , similar to the first example above, are indeed called φ Josephson junction, while the junction with non-degenerate ground state, similar to the second example above, are called $\varphi _{0}$ Josephson junctions.

Realization of φ junctions

The first indications of φ junction behavior (degenerate ground states^[2] or unconventional temperature dependence of its critical current^[3]) were reported in the beginning of the 21st century. These junctions were made of d-wave superconductors.

The first experimental realization of controllable φ junction was reported in September 2012 by the group of Edward Goldobin at University of Tübingen.^[4] It is based on a combination of 0 and π segments in one superconducting-insulator-ferromagnetic-superconductor hybrid device and clearly demonstrates two critical currents corresponding to two junction states $\phi =\pm \varphi$ . The proposal to construct a φ Josephson junction out of (infinitely) many 0 and π segments has appeared in the works by R. Mints and coauthors,^[5]^[6] although at that time there was no term φ junction. For the first time the word φ Josephson junction appeared in the work of Buzdin and Koshelev,^[1] whose idea was similar. Following this idea, it was further proposed to use a combination of only two 0 and π segments.^[7]

In 2016, a $\varphi _{0}$ junction based on the nanowire quantum dot was reported by the group of Leo Kouwenhoven at Delft University of Technology. The InSb nanowire has strong spin-orbit coupling, and magnetic field was applied leading to Zeeman effect. This combination breaks both inversion and time-reversal symmetries creating finite current at zero phase difference.^[8]

Other theoretically proposed realization include geometric φ junctions. There is a theoretical prediction that one can construct the so-called geometric φ junction based on nano-structured d-wave superconductor.^[9] As of 2013, this was not demonstrated experimentally.

Properties of φ junctions

Two critical currents related to the escape (depinning) of the phase from two different wells of the Josephson potential. The lowest critical current can be seen experimentally only at low damping (low temperature). The measurements of the critical current can be used to determine the (unknown) state (+φ or -φ) of φ junction.
In the case of φ junction constructed out of 0 and π segments, magnetic field can be used to change the asymmetry of the Josephson energy profile up to the point that one of the minima disappears. This allows to prepare the desired state (+φ or -φ). Also, asymmetric periodic Josephson energy potential can be used to construct ratchet-like devices.
Long φ junctions allow special types of soliton solutions --- the splintered vortices^[10] of two types: one carries the magnetic flux $Φ 1 < Φ 0$ , while the other carries the flux $Φ 2 = Φ 0 - Φ 1$ . Here $Φ 0$ is the magnetic flux quantum. These vortices are the solitons of a double sine-Gordon equation.^[11] They were observed in d-wave grain boundary junctions.^[6]

Applications

Similar to π Josephson junction, φ junctions can be used as a phase battery.
Two stable states +φ and -φ can be used to store a digital information. To write the desired state one can apply magnetic field, so that one of the energy minima disappears, so the phase has no choice as to go to the remaining one. To read out an unknown state of the φ junctions one can apply the bias current with value between the two critical currents. If the φ junctions switches to the voltage state, its state was −φ, otherwise, it was +φ. The use of φ junctions as a memory cell (1 bit) was already demonstrated.^[12]
In quantum domain the φ junction can be used as a two-level system (qubit).

Related Research Articles

A SQUID is a very sensitive magnetometer used to measure extremely weak magnetic fields, based on superconducting loops containing Josephson junctions.

The magnetic flux, represented by the symbol $Φ$ , threading some contour or loop is defined as the magnetic field $B$ multiplied by the loop area $S$ , i.e. $Φ = B \cdot S$ . Both $B$ and $S$ can be arbitrary, meaning $Φ$ can be as well. However, if one deals with the superconducting loop or a hole in a bulk superconductor, the magnetic flux threading such a hole/loop is quantized. The (superconducting) magnetic flux quantum $Φ 0 = h /(2 e)$ ≈ 2.067833848...×10⁻¹⁵ Wb is a combination of fundamental physical constants: the Planck constant $h$ and the electron charge $e$ . Its value is, therefore, the same for any superconductor. The phenomenon of flux quantization was discovered experimentally by B. S. Deaver and W. M. Fairbank and, independently, by R. Doll and M. Näbauer, in 1961. The quantization of magnetic flux is closely related to the Little–Parks effect, but was predicted earlier by Fritz London in 1948 using a phenomenological model.

