Superconductor Electronics: Status and Outlook
- 833 Downloads
Superconductor electronics combines passive and active superconducting components and sometimes normal resistors into functional circuits and systems that also include room-temperature electronics for amplification, power sources, necessary controls, etc., usually computer operated. Furthermore, complete systems include magnetic and electromagnetic shielding, cryogenic enclosures, and increasingly a cryocooler in self-contained units. Components or devices of low or high critical temperature superconductors include inductances (coils), passive transmission lines, resonators, antennae, filters, as well as active elements: Josephson junctions, Josephson oscillators, and superconducting quantum interference devices. Of multiple demonstrated applications, mostly but not only in science and metrology, currently most successful are voltage standards, astronomy detectors and large telescope cameras, instruments for material characterization, and magnetometers for geomagnetic prospecting. Major current efforts concentrate on energy-efficient high-end computing and quantum computing. The outcomes of these efforts are likely to be known in the course of the following decade.
KeywordsSuperconductor electronics Flux quantum Transmission line Resonator Josephson junction Josephson oscillator SQUID SQUID magnetometer Astronomy detector Multiplexer Single flux quantum logic SFQ: rapid SFQ RSFQ Energy-efficient SFQ ERSFQ Reciprocal quantum logic RQL Adiabatic quantum flux parametron AQFP Qubit Transmon Quantum processor Quantum computing Adiabatic annealing
Superconductor electronics (SCE) are functional electronic circuits incorporating active (nonlinear) and passive (linear) elements that are superconducting below the critical temperature (Tc). Normal conducting resistors may be included, if functionally necessary. In applications and when operated sufficiently below Tc, such circuits may exhibit unique characteristics and unsurpassed performance not attainable by conventional semiconductor electronics. However, the necessity of cooling to and operating at cryogenic temperatures is the main penalty that severely limits practical uses and thus far constrains SCE to remain a niche technology. It is and will be used only when nothing else will do or when operation at very low temperatures is necessary for quantum performance, independent of the device and circuit technology applied.
This short overview refrains from tutorial descriptions and most technical details; given are only few examples. I attempt instead to briefly characterize the status of devices, circuits, and their practical applications, including my views on the possible future developments. Fundamentals of SCE are given in [1, 2]. My earlier overview of SCE dates back to 1996/1997 and could still include many examples . Thanks to the impressive development of this field, now I attempt mostly to highlight the progress of the last two decades in what amounts to little more than a topical guide to the SCE abundant literature. I should note that a recent handbook on superconductor devices and applications devotes about 400 pages to a relatively up-to-date coverage of SCE .
Superconductor electronics exploit basic properties of superconductors such as the magnetic flux exclusion (the Meissner effect), flux quantization, extremely low absorption (power dissipation) in AC fields up to the gap frequency (fg), and both DC and AC Josephson effects with related macroscopic quantum behavior. For SCE, the superconductor material workhorse is niobium (Nb), having Tc ≈ 9 K and fg ≈ 0.7 THz. It is a conventional (s band) low-temperature superconductor (LTS) suitable for circuit operation typically at 4 to 5 K. Other LTSs, with much lower Tc, are widely used in radiation and particle detectors and in quantum bits (qubits). Conventional superconductors with Tc higher than that of Nb, such as niobium nitrides, carbonitrides and MgB2 are rarely used, although the latter may have some future in applications.
In spite of their Tc and fg, an order of magnitude higher than in the case of Nb, the unconventional (d band) superconductors (HTS materials), typically represented by rare-earth cuprates such as YBa2Cu3O7−δ (YBCO), found thus far only limited use in electronics, mostly in passive devices such as high-frequency resonators and filters and in portable magnetometers. This is due primarily to the difficult, expensive, and less-reliable fabrication technology of epitaxial films and multilayers necessitated by the extremely short coherence lengths, high anisotropy of properties and, caused by thermal energy, inferior performance at higher temperatures, typically at or around that of liquid nitrogen (LN2). Low price and wide availability of this coolant, as well as much higher efficiency and reliability of cryocoolers used in this temperature range, did not outweigh thus far the disadvantages indicated above, except in mobile applications, such as minerals exploration, for example.
In SCE components, devices, and circuits, superconductor thin films are incorporated into multilayered patterned structures together with other thin film materials, primarily insulators and normal conductors (e.g., for resistors and contact metallization) on suitable low dielectric permittivity (ε) and loss (tanδ) substrates such as Si, quartz, glass, and sapphire in the case of LTS electronics. Of insulators, most typical are SiO2, SiO, Al2O3, and anodized Nb. The HTS electronics requires epitaxial films and lattice-matching substrates such as SrTiO3 for low-frequency circuits, MgO, LiNbO3, and similar perovskites for high-frequency (HF) applications. The integrated circuit (IC) fabrication is largely analogous to that of semiconductors while much less advanced than the latter. Nanotechnology and large-scale integration (LSI) needs, driven mostly by current efforts into digital SCE development, thus far resulted in demonstrated integration levels approaching 106 functional devices on the chip, with 104 being typical in existing applications. The IC fabrication with eight to ten photolithography-patterned levels represents the current state of the art and is practiced mostly in commercial and laboratory foundries of which currently most advanced are those operated by D-Wave and MIT Lincoln Labs (LL) in the USA and the National Institute of Advanced Industrial Science and Technology (AIST, located in Tsukuba, Ibaraki) in Japan.
The basic linear components and devices in superconductor circuits are low-loss transmission lines, antennae, high-quality-factor (Q) resonators, and filters. The most basic nonlinear two-terminal device with unique properties is the Josephson junction. The superconducting quantum interference devices (SQUIDs) incorporating JJs are also components of many electronic circuits.
Each practical SCE electronics system consists of the following essential building blocks: (1) the SCE device circuitry, (2) magnetic and electromagnetic shielding of that circuitry, (3) cryogenics assuring the very low and stable operating temperature (or selected temperature levels), and (4) the room temperature electronics for readout and control including the necessary power supply with various biasing sources. (5) In mobile systems, motion and vibration compensation measures are usually required. Below, the first three are briefly addressed.
2 Building Blocks
2.1 Passive Superconductor Devices
Linear devices are best characterized by the frequency dependence of the superconductor surface impedance (Zs). Its real (loss) part is the surface resistance (Rs), the imaginary part is the kinetic inductance. At frequencies with photon energies sufficiently below that of energy gap (fg) and below Tc, these losses are orders of magnitude lower than in a cooled normal conductor such as high-purity copper at the same T. For reasons probably related to d-wave symmetry, the losses in HTS are somewhat higher than in niobium at the same reduced temperature (tr) = T/Tc and frequency (fr) = f/fg, but still low enough. At microwave frequencies, the practical consequence of the low superconductor losses, and of the fact that tanδ of crystalline substrates decreases rapidly with T, are low-loss transmission lines and high-quality factor (Q) resonators. For thin film planar (microstrip, stripline, coplanar, or lumped element) HTS resonator, Q values between 103 and 105 are readily obtainable at 77 K and mostly HTS resonators are used today, although novel applications emerge for LTS resonators. Three-dimensional dielectric resonators shielded with HTS films attain 105 to 107 in low-intensity electromagnetic fields. In higher radiofrequency (RF) fields, when the RF current exceeds its critical value, losses increase dramatically and nonlinear effects appear. Similarly, transmission lines exhibit extremely low insertion losses and are nondispersive, i.e., their phase velocity remains frequency-independent up to fg. Fundamentals of resonators and transmission lines can be found in textbooks on microwave engineering, for example .
2.2 Active Superconductor Devices
The essential superconductor nonlinear devices are the Josephson junctions (JJs) and SQUIDs. Both exhibit rich quantum physics at a macroscopic scale and have no equivalents in semiconductor electronics. Theory and properties of JJs are systematically presented in textbooks [1, 2]. Other two textbooks [6, 7, 8] cover SQUIDs and their applications in considerable detail, but are somewhat dated, especially with respect to applications. More up to date on these is . The theory and experiments elucidating macroscopic quantum properties of JJs are reviewed and summarized in .
