Advances in superconductor quantum and thermal detectors for analytical instruments

The superconductor detector technology utilizes superconductivity of at least one of three detector components: an absorber that interacts with particles, a device for electric signal generation, and a readout circuit. In conventional analytical instruments, the signal generation device typically consists of a semiconductor device. Replacement of the semiconductor detector with the superconductor detector leads to ultimate instrumental performance, overcoming the inherent limits of conventional analytical instruments. Although subsequent digital signal processing is conventionally performed by semiconductor digital circuits, the latest superconductor digital technology can also replace semiconductor circuits in future. The state-of-art superconductor electronics for both analog and digital circuits appears in Ref. 1. The superconductor sensors and detectors accounted for a large part of the superconductivity centennial conference—EUCAS-ISEC-ICMC in 2011.² 

This article begins with the general aspect of analytical instrumentation and the role of detectors. In the same manner as the conventional particle detectors, the superconductor detectors are principally categorized into two detection schemes: “quantum detection,” in which electric signals are directly produced by electronic excitation, and “thermal detection,” in which temperature rise due to radiation is measured with a thermometer. It should be noted that this categorization is based on how the output signal is created. Both superconductor quantum and thermal detection schemes utilize superconductivity; therefore, all superconductor detector types can be also categorized as quantum detection.

After mentioning the components of analytical instruments, the general operation principles of quantum and thermal detection are explained before deepening our understanding of the superconductor detector operation. Next, we describe the theoretical and experimental efforts to achieve expected high performance of superconductor detectors, followed by engineering advances for equipping analytical instruments with superconductor detectors. Finally, unconventional analytical data obtained mostly by our group at the National Institute of Advanced Industrial Science and Technology (AIST), part of which was formerly Electrotechnical Laboratory (ETL), are highlighted: x-ray spectroscopy (XRS) and mass spectrometry (MS). We have developed superconductor tunnel junction detectors (STJDs), transition edge sensor detectors (TESDs), and superconductor strip particle detectors (SSPDs) for x-ray photons and particles like ions and neutral molecules. They provide novel analytical data concerning dentistry, power semiconductors, atmosphere escape from planets, and prebiotic organic molecules in interstellar space as an origin of life. This article also covers other major detector types: microwave kinetic inductance detectors (MKIDs) and magnetic metallic calorimetric detectors (MMCDs).

Superconductor detectors were initially developed for detecting neutrinos and then applied to other particles. Topical reviews summarized the status in 1996.^3,4 A comprehensive review for thermal detectors appears in Ref. 5. This article is arranged such that one can trace a difference between quantum and thermal detectors, merits and demerits, and similarity and dissimilarity between semiconductor and superconductor detectors. For nonexperts and students who newly join in this research field,  Appendix A describes the definition of energy resolution and related probability distribution functions,  Appendix B explains characteristic x-ray emission lines and excitation inside x-ray detector absorber materials, and  Appendix C deals with terminology and nomenclature for superconductor detectors. We use the standard abbreviations for the superconductor detector types, considering the recommendation of the international electrotechnical commission (IEC) standards⁶ except the term of “superconducting detector” that will be changed to “superconductor detector” (see  Appendix C 2). The experts can focus on a specific chapter or sections of interest by skipping general sections.

Analytical instruments are typically divided into three components: a probe beam to excite analytes, a spectrometer to separate particles, such as x-ray photons emitted from the analytes, and a detector to produce spectral data. The spectra often consist of a number of particles for the y axis as a function of particle energies for the x axis. We emphasize that detectors often determine the quality of spectra and, thus, the performance of analytical instruments. In contrast to the precise detection of rare events such as dark matter,⁷ a higher count rate is vastly preferable for high throughput analysis or imaging analysis.

Figure 1(a) shows one example of the typical analytical instruments: electron probe microanalyzer (EPMA), which is equipped with an electron gun with an optics device focusing electrons onto a sample surface and x-ray spectrometers for analyzing characteristic x rays from atoms. Elemental mapping is realized by scanning the electron beam while measuring characteristic x rays. The x-ray spectrometer for EPMA consists of a combination of an analyzing diffraction crystal or grating and an x-ray photon counter, which is called wavelength dispersive spectroscopy (WDS). WDS spectrometers have a typical energy resolution (ΔE) of 10 eV, which is sufficient to separate different characteristic x-ray lines. However, the WDS spectrometers require collimators or slits to fix the diffraction angle, leading to a small solid angle. Therefore, to compensate the small solid angle, a high electron beam current is required, which tends to create sample damage. Another disadvantage of EPMA is that the upper limit of the number of WDS spectrometers is about five because of a geometrical limit. It results in the restriction on the number of simultaneously measurable elements.