The sine-Gordon equation is a nonlinear hyperbolic partial differential equation for a function $dependent on two variables typically denoted and, involving the wave operator and the sine of .$

<span class="mw-page-title-main">Josephson effect</span> Quantum physical phenomenon

In physics, the Josephson effect is a phenomenon that occurs when two superconductors are placed in proximity, with some barrier or restriction between them. The effect is named after the British physicist Brian Josephson, who predicted in 1962 the mathematical relationships for the current and voltage across the weak link. It is an example of a macroscopic quantum phenomenon, where the effects of quantum mechanics are observable at ordinary, rather than atomic, scale. The Josephson effect has many practical applications because it exhibits a precise relationship between different physical measures, such as voltage and frequency, facilitating highly accurate measurements.

In particle physics, the Peccei–Quinn theory is a well-known, long-standing proposal for the resolution of the strong CP problem formulated by Roberto Peccei and Helen Quinn in 1977. The theory introduces a new anomalous symmetry to the Standard Model along with a new scalar field which spontaneously breaks the symmetry at low energies, giving rise to an axion that suppresses the problematic CP violation. This model has long since been ruled out by experiments and has instead been replaced by similar invisible axion models which utilize the same mechanism to solve the strong CP problem.

The Little–Parks effect was discovered in 1962 by William A. Little and Ronald D. Parks in experiments with empty and thin-walled superconducting cylinders subjected to a parallel magnetic field. It was one of the first experiments to indicate the importance of Cooper-pairing principle in BCS theory.

The Berezinskii–Kosterlitz–Thouless (BKT) transition is a phase transition of the two-dimensional (2-D) XY model in statistical physics. It is a transition from bound vortex-antivortex pairs at low temperatures to unpaired vortices and anti-vortices at some critical temperature. The transition is named for condensed matter physicists Vadim Berezinskii, John M. Kosterlitz and David J. Thouless. BKT transitions can be found in several 2-D systems in condensed matter physics that are approximated by the XY model, including Josephson junction arrays and thin disordered superconducting granular films. More recently, the term has been applied by the 2-D superconductor insulator transition community to the pinning of Cooper pairs in the insulating regime, due to similarities with the original vortex BKT transition.

Superconducting quantum computing is a branch of solid state quantum computing that implements superconducting electronic circuits using superconducting qubits as artificial atoms, or quantum dots. For superconducting qubits, the two logic states are the ground state and the excited state, denoted $respectively. Research in superconducting quantum computing is conducted by companies such as Google, IBM, IMEC, BBN Technologies, Rigetti, and Intel. Many recently developed QPUs use superconducting architecture.$

In superconductivity, a semifluxon is a half integer vortex of supercurrent carrying the magnetic flux equal to the half of the magnetic flux quantum $Φ 0$ . Semifluxons exist in the 0-π long Josephson junctions at the boundary between 0 and π regions. This 0-π boundary creates a π discontinuity of the Josephson phase. The junction reacts to this discontinuity by creating a semifluxon. Vortex's supercurrent circulates around 0-π boundary. In addition to semifluxon, there exist also an antisemifluxon. It carries the flux $-Φ 0 /2$ and its supercurrent circulates in the opposite direction.

In superconductivity, a Josephson vortex is a quantum vortex of supercurrents in a Josephson junction. The supercurrents circulate around the vortex center which is situated inside the Josephson barrier, unlike Abrikosov vortices in type-II superconductors, which are located in the superconducting condensate.