The supercurrent I through a JJ is given by the 1st equation, where I0 is the junction’s critical current and φ is the difference φ1 − φ2 between phases of the superconductor’s function of state in two weakly linked superconducting electrodes. Equation (1) defines the static properties of a JJ. At currents exceeding I0, voltage drop (V ) appears across the JJ and phases of the functions of state are slipping relative to each other at a rate determined by (2), where e is the electron charge, h the Planck constant, and Φ0 the flux quantum. Hence, in the voltage state, the JJ emits radiation of frequency fJ = 2eV /h ≈ 483 V (MHz/μV) and the correlation between V and fJ is defined only by fundamental constants. The idea of V -tunable oscillators has been intensely pursued experimentally but without convincing technical success for LTS.1 When a voltage-biased JJ is exposed to an external rf (microwave) field of frequency (f), its I − V characteristic exhibits sharp current steps, the so-called Shapiro steps, at voltages V = nhf /2e, where n = 0, 1, 2, 3 ... is an integer.2
Here, Φ0 = h/2e= 2.0679 × 10− 15 Wb is the flux quantum, h the Planck’s constant and e the electron charge. The cursory description above shows that JJs are not simply superconducting analogs of semiconductor diodes, just having the energy gap orders of magnitude lower than semiconductors.
In contrast to semiconductor electronics, a single-element three-terminal device fully analogous to a transistor in being capable of power gain and fan-out does not exist. Devices with either current or voltage gain are possible. However, SQUIDs can be considered to be three-terminal devices, with the gate signal being the electromagnetic (EM) field threading the loop. SQUIDs are capable of power amplification up to RF frequencies in gigahertz range. Properties and characteristics of devices and circuits incorporating DC SQUIDs are addressed in conjunction with their various applications.
The most widely used DC SQUID incorporates two JJs interrupting a closed superconducting loop biased by external current or voltage source. The RF SQUID includes only one JJ and in most cases is inductively RF biased via a resonant tank circuit. SQUIDs with more than two JJs are used as some types of qubits. The general property of any SQUID is that, when biased into the voltage state, its output signal (voltage or current) is a periodical function of the magnetic flux threading the SQUID loop, with the period equal Φ0. In the following sections, only DC SQUIDs are covered while RF SQUIDs are described in .
Magnetic field and flux vortices, mostly of Abrikosov type, as well as electromagnetic fields, can penetrate superconducting films and devices in several ways. Static (DC) field biasing causes changes in operating parameters, electromagnetic interference (EMI), electromagnetic crosstalk, low-frequency flux noise, motion in static Earth’s field due to vibration and displacement of mobile SCE instrumentation, etc., may result in parameter change, noise, or loss of flux lock in SQUID magnetometers. All these deleterious effects affect properties and behavior of devices, circuits, and systems and must be prevented or at least adequately minimized by shields and other measures such as on-chip local pinning of penetrating vortices by antidots, moats, and other suitable local discontinuities in the superconducting layers, for example. Obviously, SQUIDs and SQUID-based devices such as SFQ cells, gates, and qubits are particularly vulnerable and therefore require particularly effective on-chip vortex traps, magnetic and electromagnetic shielding enclosures, preferably superconducting, and very careful electromagnetic shielding, e.g., by very thin metallized Mylar foils and multilayers. Shielding of scientific experiments, e.g., nuclear magnetic resonance (NMR) or search for electric dipole moment (EDM) and biomagnetic diagnostic systems require magnetic shielded rooms (MSRs) with walls of several sheet layers of high-permeability soft alloy (e.g., Mu-metal) and aluminum, compensation of static and low-frequency fields by the, so-called, active shielding, etc. General principles of magnetic and electromagnetic shielding are treated by textbooks, see for example [11, 12], which include design and calculations of shielding enclosures and their effectiveness. Shielding of large superconducting systems is discussed, e.g., in , of qubits in , and shielding by superconductors in . With the advent of HTS, tubular can ceramic superconductor shields, especially those fabricated of Bi-2223, became popular and are available commercially. In practice, effective electromagnetic shielding often involves also a considerable amount of art based on acquired experience. Shielding measures for specific applications are mentioned in the following sections.
Cryogenics is an auxiliary part of any electronic system incorporating SCE. The usual cryogenic enclosures are commercial vacuum and superinsulation-insulated dewars, usually filled with a liquid cryogen, either liquid helium (LHe) or liquid nitrogen (LN2), providing two standard operating temperature levels—4.2 K for LHe and 77 K for LN2. Commercial mechanical cryocoolers of various types can also provide wide intermediate T ranges. In the range below 77 K, one-stage Stirling cryocoolers are the most typical unless minimum of noise is required. In the range around 4.2 K, multistage Gifford-McMahon and pulse-tube cryocoolers are used, the latter if minimum of noise is essential. Temperatures somewhat lower than those of the liquid boiling point at one atmosphere can be obtained by pumping, i.e., reducing the pressure over the liquid surface, a rather inconvenient laboratory method. Quantum computing (QC) processors and communication links require temperatures much lower than that of LHe, for QC in the 10–50-millikelvin range, assured by commercial dilution refrigerators.
In the past two decades, the use of cryogenic cryocoolers has been expanding for LTS and HTS due both to technology and reliability improvements and, for LTS, to steadily increasing prices of LHe and its paucity. Cryocooling is likely to become the dominant SCE cooling mode of the future, with a trend towards integration into the equipment’s housing to promote its “invisibility” and miniaturization for electronic applications, including even cooling on the chip. Generally, small cryocoolers typical for most LTS SCE systems are much less efficient than large plants used for liquefaction of LHe and cooling of various large-scale systems, such as particle accelerators, for example. For small machines, order of 1% of Carnot efficiency is typical. Relatively recent surveys of cryocooler performance are [16, 17, 18, 19], but an up-to-date survey of small cryocoolers specifically for SCE is sorely needed.
3 Antennae and Filters
Wireless RF communication in the wide range from hertz to terahertz requires signal transmitter and receiver antennae usually shaped as three-dimensional structures. Especially since the advent of HTS, very low surface (loss) resistance of superconductors leading to improved antenna efficiency stimulated studies of electrically small passive antennae (ESA), and superdirective small arrays of these, essentially for superconducting receivers.4 Such antennae are usually used in conjunction with lenses or reflectors. Geometrical, electrical, and thermal limitations imposed by the necessity to use HTS epitaxial films on relatively high electric permittivity crystal substrates, and the necessity of using a cryogenic vacuum radome with a window transparent to incoming radiation in the operational RF band, but highly reflective to thermal radiation, limited such studies mainly to efficiently matching a normal antenna to the remaining superconductor circuitry. Small planar HTS antennae have sometimes been used in laboratory demonstrations of radiation and particle detectors and HTS intrinsic JJ oscillators. Fundamentals of passive small antennae, also superconducting, are given in existing textbooks [20, 21, 22]. A particular example of passive small superconducting antenna for the low-frequency range is the pickup coil of a SQUID magnetometer. It is sensitive to the electromagnetic field and, for example, effectively detects weak far-field signals induced underground in deep layers of mineral ores by electrical pulses of an external transmitting antenna in the TEM geomagnetic exploration method (see Section 6.2). I am not aware of any existing microwave ESA application in operating civilian systems of any kind.
A rather new development are superconducting quantum arrays (SQAs) or superconducting interference filters (SQUIFs) , first demonstrated in the past decade as filters and amplifiers. With proper bias, their output signal is a steep linear function of the input signal applied simultaneously to the whole spectrum of random loop dimensions. This and other properties are succinctly described in Section 6.3. Kornev and collaborators were first to investigate their usefulness as signal amplifying broadband antennae . A summary of their multiple publications is included in . Others have shown the broadband detection of far-field FM radio stations by such arrays made of Nb . Such SQAs should be operable in the bandwidth from Hz to several or even tens of GHz in conjunction with SC analog to digital converters (ADCs). The next decade may bring reports of successful implementation of SQA antennae, e.g., in operating microwave satellite signal receivers.
Historically, in late 1990s, multipole superconducting filters became the first commercial HTS SCE industrial product for US mobile (cellular) telephony stations, due to a particularly dense and complicated allocation of commercial carrier sub-band, and thus greater need for high-performance multipole bandpass and bandstop filters with very sharp skirts (narrow transition widths δfs between bandstop and bandpass frequencies). However, and rather soon, the advent of newer generations of mobile telephony obviated the need for superconducting filters, and their lasting acceptance was not attained. Today, RF filters are mostly of interest for superconducting digital receivers and other HF analog/digital (mixed) circuitry.