<span class="mw-page-title-main">Flux qubit</span> Superconducting qubit implementation

In quantum computing, more specifically in superconducting quantum computing, flux qubits are micrometer sized loops of superconducting metal that is interrupted by a number of Josephson junctions. These devices function as quantum bits. The flux qubit was first proposed by Terry P. Orlando et al. at MIT in 1999 and fabricated shortly thereafter. During fabrication, the Josephson junction parameters are engineered so that a persistent current will flow continuously when an external magnetic flux is applied. Only an integer number of flux quanta are allowed to penetrate the superconducting ring, resulting in clockwise or counter-clockwise mesoscopic supercurrents in the loop to compensate a non-integer external flux bias. When the applied flux through the loop area is close to a half integer number of flux quanta, the two lowest energy eigenstates of the loop will be a quantum superposition of the clockwise and counter-clockwise currents. The two lowest energy eigenstates differ only by the relative quantum phase between the composing current-direction states. Higher energy eigenstates correspond to much larger (macroscopic) persistent currents, that induce an additional flux quantum to the qubit loop, thus are well separated energetically from the lowest two eigenstates. This separation, known as the "qubit non linearity" criteria, allows operations with the two lowest eigenstates only, effectively creating a two level system. Usually, the two lowest eigenstates will serve as the computational basis for the logical qubit.

In physics, a quantum vortex represents a quantized flux circulation of some physical quantity. In most cases, quantum vortices are a type of topological defect exhibited in superfluids and superconductors. The existence of quantum vortices was first predicted by Lars Onsager in 1949 in connection with superfluid helium. Onsager reasoned that quantisation of vorticity is a direct consequence of the existence of a superfluid order parameter as a spatially continuous wavefunction. Onsager also pointed out that quantum vortices describe the circulation of superfluid and conjectured that their excitations are responsible for superfluid phase transitions. These ideas of Onsager were further developed by Richard Feynman in 1955 and in 1957 were applied to describe the magnetic phase diagram of type-II superconductors by Alexei Alexeyevich Abrikosov. In 1935 Fritz London published a very closely related work on magnetic flux quantization in superconductors. London's fluxoid can also be viewed as a quantum vortex.

A Josephson junction (JJ) is a quantum mechanical device which is made of two superconducting electrodes separated by a barrier. A $π$ Josephson junction is a Josephson junction in which the Josephson phase φ equals $π$ in the ground state, i.e. when no external current or magnetic field is applied.

In a standard superconductor, described by a complex field fermionic condensate wave function, vortices carry quantized magnetic fields because the condensate wave function $is invariant to increments of the phase by . There a winding of the phase by creates a vortex which carries one flux quantum. See quantum vortex.$

In quantum computing, and more specifically in superconducting quantum computing, the phase qubit is a superconducting device based on the superconductor–insulator–superconductor (SIS) Josephson junction, designed to operate as a quantum bit, or qubit.

Type-1.5 superconductors are multicomponent superconductors characterized by two or more coherence lengths, at least one of which is shorter than the magnetic field penetration length $, and at least one of which is longer. This is in contrast to single-component superconductors, where there is only one coherence length and the superconductor is necessarily either type 1 or type 2. When placed in magnetic field, type-1.5 superconductors should form quantum vortices: magnetic-flux-carrying excitations. They allow magnetic field to pass through superconductors due to a vortex-like circulation of superconducting particles. In type-1.5 superconductors these vortices have long-range attractive, short-range repulsive interaction. As a consequence a type-1.5 superconductor in a magnetic field can form a phase separation into domains with expelled magnetic field and clusters of quantum vortices which are bound together by attractive intervortex forces. The domains of the Meissner state retain the two-component superconductivity, while in the vortex clusters one of the superconducting components is suppressed. Thus such materials should allow coexistence of various properties of type-I and type-II superconductors.$

In quantum field theory, a non-topological soliton (NTS) is a soliton field configuration possessing, contrary to a topological one, a conserved Noether charge and stable against transformation into usual particles of this field for the following reason. For fixed charge Q, the mass sum of Q free particles exceeds the energy (mass) of the NTS so that the latter is energetically favorable to exist.