Hence, a narrowband filter with set w, sufficiently high n, and low insertion loss must be build using very high Qu resonators. Most of these filters are of Chebyshev or Cauer (elliptic) type and use either microstrip or lumped element technology.
The power handling capability of planar filters is limited to about 100 mW and sufficient for receiving devices, but not transmitters, where tens of watts are needed. Coupled dielectric disc 3D resonators can fulfill this requirement. Generally, at too high-power levels, nonlinear and other spurious effects appear and the filtering performance collapses.
4 Amplifiers, Oscillators, and their Applications
The microstrip SQUID where the input spiral coil has one free end. Thanks to Cs, the washer serves as the ground plane in which mirror image RF current is induced ( and refs. therein). At very low T, this type of SQUID is capable of approaching the quantum limit. This type is favored, e.g., in cosmology experiments, such as search for axion dark matter.
An optimized variant of SQUID named superconducting low-inductance undulating galvanometer (SLUG) of extremely low L, that was first created by John Clarke in the mid-1960s and recently optimized for qubit readout [34, 35, 36]. At 6 to 7 GHz, it can provide G≈ 20 dB in a rather wide bandwidth, while providing reverse isolation superior to that of microwave isolators . It is currently favored as a preamplifier for qubit readout. Its disadvantage is the thermal noise backaction, which can, however, be obviated by pulsed operation.
The parametric amplifier (PA), and especially the Josephson PA (JPA), that can approach or attain the quantum limit, exhibit the lowest noise demonstrated thus far, and a reasonable gain, with G on the order of 10 dB, but suffers of a narrow bandwidth, limited dynamic range, and the need for a microwave pump source. The first can be overcome, e.g., by a JPA of single-ended design that includes an on-chip, high-bandwidth flux bias line  or by traveling wave geometry exploiting the nonlinear kinetic inductance of a superconducting transmission line . However, the microwave pump source is unavoidable in any case, and this makes PA or JPA rather less attractive for quantum circuits than the SLUG, in spite of its somewhat higher noise. Nevertheless, JPA can be invaluable in scientific quantum experiments.
There are no practical LTS oscillators exploiting the 2nd Josephson equation, mainly due to the difficulty in assuring needed output power of coherent radiation of large and highly uniform JJ arrays with a sufficiently narrow linewidth. The only type of superconducting oscillator currently operating in a realistic, in-field scientific application is that based on LTS long JJ flux-flow device (Section 2.2). Thanks to successful phase locking, this flux-flow oscillator (FFO) has linewidth between 1 and 5 MHz and output power sufficient to pump a matched SIS mixer. The FFO is incorporated as into an on-chip integrated, phase-locked superconducting heterodyne receiver (SIR) [40, 41]. Such a device is periodically flying in a stratospheric balloon  of the mission terahertz and submillimeter limb sounder (TELIS). It is a three-channel superconducting heterodyne spectrometer for atmospheric research use of which one channel is using SIR. In its 2009 maiden flight over Sweden, that channel collected data proving the capability for high-resolution molecular spectroscopy of multiple gaseous species in the troposphere/stratosphere and is since used for collecting date in the 480- to 650-GHz frequency range . One can expect insertion of SIR-type receivers in future such missions. Demonstrations of SIR usefulness included also analysis of exhaled air in medical surveys and measurement of terahertz radiation emitted from intrinsic JJ stacks .
In the c-direction of single crystals, the extreme anisotropy of HTS cuprates, such as bismuth–strontium–calcium cuprate (BSCCO, Bi2Sr2CaCu2O8 + x), results in the formation of natural stacks of intrinsic JJs capable of emitting coherent radiation at terahertz frequencies from stacks containing several hundreds to thousands of JJs . In the past decade, considerable effort was devoted to investigating properties of such stacks and possibilities of fabricating intrinsic oscillators for practical applications. The fabrication complexity is shifted in this case from junction fabrication to extraction of uniform BSCCO platelets from a bulk crystal, facilitated by ease of such crystal cleaving along the a − b planes, and ingenuous carving out of a functional JJ stack [44, 45] ( and refs. therein). Work by several in-part collaborating groups indicated that coherence is promoted by cavity resonance of the stack. However, self-heating and fabrication constraints limit the height of a single mesa. Self-heating becomes more severe with the number of JJs and is aggravated by poor thermal conductivity along the c-axis of the BSCCO crystal. The understanding of various coherence promoting and emission-limiting mechanisms is still in progress. Best-attained coherent emission data are frequencies up to 2 THz, linewidths down to several megahertz, and emitted power of over 0.2 mW from a single mesa  and up to 0.6 mW from an array of three synchronized mesas . By now, efforts in application as terahertz sources reached the stage of laboratory demonstrations, e.g., of terahertz imaging  or portable terahertz source ( and refs. therein). A relatively up-to-date review of HTS emitters exists .
5 Radiation and Particle Detectors for Astronomy and Optical Quantum Communication
5.1 Detector Types
Two decades ago, the main and important contribution of superconducting devices to astronomical activities was the use of niobium SIS tunnel junction heterodyne mixers for astronomy receivers in the radiation range of about 0.1 to 1 THz, in spite of their rather low saturation level [52, 53]. The importance of mixers did not diminish, but a paradigm change occurred thanks to the invention, development and deployment of novel bolometric radiation and particle sensors, able to cover a wide f range from sub-infrared up to X and γ-rays, namely voltage-biased transition edge sensors (TES) and microwave kinetic inductance detectors (MKIDs) [54, 55]. By now, they supplanted semiconductor detectors in large radio astronomy telescopes with multipixel cameras. This became possible due to the development and introduction of SQUID readout and multiplexing, which are dramatically reducing the heat load to cryocoolers. Currently, the largest such cameras, installed in dishes 10 m in diameter, have 1–2 × 104 pixels, more with each new camera generation, with up to 106 in perspective. Another detector increasingly used in various scientific experiments is the metallic magnetic calorimeter (MMC) using a paramagnetic alloy, such as Au/Er and Ag/Er, as an energy absorber [56, 57]. All these detectors have excellent energy resolution, fast response time, and good linearity but are relatively slow. When incident energetic photons break Cooper pairs, the resulting quasiparticles (QPs) remain in thermodynamic equilibrium with excited phonons dissipating in the thin film substrate that acts as the thermal sink. A recent review of various bolometric (thermal) detectors for scientific cosmological searches can be found in .
At the same time, superconducting hot electron bolometers emerged that can be used both as bolometers and single photon detectors. At low temperatures, the latter (SNSPD5) feature both a very high quantum efficiency and extremely fast response, on the order of a few picoseconds in the case of LTS thin films . These devices, arguably the most important new contribution to the SCE detector field, are suitable for very efficient detection of optical single photons with negligible dark counts. SNSPDs enabled the completely new field of long-distance optical quantum communication  while simultaneously extending frequency range of mixers up to 5 GHz. Such mixers were promptly tested and used in space . Below, TES, MKID, and SNSPD devices are briefly highlighted.
Of the three worldwide largest cameras of the currently operating radio telescopes, two, namely the SCUBA-27 in Hawaii (wavelengths 0.45 and 0.85 mm) and the South Pole Telescope (SPT-SZ, SPT-3G, wavelengths 3.331, 1.99, and 1.362 mm), are equipped with TES pixel cameras. SCUBA utilizes time-domain multiplexing while SPT the frequency-domain alternative (see Section 5.2). Both have to their credit large numbers of new discoveries. SCUBA-2, operated by large international collaboration, has the all-sky astronomical survey as its main goal, while SPT already discovered hundreds of galaxy clusters and is concentrating on studies of cosmic microwave background (CMB) and determination of neutrino mass.
The third and youngest (2017) of the currently largest radio telescope with 104-pixel cameras, the DARKNESS,8 is installed in the California’s Mt. Palomar observatory, operates in the frequency range between 0.4 and 1.1 μm (infrared), and has as objective the discovery of dark exoplanets of relatively small stars. This camera, considered a pathfinder for future technologies of large cameras, utilizes MKID detectors.