The Aharonov–Casher effect is a quantum mechanical phenomenon predicted in 1984 by Yakir Aharonov and Aharon Casher, in which a traveling magnetic dipole is affected by an electric field. It is dual to the Aharonov–Bohm effect, in which the quantum phase of a charged particle depends upon which side of a magnetic flux tube it comes through. In the Aharonov–Casher effect, the particle has a magnetic moment and the tubes are charged instead. It was observed in a gravitational neutron interferometer in 1989 and later by fluxon interference of magnetic vortices in Josephson junctions. It has also been seen with electrons and atoms.

<span class="mw-page-title-main">Alexandre Bouzdine</span> French and Russian theoretical physicist

Alexandre Bouzdine (Buzdin) (in Russian - Александр Иванович Буздин; born March 16, 1954) is a French and Russian theoretical physicist in the field of superconductivity and condensed matter physics. He was awarded the Holweck Medal in physics in 2013 and obtained the Gay-Lussac Humboldt Prize in 2019 for his theoretical contributions in the field of coexistence between superconductivity and magnetism.

References

1 2 Buzdin, A.; Koshelev, A. (June 2003). "Periodic alternating 0- and π-junction structures as realization of φ-Josephson junctions". Physical Review B. 67 (22): 220504. arXiv: cond-mat/0305142 . Bibcode:2003PhRvB..67v0504B. doi:10.1103/PhysRevB.67.220504. S2CID 119407977.
↑ Il'ichev, E.; Grajcar, M.; Hlubina, R.; IJsselsteijn, R. P. J.; Hoenig, H. E.; Meyer, H.-G.; Golubov, A.; Amin, M. H. S.; Zagoskin, A. M.; Omelyanchouk, A. N.; Kupriyanov, M. Yu. (4 June 2001). "Degenerate Ground State in a Mesoscopic Grain Boundary Josephson Junction". Physical Review Letters. 86 (23): 5369–5372. arXiv: cond-mat/0102404 . Bibcode:2001PhRvL..86.5369I. doi:10.1103/PhysRevLett.86.5369. PMID 11384500. S2CID 24036125.
↑ Testa, G.; Monaco, A.; Esposito, E.; Sarnelli, E.; Kang, D.-J.; Mennema, S. H.; Tarte, E. J.; Blamire, M. G. (2004). "Midgap state-based π-junctions for digital applications". Applied Physics Letters. 85 (7): 1202. Bibcode:2004ApPhL..85.1202T. doi:10.1063/1.1781744.
↑ Sickinger, H.; Lipman, A.; Weides, M.; Mints, R. G.; Kohlstedt, H.; Koelle, D.; Kleiner, R.; Goldobin, E. (September 2012). "Experimental Evidence of a φ Josephson Junction". Physical Review Letters. 109 (10): 107002. arXiv: 1207.3013 . Bibcode:2012PhRvL.109j7002S. doi:10.1103/PhysRevLett.109.107002. PMID 23005318. S2CID 15055676.
↑ Mints, R. (February 1998). "Self-generated flux in Josephson junctions with alternating critical current density". Physical Review B. 57 (6): R3221–R3224. Bibcode:1998PhRvB..57.3221M. doi:10.1103/PhysRevB.57.R3221.
1 2 Mints, R.; Papiashvili, Ilya (August 2001). "Josephson vortices with fractional flux quanta at YBa2Cu3O7-x grain boundaries". Physical Review B. 64 (13): 134501. Bibcode:2001PhRvB..64m4501M. doi:10.1103/PhysRevB.64.134501.
↑ Goldobin, E.; Koelle, D.; Kleiner, R.; Mints, R. G. (November 2011). "Josephson Junction with a Magnetic-Field Tunable Ground State". Physical Review Letters. 107 (22): 227001. arXiv: 1110.2326 . Bibcode:2011PhRvL.107v7001G. doi:10.1103/PhysRevLett.107.227001. PMID 22182037. S2CID 15019215.
↑ Szombati, D. B.; S. Nadj-Perge; D. Car; S. R. Plissard; E. P. A. M. Bakkers; L. P. Kouwenhoven (2 May 2016). "Josephson ϕ0-junction in nanowire quantum dots". Nature Physics . 12 (6): 568–572. arXiv: 1512.01234 . Bibcode:2016NatPh..12..568S. doi:10.1038/nphys3742. S2CID 38016105.
↑ Gumann, A.; Iniotakis, C.; Schopohl, N. (2007). "Geometric π Josephson junction in d-wave superconducting thin films". Applied Physics Letters. 91 (19): 192502. arXiv: 0708.3898 . Bibcode:2007ApPhL..91s2502G. doi:10.1063/1.2801387. S2CID 119119995.
↑ Mints, R.; Papiashvili, Ilya; Kirtley, J.; Hilgenkamp, H.; Hammerl, G.; Mannhart, J. (July 2002). "Observation of Splintered Josephson Vortices at Grain Boundaries in YBa2Cu3O_7−δ". Physical Review Letters. 89 (6): 067004. Bibcode:2002PhRvL..89f7004M. doi:10.1103/PhysRevLett.89.067004. PMID 12190605.
↑ Goldobin, E.; Koelle, D.; Kleiner, R.; Buzdin, A. (December 2007). "Josephson junctions with second harmonic in the current-phase relation: Properties of φ junctions". Physical Review B. 76 (22): 224523. arXiv: 0708.2624 . Bibcode:2007PhRvB..76v4523G. doi:10.1103/PhysRevB.76.224523. S2CID 55468272.
↑ Goldobin, E.; Sickinger, H.; Weides, M.; Ruppelt, N.; Kohlstedt, H.; Kleiner, R.; Koelle, D. (2013). "Memory cell based on a ϕ Josephson junction". Applied Physics Letters. 102 (24): 242602. arXiv: 1306.1683 . Bibcode:2013ApPhL.102x2602G. doi:10.1063/1.4811752. S2CID 113004268.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Buzdin:2003:phi-LJJ-1] 1 2 Buzdin, A.; Koshelev, A. (June 2003). "Periodic alternating 0- and π-junction structures as realization of φ-Josephson junctions". Physical Review B. 67 (22): 220504. arXiv: cond-mat/0305142 . Bibcode:2003PhRvB..67v0504B. doi:10.1103/PhysRevB.67.220504. S2CID 119407977.

[Ilichev:2001-2] Il'ichev, E.; Grajcar, M.; Hlubina, R.; IJsselsteijn, R. P. J.; Hoenig, H. E.; Meyer, H.-G.; Golubov, A.; Amin, M. H. S.; Zagoskin, A. M.; Omelyanchouk, A. N.; Kupriyanov, M. Yu. (4 June 2001). "Degenerate Ground State in a Mesoscopic Grain Boundary Josephson Junction". Physical Review Letters. 86 (23): 5369–5372. arXiv: cond-mat/0102404 . Bibcode:2001PhRvL..86.5369I. doi:10.1103/PhysRevLett.86.5369. PMID 11384500. S2CID 24036125.

[Testa:2004-3] Testa, G.; Monaco, A.; Esposito, E.; Sarnelli, E.; Kang, D.-J.; Mennema, S. H.; Tarte, E. J.; Blamire, M. G. (2004). "Midgap state-based π-junctions for digital applications". Applied Physics Letters. 85 (7): 1202. Bibcode:2004ApPhL..85.1202T. doi:10.1063/1.1781744.