Of developed superconductor multiplexers, two types are used in large multipixel cameras: time- and frequency-domain circuits. In time-domain multiplexers, TES pixels are continuously biased, with readout SQUIDs of each column of an array connected in series and sequentially probed in time by switching bias from one SQUID to the next. In the frequency-domain approach, each pixel’s resonator has a different f0 and all pixels can be probed simultaneously by a comb generator signal. Signal from each pixel is then recovered by frequency-selective demodulators of room temperature electronics. An in-part comparative discussion of both approaches in application to TES arrays is given in chapter 8 of  while frequency-domain readout of large MKID arrays is discussed in .
5.3 Superconductor Nanostripe Single-Photon Detectors
The SNSPD detectors are based on a localized hotspot formation in a two-dimensional superconducting nanostripe [58, 65] and are directly related to hot electron bolometers (HEBs) pioneered by Gershenson et al. [66, 67, 68]. Their detection mechanism can be qualitatively described as follows . An absorbed visible light quantum (E ≥ 2 eV) excites an energetic QP of comparable energy which creates many secondary excited electrons, initially through electron-electron fast scattering and later, with falling energy, by electron-phonon (e-p) emission by QPs breaking even more Cooper pairs. This cascading QP multiplication continues until the secondary QPs approach the energy level of upper edge of the energy gap 2Δ of the superconductor. Very large internal gain, sufficient to attain single-photon energy resolution, results from this QP multiplication mechanism. The whole QP thermalization process takes about 7 ps for NbN, the generally used LTS nanostripe material , and less than 1 ps for YBCO . Advantages of this type of detectors are many, including broadband performance and very low dark count. Hot electron bolometers can also be advantageously used as superior TES and MKID devices.
6 SQUID Magnetometers and other SQUID Applications
6.1 Signal Pickup and Readout
6.2 Magnetometric Applications
Only a few examples of multiple magnetometric applications can be mentioned here. These and some other are properly covered in [4, 7, 8]. The first possible SQUID application was proposed in geomagnetism, already in 1960s, and was demonstrated by the team that discovered quantum interference and pioneered SQUIDs.9 Also, the first truly successful industrial venture has been a small US company manufacturing rock magnetometers.10 It is thus interesting that after three decades of investigating various geomagnetic prospecting methods, the first application with a very significant albeit indirect economic impact emerged in geomagnetic prospecting of mineral metallic ore deposits. It is the transient electromagnetics (TEM) method of contactless surveying, both ground-based and airborne [73, 74]. That success resulted mainly from pioneering efforts of Commonwealth Scientific and Industrial Research Organization (CSIRO) in Australia, and IPHT-Jena in Germany.11 These teams developed portable HTS and LTS SQUID magnetometer systems capable of operating in motion in the presence of ambient static and electromagnetic fields orders of magnitude stronger than the measured signals, thanks also to judicious use of electromagnetic shielding. TEM is an active time-domain method in which strong pulse sequences from a transmitter result in very weak secondary electromagnetic field B(t) response signals due to currents induced deep in the ground. Surveys hundreds of meters deeper into the ground than when using conventional room-temperature inductive coil TEM receivers that measure ∂B/∂t thus became possible. The use of SQUIDs permits to measure decaying signals N times longer which produces about N1/2 times deeper recorded data. This difference proved decisive in discoveries of deeper mineral ore deposits on three continents (Africa, Australia, North America). Only in the first decade of SQUID use, the value of these deposits has been estimated at much more than US$50 billion. The number of SQUID surveying systems in use remains small. Another increasingly successful surveying method, the tensor magnetometry, is covered in the two references cited above.
In terms of industrial instrumentation products, SQUID susceptometers  permitting one to measure magnetic susceptibility of material samples became probably the biggest commercial SQUID magnetometry success to date. The discovery of HTS cuprates and searches for other HTS materials prompted very large number of laboratories to acquire such instruments . The LTS SQUID itself is just a small component enabling sales of a relatively large self-contained instrument including a superconducting magnet, sophisticated controls, and more recently also a compact mechanical cryocooler.
An application, into which enormous effort has been and is still being invested since the 1970s, is the biomagnetic research using diagnostic instrumentation [77, 78] and the related software, especially for approximate solutions to the inverse problem . Several diagnostic methods have been developed and tested, also in realistic clinical environments. Of these, the magnetoencephalography, which images and interprets the local distribution of magnetic fields due to neuronal currents, is used in brain, psychology, and even language research. However, the total number of such systems installed worldwide is low, somewhere between one and two hundred. While clinical uses were convincingly demonstrated, e.g., in presurgical localization of epileptic foci, wide clinical acceptance is missing due to high cost including the necessary MSR and insufficiently attractive cost/benefit ratio. Even less accepted for the same reason is magnetocardiography, the imaging of local heart fields using similar systems, with only few hospitals worldwide successfully practicing it , in spite of numerous preliminary clinical studies showing clear patient benefits, e.g., in noninvasive screening of ischemia suspects or fetal (prenatal) diagnostics.
The last, but not least application of SQUID magnetometry that should be mentioned here is the nondestructive evaluation of materials and structures (NDE) using SQUID magnetometers or gradiometers [81, 82]. In many cases, it faces technical problems and uses solutions similar to those of geomagnetic SQUID applications, e.g., in requiring portability, operation while in motion, and immunity to much stronger environmental fields without using magnetic shielding of the superconducting pickup coil systems or antennae. It is important to note that SQUID NDE is usable also to test metallic materials and structures that are not magnetic. In this case, by inducing eddy currents with the help of a local transmitter, one measures local magnetic signatures of eddy current anomalies in tested objects.
The acceptance of SQUID NDE is currently limited to rare cases where the cost/benefit ratio is attractive enough, or where no other solution exists, as is the case in testing niobium sheets for very-high-Q cavities used in particle accelerators. In the recent past, SQUID NDE was in one case used in industry for testing airplane parts, e.g., turbine discs, where risk of ferromagnetic inclusions existed. In this case, the use of SQUIDs was eventually discontinued in favor of a fluxgate magnetometer, in spite of a major sensitivity reduction. In the 1990s, extremely successful were in-field demonstrations of invisible cracks detection in metallic tendons of concrete highway bridges using portable HTS SQUIDs. Nevertheless, broader introduction of SQUIDs for periodic bridge testing did not follow due to economic/cost reasons. The hour of HTS SQUIDs in such applications may still come with ageing of most bridge structures in developed countries and major disasters such as the most recent catastrophic collapse of the Genoa freeway bridge (August 2018).
6.3 New Developments, other Applications
7 Metrology Instrumentation
Of all applications of SCE devices to metrology, Josephson voltage standards (JVS) are excellent examples of very important applications where capabilities of SCE are unique. Pioneering development of JVS started in the early 1970s at NIST-Boulder13  and continues until today thanks to ever new improvement concepts contributed by several National Metrology Institutes (NMI), especially the National Institute of Standards and Technology (NIST)-Boulder and PTB-Berlin14 , partly in fruitful collaboration. JVS are also first examples of successful LSI of JJs. Of many reviews already published on the subject, the most up-to date is probably .
Conventional Josephson Voltage Standard
After a nearly quarter century of development, the present 10 V standard system consists of a cryostat with large magnetically shielded microwave-fed series/parallel array of NbAlOx JJs embedded into microstrip lines with identical terminations and the room-temperature electronics with the attending microwave circuitry, frequency counter, computer selecting the working point, etc., see  for details. Currently, turnkey commercial systems manufactured in the USA and Germany include a mechanical cryocooler. Today, CJVS are made also in Japan and China.
Programmable Voltage Standard
While CJVS are acceptable for DC voltage measurements, they have some stability problems (noise-caused jumps from one step to the adjacent one) and are not suitable for tasks of growing importance that require rapid selection of the working point, such as rapid analysis of analog-to-digital and digital-to analog converters (ADC and DAC) and synthesis of accurate AC waveforms. To address such new tasks, two new types of JVS have been conceived, developed, and demonstrated by NIST: the programmable JVS (PJVS) and the Josephson arbitrary waveform synthesizer (JAWS). Both were first demonstrated in the last decade of XX century.
To make PJVS possible, the technology of internally shunted damped JJs having βc < 1 had to be developed first to avoid parasitic inductances of external shunts. This was accomplished first with SNS junctions having rather low resistance Pd–Au barriers and practically zero capacitance. They necessitated using drive frequency of only 8 GHz. To make possible the usual drive of 70 GHz, tunable semiconductor NbxSi1−x barriers were eventually introduced with Nb content near the metal-insulator transition . Their resistance can be tuned by varying the niobium content x. Eventually, NIST and PTB demonstrated together first 10 V arrays driven by 70 GHz and containing near 70,000 JJs .
Direct comparisons between CJVS and PJVS indicated agreement better than 1 part in 1010 ( and refs. therein). Consequently, the replacement of CJVS by PJVS will have no effect on the accuracy of calibration.
Josephson Arbitrary Waveform Synthesizer
The PJVS and JAWS can be seen as complementary standards. Current direct DC comparison of the two standards indicated agreement to 1 part in 108 and permitted to identify sources of residual errors in JAWS , Further details and references to both standards can be found, e.g., in .
In 2019, the upcoming redefinition of the SI international system will make of JVS the direct realization of the unit volt rather than a representation of the volt. After that, and with frequency standard traceable to SI units base, the unit volt will be independently realized in every calibration laboratory .
Metrological applications involve also SCE devices and circuits other than the JVS, especially SQUIDs in noise thermometry, superconducting comparators, etc. These other applications are discussed in [96, 97].
8 Digital Electronics
8.1 Historical Developments
Only one relatively up-to-date review of superconductor digital electronics exists, and the reader is directed to it . Reviews published earlier are all dated, while what follows are only brief highlights.
The first concept to use superconductors as switches (cryotron gates) for computers and a demonstration of first integrated cryotron digital circuit date back to 1950s [99, 100] and precede even the birth of Josephson junctions. Voltage level gate circuitry, using JJs as much faster switches was inspired by cryotrons  and pursued in a major IBM project from 1967 until 1983.15 This “latching logic” was then abandoned as not advantageous enough compared with very large-scale integrated circuit (VLSI) semiconductor electronics. In Japan, an analogous Japanese Ministry of International Trade and Industry (1949–2001) (MITI)-sponsored effort continued until 1990 and resulted in the demonstration of an operating microprocessor with over 2 × 104 JJs and a small 4 K random access memory .
Already in 1985, a totally different concept emerged, which became known as the rapid single flux quantum (RSFQ) logic [103, 104]. This concept is based on bit coding in the single magnetic flux quantum, which assures very low energy and fast switching between binary states (≈ 10− 19 J/bit, order of ps/bit). It could be used not only for computer logic but also for mixed signal devices such as analog-to-digital (ADCs), digital-to-analog, and time-to-digital converters (DACs, TDCs) of performance far superior to their semiconductor counterparts .
8.2 Energy-Efficient Computing
In the current decade, the energy efficiency became a growing major concern for CMOS and future computer technologies beyond Moore’s law . Caused by the runaway increase of power consumption by large servers for cloud computing, “big data”, etc., it is becoming a new metrics, equally and even more critical than further improvements in computing speed. Improvements in energy efficiency are addressed by a multitude of methods, both by hardware and software means. In the case of RSFQ logic, the elimination of static power consumption became the immediate goal . Alternative approaches currently in development either modify the dc bias or replace it by ac bias. In terms of funding, pursuit of such approaches infused new life into the development of RSFQ and superconductor memory technology. The Cryogenic Computing Complexity (C3) US Intelligence Advanced Research Projects Activity (IARPA) program is currently the main driving force in this development work . It includes also a technological component necessary for up to exa-scale computing capacities: higher integration densities require a nanotechnology-level fabrication process with correspondingly higher JJ critical currents.
The energy-efficient RFSQ (ERSFQ) is one of the dc power line approaches where resistors in the dc current biasing network are replaced by inductances and current-limiting JJs of an on-chip voltage biasing scheme . This way the whole circuitry is superconducting while the library of already optimized RSFQ logic gates is still usable without modifications. However, this approach requires high inductance bias lines to maintain the operational margins and minimize the timing jitter. Small-scale large inductors may be obtained by introducing high-kinetic-inductance line layer. The clock speed of ERSFQ can still be maintained in the range of tens to possibly hundreds of GHz, as in RSFQ. Another version of resistor-free dc bias scheme is eRSFQ, which features synchronous phase balancing which eliminates phase unbalance and the timing jitter, but at the price of re-designing all logic gates.
The reciprocal quantum logic (RQL) is an AC biasing concept in which an AC transmission line is inductively powering the devices in series, with a four-phase AC clock providing directionality . Here, “1” is encoded in a pair of pulses of opposite polarity, while the absence of such a pair encodes “0.” The pulse pair resets the phase and thus avoids any phase unbalance. Another advantage is the relative simplicity of logic, which is combinational and low latency, like the CMOS semiconductor logic. Also, a low bit error rate was demonstrated for RQL. Compared with ERSFQ, the clock rate is expected to be lower, a possible disadvantage.
Dynamic power dissipation of 10− 19 J/pulse is comparable in all non-adiabatic logic approaches, whether dc or ac powered. Orders of magnitude lower dynamic dissipation can be theoretically attained in adiabatic logic, of which most developed is that of the ac powered adiabatic quantum flux parametron (AQFP) based on an early concept of quantum flux parametron  but operated adiabatically. The energy change in adiabatic switching between energy potential wells was measured to be on the order of 10− 20 J  with the possibility of further optimization by circuit design. The theoretical lower limit for irreversible switching is about 10− 23 J . Other advantages of AQFP compared with energy-efficient RFSQ concepts are easy routing without transmission lines and no-cost logic inversion. The disadvantage is the much lower operation frequency typical of the adiabatic processes and high latency. Improvement in both through multiexcitation AQFP , does not eliminate these drawbacks radically (Fourie, Univ. of Stellenbosch, private information). The AQFP approach is pursued in Japan.
Coming decade should show which of the current developmental approaches to energy-efficient superconductor logic and memory will be eventually adopted and whether any of them can offer sufficient advantages at least in some applications or not. Certainly, energy consumption per bit in SFQ and AQFP is lower by one to two orders of magnitude compared with all other prospective technologies beyond CMOS , and so is the extrapolated power consumption by future superconductor supercomputers (inclusive the cryocooling overhead) when compared with semiconductor computer technologies at the exascale performance level of 1018 floating point operations per second (flop/s) .
8.3 Random-Access Memory
Lack of scalable and real-estate-efficient concept of RAM plagued all pursued approaches to superconducting computing. In spite of major efforts, only a 4-kbit on-chip memory was attained in the MITI program , and after many years, a 64-kbit hybrid memory combining JJs with CMOS could be demonstrated . Currently, pursued are various memory concepts and structures involving JJs with weakly and strongly ferromagnetic thin layers [118, 119, 120, 121, 122]. One example is the SIS’FS device structure, where S’ is a very thin superconductor layer that may be affected by the ferromagnetic (F) layer . Such junctions exhibit hysteretic behavior where magnetization switchable by magnetic field is switching the JJ critical current between two levels and can be made compatible with logic devices. A rather new result is the possibility of switching between 0 and π JJ behavior by spin-triplet supercurrent . These various memory concepts offers the possibility of developing spintronic-like devices, such as the spin valve, for example, with promises of scalability, energy efficiency, and hopefully also acceptable real-estate demand. Some such concepts are developed in the C3 program, but none appears ripe enough to be highlighted here.
9 Qubits and Quantum Computing
Among the new applications of SCE, the superconductor quantum processors and prospective quantum computers are the most prominent novel application of Josephson junctions and SQUIDs. I offer only an arbitrary sampling of the subject and have to assume that the reader is familiar with principles of quantum mechanics and basic notions of quantum computing (QC), where N interacting (entangled) qubits can simultaneously represent 2N values. Several reviews of qubit types and QC circuits using superconductivity exist [124, 125, 126, 127].