[Sickinger:2012:varphiExp-4] Sickinger, H.; Lipman, A.; Weides, M.; Mints, R. G.; Kohlstedt, H.; Koelle, D.; Kleiner, R.; Goldobin, E. (September 2012). "Experimental Evidence of a φ Josephson Junction". Physical Review Letters. 109 (10): 107002. arXiv: 1207.3013 . Bibcode:2012PhRvL.109j7002S. doi:10.1103/PhysRevLett.109.107002. PMID 23005318. S2CID 15055676.

[Mints:1998:SelfGenFlux@AltJc-5] Mints, R. (February 1998). "Self-generated flux in Josephson junctions with alternating critical current density". Physical Review B. 57 (6): R3221–R3224. Bibcode:1998PhRvB..57.3221M. doi:10.1103/PhysRevB.57.R3221.

[Mints:2001:FracVortices@GB-6] 1 2 Mints, R.; Papiashvili, Ilya (August 2001). "Josephson vortices with fractional flux quanta at YBa2Cu3O7-x grain boundaries". Physical Review B. 64 (13): 134501. Bibcode:2001PhRvB..64m4501M. doi:10.1103/PhysRevB.64.134501.

[Goldobin:2011:H-TunableEffCPR-7] Goldobin, E.; Koelle, D.; Kleiner, R.; Mints, R. G. (November 2011). "Josephson Junction with a Magnetic-Field Tunable Ground State". Physical Review Letters. 107 (22): 227001. arXiv: 1110.2326 . Bibcode:2011PhRvL.107v7001G. doi:10.1103/PhysRevLett.107.227001. PMID 22182037. S2CID 15019215.

[8] Szombati, D. B.; S. Nadj-Perge; D. Car; S. R. Plissard; E. P. A. M. Bakkers; L. P. Kouwenhoven (2 May 2016). "Josephson ϕ0-junction in nanowire quantum dots". Nature Physics . 12 (6): 568–572. arXiv: 1512.01234 . Bibcode:2016NatPh..12..568S. doi:10.1038/nphys3742. S2CID 38016105.

[Gumann:2007:Geometric-pi-JJ-9] Gumann, A.; Iniotakis, C.; Schopohl, N. (2007). "Geometric π Josephson junction in d-wave superconducting thin films". Applied Physics Letters. 91 (19): 192502. arXiv: 0708.3898 . Bibcode:2007ApPhL..91s2502G. doi:10.1063/1.2801387. S2CID 119119995.

[Mints:2002:SplinteredVortices@GB-10] Mints, R.; Papiashvili, Ilya; Kirtley, J.; Hilgenkamp, H.; Hammerl, G.; Mannhart, J. (July 2002). "Observation of Splintered Josephson Vortices at Grain Boundaries in YBa2Cu3O_7−δ". Physical Review Letters. 89 (6): 067004. Bibcode:2002PhRvL..89f7004M. doi:10.1103/PhysRevLett.89.067004. PMID 12190605.

[Goldobin:2007:CPR2-11] Goldobin, E.; Koelle, D.; Kleiner, R.; Buzdin, A. (December 2007). "Josephson junctions with second harmonic in the current-phase relation: Properties of φ junctions". Physical Review B. 76 (22): 224523. arXiv: 0708.2624 . Bibcode:2007PhRvB..76v4523G. doi:10.1103/PhysRevB.76.224523. S2CID 55468272.

[Goldobin:2013:varphi-bit-12] Goldobin, E.; Sickinger, H.; Weides, M.; Ruppelt, N.; Kohlstedt, H.; Kleiner, R.; Koelle, D. (2013). "Memory cell based on a ϕ Josephson junction". Applied Physics Letters. 102 (24): 242602. arXiv: 1306.1683 . Bibcode:2013ApPhL.102x2602G. doi:10.1063/1.4811752. S2CID 113004268.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]