Studies of macroscopic quantum coherence, see , and of the quantization of charge in the superconducting single-electron transistors (SETs), or, the so-called Cooper pair box ( and refs. therein), suggested that at sufficiently low temperatures, typically below 50 mK, JJs and SQUIDs can behave as artificial macroscopic atoms, i.e., are two-level systems that by applying pulse excitation of electromagnetic field could be brought into controlled quantum superposition of their two states and used as an embodiment of qubits potentially suitable for quantum information processing by making such qubits quantum coupled and entangled. Indeed, in 1999 the first such device, a charge qubit, was demonstrated experimentally . Two charge states, differing by 2e and coupled by tunneling of Cooper pairs,16 were controllably brought into coherent superposition, although over a time of only τ ≈ 1 ns. In the following two decades, coherence times of various SQUID qubit types could be gradually extended by five orders of magnitude but remain still too short for effective quantum correction algorithms, thus resulting in low computing fidelity. Coherence below at least the millisecond range remains a significant disadvantage of SE compared with real atoms.17 In the past 5 years, major causes of decay (decoherence) originating in qubit materials and electromagnetic environment were one-by-one identified and eliminated. To have magnetic field tunability, most qubit types are SQUIDs with the number of JJs between 1 and 3. Tunability, the relative ease in establishing controlled inter-qubit coupling and fabrication by a scalable and rather well-established thin film multilayer processing of integrated circuits are major advantages of SQUID-based qubits. Typical very low Tc superconductors for qubit and circuit fabrication are aluminum (Al) and titanium nitride (TiN).
It is not yet certain that SCE alone will be eventually the preferred QC technology rather than, for example, some combinations of SCE with semiconductor quantum dots and integrated optics. However, SCE is presently suitable for building functional small quantum processor hardware on a demonstrated and accessible to the public test user level of 20 interacting qubits  and between 49 and over 72 qubit level reportedly attained in 2017/2018 by the leading QC R&D industry teams, respectively, of IBM, Intel, and Google.18 The latest claim is from Rigetti to having attained 128 qubits/chip. While the 20 qubit level is too low to demonstrate any advantage in calculation speed over classical quantum simulations (the “quantum supremacy”), it is believed that, for selected problems not requiring full-fledged quantum correction, a clear advantage can be demonstrated with an about 50 qubit processor . At present, superconductor technology of qubits thus appears to be the preferred one, attracting the relatively largest institutional and industrial funding.
An intriguing perspective for the more distant future of QC is topological QC with non-Abelian anyons, quasiparticle excitations that obey non-Abelian (commutative) statistics . To that class of excitations belong the theoretically predicted localized Majorana20 zero modes (MZM) in semiconductor nanowires that may be tuned to become topological superconductors when coupled to an s-wave superconductor [137, 138, 139]. Majorana fermions are postulated quasiparticles that are their own antiparticles. Degenerate states associated with their zero modes should result in a topologically protected quantum memory and long-lasting coherence leading to an ideal embodiment for fault-tolerant quantum computation. Thus far, only indirect signatures of such degenerate states have been found experimentally through measurements of quantized conductance in InSb , or Shapiro steps in heterostructures of HgTe and Bi-doped Sb2Te3 with Al and Nb superconductors [141, 142]. Nevertheless, awaiting harnessing of MZM, scalable designs for topological QC have already been proposed . Topological computation studies currently enjoy industrial support, mainly from Microsoft.
In closing, I should mention that very high numbers of superconducting circuits are also used in commercial d-wave quantum annealers, computing machines that may offer quantum speedup in solving certain classes of problems such as optimization, see for example [144, 145]. To date, there is no unambiguous and generally accepted confirmation of computing advantages claimed by the manufacturer . Nevertheless, a few such relatively large machines incorporating 105 JJs have been sold and studies of quantum annealing continue at academia, research organizations, and industry. There is also an IARPA-sponsored effort to develop quantum enhanced computers.21
10 Concluding Summary and Outlook
As of today, SE is applied predominantly and very successfully in science, e.g., radio astronomy and cosmology, also in metrology, while gaining ground in exploration of minerals. Military, space, and quantum communication uses are slowly emerging and have assured bright future. Considerable efforts in developing industrial and medical applications, coinciding with the first 10–15 years after the discovery of HTS, resulted in many quite successful demonstrations of various industrial and clinical test instruments and large diagnostic systems. However, most has not been sufficiently advantageous and economically competitive to pass the test of time when used in realistic environments. Eventually, they did not gain widespread acceptance, although some still have some chances.
Until now, SCE has not matched the immense commercial success of MRI systems with superconducting magnets. Time will tell if the current efforts to save power in high-end computers will succeed with low-energy SCE digital logic gates. In the race to build quantum computers, SCE is the only technology that can produce both qubits and the high-speed low-power digital and mixed signal circuits that can control the qubits. This combination could prove decisive. Considering the high level of funding and effort, the outcome of this race, and of efforts towards high-end superconducting computers, is likely to be known in the next decade.
Best to date has been the demonstration of THz oscillators in intrinsic HTS JJ stacks of BSCCO, with output power limited by internal heating effect.
International Electrotechnical Commission: IEC 61788-22-1 standard on “Superconductivity—part 22-1: superconducting electronic devices—generic specification for sensors and detectors”: https://webstore.iec.ch/publication/26674; IEC 60617—graphical symbols for diagrams: superconducting region, one superconducting connection (S01924), normal-superconducting boundary (S01925), and Josephson junction (S01926), https://webstore.iec.ch/publication/2723. Free access: http://snf.ieeecsc.org/sites/ieeecsc.org/files/documents/snf/abstracts/edOhkuboM_Report_of_IEC-IEEE_joint_std_IWSSD20160129-final-4_040617.pdf
Small compared with the wavelengths received.
Acronym SNSPD most often stands for Superconducting Nanowire Single-photon Detector. However, more correctly, it should stand for Superconducting Nanostripe Single-Photon Detector, since the active element is a two-dimensional superconductor and not a one-dimensional nanowire.
Just large enough to couple to the SQUID’s pickup coil.
Submillimeter common use bolometer array.
Dark-speckle near-infrared energy-resolved superconducting spectrophotometer
Ford Scientific Laboratories, Dearborn, MI, USA.
2G Enterprises, founded by the late Bill Goree in 1978.
IPHT is the German acronym of the Institute for Photonics and Advanced Technology; IPHT is located in Jena, Germany.
At the University of Tübingen, Germany.
Formerly the National Bureau of Standards, NBS, USA.
PTB is the German acronym of the Physikalisch-Technische Bundesanstalt, the German National Metrology Institute.
Also, at two other US companies (Sperry and TRW), superconductor computer technology was pursued with different materials and JJ fabrication approaches.
Tunneling between their differing potential wells.
Presently, 10 to 100 μs compared with minutes in the case of trapped ions. These are not readily scalable, but entanglement of several qubits was experimentally attained with trapped ions and simple quantum algorithms performed with high fidelity. In superconductor qubits, by one more order of magnitude, longer coherence time will be needed to implement quantum correction algorithms necessary for universal quantum computation.
Thus far, there are no publications documenting this, only press releases.
Advantages of the IBM transmon gates are insensitivity to flux noise and simpler control circuitry. Among disadvantages are the required very narrow tolerances on JJ I0 and reduced speed of the gate.
Majorana fermions are postulated quasiparticles that are their own antiparticles.
ODNI News Release No. 5-17, 24 April 2017, IARPA launches “QEO” program to develop quantum enhanced computers.
I am indebted to the following colleagues for critically reading and commenting on the whole or part of the manuscript: David DiVinzenzo, John X. Przybysz, Roman Sobolewski, Elie K. Track, and Huabing Wang. Some of the references were kindly provided by Sam P. Benz, David DiVinzenzo, Valery Koshelets, John X. Przybysz, Allard Schnabel, Thomas Schurig, Marcel ter Brake, and Huabing Wang. Most of the figures were computer drawn by Dieter Lomparski. I owe sincere thanks to all those listed above.
- 1.Van Duzer, T., Turner, C.W.: Principles of Superconductive Devices and Circuits, 2nd edn. Prentice Hall. ISBN-13: 978-0132627429, ISBN-10: 0132627426 (1999)Google Scholar
- 2.Likharev, K.K.: Dynamics of Josephson Junctions and Circuits Gordon & Breach, Taylor & Francis, Ltd. (1992, 1996). ISBN: 3-88124-042-9 (1986)Google Scholar
- 3.Braginski, A.I.: Superconducting electronics: A dream or reality. Bull. Polish Acad. Sci. 48(1), 57 (1997)Google Scholar
- 4.Seidel, P. (ed.): Applied Superconductivity: Handbook on Devices and Applications, vol. 2. Wiley (2015). ISBN-13: 978-3-27-741209-9, ISBN-10: 3527412093Google Scholar
- 5.Pozar, D.M.: Microwave Engineering, 4th edn. Wiley (2011)Google Scholar
- 6.Barone, A. (ed.): Principles and Applications of Superconducting Quantum Interference Devices. World Scientific Pub Co Inc, Singapore (1992). ISBN-13: 978-9810209117, ISBN-10: 9810209118Google Scholar
- 7.Clarke, J., Braginski, A.I. (eds.): The SQUID Handbook, vol. 1. Wiley (2004). ISBN: 3-527-40229-2Google Scholar
- 8.Clarke, J., Braginski, A.I. (eds.): The SQUID Handbook, vol. 2. Wiley (2006). ISBN-13: 978-3-527-40408-7, ISBN-10: 3-527-40408-2Google Scholar
- 9.Clarke, J., Cleland, A.N., Devoret, M.H., et al.: Quantum mechanics of a macroscopic variable: the phase difference of a Josephson junction. Science 239, 992 (1988)Google Scholar
- 10.Braginski, A.I., Zhang Y.: Practical RF SQUIDs: Configuration and performance. In: Clarke, J., Braginski, A.I. (eds.) The SQUID Handbook. ISBN 3-527-40229-2, vol. 1. Wiley (2004)Google Scholar
- 11.Rikitake, T.: Magnetic and Electromagnetic Shielding. Springer Science & Business Media, Kluwer (1987). ISBN-13: 978-9027724069, ISBN-10: 9027724067Google Scholar
- 12.Celozzi, S., Araneo, R., Lovat, G.: Electromagnetic Shielding. IEEE Press and Wiley Interscience (Wiley). ISBN: 978-0-470-05536-6 (2008)Google Scholar
- 13.Stuiber, S.A.: Creation of ultra-low remnant fields and homogeneous NMR fields for precision experiments. Ph. D. dissertation, Techn. Univ. München (2018)Google Scholar
- 17.Kittel, P.: Cryocooler performance estimator. In: Miller, S.D., Ross, R.G. Jr. (eds.) Cryocoolers 14. International Cryocooler Conference, Inc (2007)Google Scholar
- 19.Ladner, D.R.: Performance and mass vs. operating temperature for pulse tube and Stirling cryocoolers. In: Miller, S.D., Ross, R.G. Jr (eds.) Cryocoolers 16. International Cryocooler Conference, Inc (2011)Google Scholar
- 20.Chaloupka, H.: Antennae. In: Cardwell, D., Ginley, D. (eds.) Handbook of Superconducting Materials. ISBN-13: 978-0750308984, ISBN-10: 0750308982. CRC Press (2003)Google Scholar
- 21.Hansen, R.C.: Electrically Small, Superdirective, and Superconducting Antennas. Wiley. ISBN-13: 978-0471782551 (2006)Google Scholar
- 22.Hansen, R.C., Collin, R.E.: Small Antenna Handbook. Wiley. ISBN: 0470890835 (2011)Google Scholar
- 25.Chaloupka, H.J., Kornev, V.K.: Antennae. In: Cardwell, D., Larbalestier, D. (eds.) Handbook of Superconducting Materials. ISBN-13: 978-1439817308, ISBN-10: 1439817308. CRC Press (2019)Google Scholar
- 27.Hong, J.-S., Lancaster, M.J.: Microstrip Filters for RF/Microwave Applications. Wiley (2001)Google Scholar
- 28.Shen, Z.-Y.: Microwave resonators and filters. In: Cardwell, D., Ginley, D. (eds.) Handbook of Superconducting Materials. ISBN-13: 978-0750308984, ISBN-10: 0750308982. CRC Press (2003)Google Scholar
- 29.Chaloupka, H., Kolesow, S.: Design of lumped-element 2D RF devices. In: Weinstock, H., Nisenoff, M. (eds.) Microwave Superconductivity. ISBN-13: 978-1402004469, ISBN-10: 140200446. Kluwer, Acad. Publ. (2001)Google Scholar
- 30.Oates, D.E.: Microwave resonators and filters. In: Cardwell, D., Larbalestier, D. (eds.) Handbook of Superconducting Materials. ISBN-13: 978-1439817308, ISBN-10: 1439817308. CRC Press (2019)Google Scholar
- 31.Clarke, J, Lee, A.T, Mück, M, Richards, P.L.: SQUID voltmeters and amplifiers. In: Clarke, J., Braginski, A.I. (eds.) The SQUID Handbook. ISBN-13: 978-3-527-40408-7; ISBN-10: 3-527-40408-2, vol. 2. Wiley (2006)Google Scholar
- 34.Ribeill, G.J., Hover, D., Chen, Y. -F., et al.: Superconducting low-inductance undulatory galvanometer microwave amplifier: Theory. Appl. Phys. Lett. 110, 103901 (2011)Google Scholar
- 41.Dmitriev, P.N., Filippenko, L.V., Koshelets, V.P.: Applications in superconducting SIS mixers and oscillators: Toward integrated receivers. In: Wolf, E., Arnold, G., Gurvitch, M., Zasadzinski, J. (eds.) Josephson Junctions. History, Devices, and Applications. ISBN-13: 978-981-4745-47-5, ISBN-10: 978-1-315-36452-0, pp. 185–244. Pan Stanford Publ. (2017)Google Scholar
- 44.Ishida, H, Okanoue, K., Hamasaki, H., et al.: Self-planarizing process for the fabrication of Bi2Sr2CaCu2 O x stacks. Appl. Phys. Lett. 122503, 86 (2005)Google Scholar
- 52.Blundell, R., Winkler, D.: The superconductor-insulator-superconductor mixer receiver—a review. In: Costabile, G., Pagano, S., Pedersen, N.F., Russo, M. (eds.) Nonlinear Superconductive Electronics and Josephson Devices. ISNB: 13: 978-14613-6719-2, ISBN-10: ISBN-13: 978-14615-3852-3, pp. 55–72. Plenum Press, New York (1991)Google Scholar
- 53.Winkler, D.: Superconducting mixers. In: Cardwell, D., Ginley, D. (eds.) Handbook of Superconducting Materials. ISBN-13: 978-0750308984, ISBN-10: 0750308982. CRC Press (2003)Google Scholar
- 56.Fleischmann, A, Enss, C., Seidel, G.: Metallic magnetic calorimeters. In: Enss, C. (ed.) Cryogenic Particle Detection. Topics Appl. Phys., vol. 99, pp. 149–201. Springer (2005)Google Scholar
- 60.Miki, S., Fujiwara, M., Jin, R.B.: Quantum information networks with superconducting nanowire single-photon detectors. In: Hadfield, R.H., Johansson, G. (eds.) Superconducting Device in Quantum Optics. ISBN-13: 97-3-319-24089-3, ISBN-10: 978-3-319-24091-6. Springer Intl. Publ. (2016)Google Scholar
- 65.Korneev, A., Semenov, A., Vodolazov, G.N., et al.: Physics and operation of superconducting single-photon detectors. In: Wördenweber, R., Vanacken, J. (eds.) Superconductors at the Nanoscale: from Basic Research to Applications, pp. 279–308. De Gruyter Press (2017)Google Scholar
- 66.Gershenson, E.M., Gershenson, M.E., Gol’tsman, G.N., et al.: Heating of quasiparticles in a superconducting film in the resistive state. Pis’ma Zh Eksp. Teor. Fiz. 34(5), 281–285 (1981)Google Scholar
- 67.Gershenson, E.M., Gershenson, M.E., Gol’tsman, G.N., et al.: Electron-phonon interaction in ultrathin Nb films. Zh. Eksp. Teor. Fiz. 97(3), 901–911 (1990)Google Scholar
- 69.Sobolewski, R.: Optical sensors. In: Cardwell, D., Larbalestier, D. (eds.) Handbook of Superconducting Materials. ISBN-13: 978-1439817308, ISBN-10: 1439817308. CRC Press (2019)Google Scholar
- 71.Drung, D., Mück, M.: SQUID electronics. In: Clarke, J., Braginski, A.I. (eds.) The SQUID. ISBN: 3-527-40229-2, vol. 1. Wiley (2004)Google Scholar
- 72.Zhang, Y., Dong, H., Krause, H.J., Xie, X.: SQUID Readout Electronics and Magnetometric Systems for Practical Applications. Wiley. ISBN-13: 978-3-527-34488-8 (2020)Google Scholar
- 73.Clem, T., Foley, C.P., Keene, M.N.: SQUIDs for geophysical survey and magnetic anomaly detection. In: Clarke, J., Braginski, A.I. (eds.) The SQUID Handbook. ISBN-13: 978-3-527-40408-7; ISBN-10: 3-527-40408-2, vol. 2. Wiley (2006)Google Scholar
- 74.Stolz, R.: Geophysical exploration. In: Seidel, P. (ed.) Applied Superconductivity: Handbook on Devices and Applications. ISBN-13: 978-3-27-741209-9, ISBN-10: 3527412093, vol. 2. Wiley (2015)Google Scholar
- 75.Blake, R, Wellstood, F.: Measurements of magnetism and magnetic properties of matter. In: Clarke, J., Braginski, A.I. (eds.) The SQUID Handbook. ISBN-13: 978-3-527-40408-7; ISBN-10: 3-527-40408-2, vol. 2. Wiley (2006)Google Scholar
- 76.Sager, R.: SQUIDs-from laboratory devices to commercial products. In: Rogalla, H., Kes, P. (eds.) 100 Years of Superconductivity. ISBN-13: 978-1-4398-4946-0. Taylor & Francis Ltd (2011)Google Scholar
- 77.Vrba, J., Nenonen, J., Trahms, L.: Biomagnetism. In: Clarke, J., Braginski, A.I. (eds.) The SQUID Handbook. ISBN-13: 978-3-527-40408-7; ISBN-10: 3-527-40408-2, vol. 2. Wiley (2006)Google Scholar
- 78.Nowak, H.: SQUIDs in biomagnetism. In: Seidel, P. (ed.) Applied Superconductivity: Handbook on Devices and Applications. ISBN-13: 978-3-27-741209-9, ISBN-10: 3527412093, vol. 2. Wiley (2015)Google Scholar
- 79.Lima, E.A., Irimia, A., Wikswo, J.P.: The magnetic inverse problem. In: Clarke, J., Braginski, A.I. (eds.) The SQUID Handbook. ISBN-13: 978-3-527-40408-7; ISBN-10: 3-527-40408-2, vol. 2. Wiley (2006)Google Scholar
- 81.Krause, H.-J., Donaldson, G.: Nondestructive evaluation of materials and structures using SQUID. In: Clarke, J., Braginski, A.I. (eds.) The SQUID Handbook. ISBN-13: 978-3-527-40408-7; ISBN-10: 3-527-40408-2, vol. 2. Wiley (2006)Google Scholar
- 82.Krause, H.-J., Mück, M., Tanaka, S.: SQUIDs in nondestructive evaluation. In: Seidel, P. (ed.) Applied Superconductivity: Handbook on Devices and Applications. ISBN-13: 978-3-27-741209-9, ISBN-10: 3527412093, vol. 2. Wiley (2015)Google Scholar
- 86.Niemeyer, J.: How the DC array standards were developed. In: Rogalla, H., Kes, P. (eds.) 100 Years of Superconductivity. ISBN-13: 978-1-4398-4946-0. Taylor & Francis Ltd (2011)Google Scholar
- 87.Kohlmann, J.: Application to Josephson voltage standards. In: Wolf, E., Arnold, G., Gurvitch, M., Zasadzinski, J. (eds.) Josephson Junctions. History, Devices, and Applications. ISBN-13: 978-981-4745-47-5, ISBN-10: 978-1-315-36452-0, pp. 359–383. Pan Stanford Publ. (2017)Google Scholar
- 92.Flowers-Jacobs, N.E., Waltman, S.B., Fox, A.E., et al.: Josephson arbitrary waveform synthesizer with two layers of Wilkinson dividers and a FIR filter. IEEE Trans. Appl. Supercond. 26(6), 1400307 (2016)Google Scholar
- 94.Flowers-Jacobs, N.E., Rüfenacht, A., Fox, A.E., et al.: Three volt pulse-driven Josephson arbitrary waveform synthesizer. CPEM 2018 digest (to be accessible online via IEEE Xplore) (2018)Google Scholar
- 95.Rüfenacht, A., Flowers-Jacobs, N.E., Fox, A.E., et al.: DC comparison of a programmable Josephson voltage standard and a Josephson arbitrary waveform synthesizer. CPEM 2018 digest (to be accessible online via IEEE Xplore) (2018)Google Scholar
- 96.Gallop, J., Piquemal, F.: SQUIDs for standards and metrology. In: Clarke, J., Braginski, A.I. (eds.) The SQUID Handbook. ISBN-13: 978-3-527-40408-7; ISBN-10: 3-527-40408-2, vol. 2. Wiley (2006)Google Scholar
- 97.Gallop, J., Hao, L.: Applications: Other devices—metrology. In: Cardwell, D., Larbalestier, D. (eds.) Handbook of Superconducting Materials. ISBN-13: 978-1439817308, ISBN-10: 1439817308. CRC Press (2019)Google Scholar
- 98.Przybysz, J.X., Miller, D.L., Hannes Toepfer, H., et al.: Superconductor digital electronics. In: Seidel, P (ed.) Applied Superconductivity: Handbook on Devices and Applications. ISBN: 978-3-527-41209-9. Wiley (2015)Google Scholar
- 100.Bremer, J.: The invention of superconducting integrated circuit. IEEE History Center Newslett. 75, 6–7 (2007)Google Scholar
- 102.Hasuo, S.: Josephson microprocessors. In: Weinstock, H (ed.) SQUID Sensors: Fundamentals, Fabrication and Application. NATO ASI Ser. E-Appl. Sci, p. 329. Kluwer Acad. Publ (1996)Google Scholar
- 103.Likharev, K.K., Mukhanov, O.A., Semenov, V.K.: Resistive single flux quantum logic for the Josephson junction technology. In: Hahlbohm, H.D., Lübbig, H. (eds.) SQUID ‘85: Superconducting Quantum Interference Devices and Their Applications, pp. 1103–1108. W. de Gruyter Pbl (1985)Google Scholar
- 106.Mukhanov, O.A.: Digital electronics. In: Cardwell, D., Larbalestier, D. (eds.) Handbook of Superconducting Materials. ISBN-13: 978-1439817308, ISBN-10: 1439817308. CRC Press (2019)Google Scholar
- 123.Glick, J.A., Edwards, S., Korucu, D., et al.: Spin triplet supercurrent in JJs containing a synthetic antiferromagnet with perpendicular magnetic anisotropy. arXiv:1710.07247v1 [cond.mat.supr-con] (2017)
- 125.Devoret, M.H., Walraff, A., Martinis, J.M.: Superconducting qubits: A short review. arXiv:cond-mat/0411174 (2008)
- 127.Mooij, J.E.: Qubits. In: Rogalla, H., Kes, P.H. (eds.) 100 Years of Superconductivity, pp. 375–388. CRC Press (2011)Google Scholar
- 130.IBM Online Quantum Experience (IBM Q): https://quantumexperience.ng.bluemix.net/qx/experience
- 140.Zhang, H., Liu, Ch.-X., Gazibegovic, S., et al.: Quantized Majorana conductance. Nature 556, 26142 (2018)Google Scholar
- 142.Schüffelgen, P., Rosenbach, D., Li, C., et al.: Boosting transparency in topological Josephson junctions via stencil lithography. arXiv:1711.01665 (2017)
- 144.Rønnow, T.F., Wang, Z, Job, J., et al.: Defining and detecting quantum speedup. arXiv:1401.2910v1 (2014)
- 145.Denchev, V.S., Boixo, S., Isakov, S.V., et al.: What is the computational value of finite-range tunneling? Phys. Rev. X 6, 031015 (2016)Google Scholar