Another example of similar analytical instruments is the popular scanning electron microscopy (SEM) in Fig. 1(b). SEM tolerates a low electron beam current compared with EPMA, which is effective to avoid radiation damage. X-ray spectra are acquired by another spectroscopic type called energy dispersive spectroscopy (EDS). X-ray detectors for EDS enable energy measurement of individual x-ray photons without mechanical or electrical scanning. The EDS detector can be placed near the sample surface and accept a large portion of x rays emitted at 4π steradian from the sample, realizing a significantly larger solid angle than WDS. Furthermore, simultaneous analysis of multiple elements more than five can be performed. However, the latest semiconductor EDS detectors have an energy resolution of ∼100 eV, which is insufficient for separating important characteristic x-ray lines. Natural linewidths of characteristic x rays from elements embedded in a matrix are typically 10–20 eV, broadened by chemical bonds or electronic band structure. Superconductor detectors enable to combine a WDS-grade high energy-resolution and an EDS-grade high detection efficiency. This is one of the motivations to employ superconductivity for innovative analytical instruments.

Before describing the details, let us glance at a performance difference between semiconductor and superconductor detectors in Fig. 2. We analyzed a hen eggshell and a porcine dentin, which mainly consist of calcite (CaCO₃) and hydroxyapatite [Ca₅(PO₄)₃(OH)] reinforced by collagen fibers, respectively. These biological composites have similar compositions for major elements, such as Ca, P, Mg, Na, O, N, and C. There is a small difference in trace mineral elements. Figure 2 compares the x-ray spectra acquired by a silicon drift detector (SDD)⁸ and an STJD. The x-ray spectra were acquired for 10 min with the SDD and STJD simultaneously. The STJD spectra for the eggshell and the dentin show fine features due to Ca, Cr, and F, drawing a distinction between the two samples. On the other hand, the SDD spectrum exhibits only broad C, N, and O peaks with a subtle sign of F. Figure 2 clearly demonstrates the advantage of the STJD over the SDD in this x-ray energy range.

A research trend of superconductivity is normally searching higher critical temperatures even near or above room temperature. For superconductor particle detectors, on the other hand, low temperature superconductors (LTSs) are normally preferable to achieve a better energy resolution; for example, Nb and Al. In addition, superconductor/normal metal bilayers, such as Nb/Al, Mo/Cu, Ti/Au, are utilized for controlling T_c with the proximity effect.⁹ Most physical aspects of detector operation using LTSs is understandable based on the standard Bardeen–Cooper–Schrieffer (BCS) theory,¹⁰ which postulates a single isotropic superconducting energy gap or s-wave symmetry.

In quantum detection, a superconducting energy gap is utilized as the scale for measuring a particle kinetic energy through the number of quasiparticles created by breaking Cooper pairs. Operating temperatures shall be well below T_c to suppress thermally excited quasiparticles. In thermal detection, superconductor films are held in the middle of superconducting transition or biased just below I_c. Superconductivity requires refrigerators, such as adiabatic demagnetization and ³He–⁴He dilution for a temperature range below 0.1 K and ³He sorption refrigerators at ∼0.3 K, which are significantly larger than liquid nitrogen cryostats or Peltier coolers for semiconductor detectors.

A question may arise for the practicability of high temperature superconductors (HTSs), which are advantageous to the cryogenic requirement. In 1986, immediately after the first workshop on low temperature detectors (LTD-1), an HTS system of LaBaCuO was discovered.¹¹ The possibility of using HTSs for detectors was already discussed in 1987.¹² Since then, many HTSs have been discovered. SQUIDs with HTS Josephson junctions (JJs) were fabricated soon after the HTS discovery.^13,14 However, the detection scheme relying on quasiparticle excitation is inapplicable to HTSs, because most HTSs have d-wave symmetry that has a zero energy gap along some crystallographic directions. HTS devices face another difficulty of thin-film growth and microfabrication. T_c values easily change with oxygen content in the cuprate HTSs, such as YBa₂Cu₃O_x (YBCO), depending on thin-film preparation conditions.¹⁵ Moreover, its multi-elemental oxide nature prohibits a high-degree of uniformity. In addition to the material difficulties, HTS tunnel junctions or superconductor–insulator–superconductor (SIS) junctions with a similar quality to LTSs have never been realized because of an extremely short coherence length of less than 1 nm. In most cases, HTS JJs operate in the mode explained by the resistively shunted junction (RSJ) model.¹⁶ 

BiSrCaCuO (BSCCO) crystals have intrinsic JJs along the c axis, forming a stuck of SIS alternation.¹⁷ The intrinsic JJs were utilized to generate and detect THz radiation.^18,19 THz emission was also observed from YBCO.²⁰ In addition to the THz range, we had an idea for utilizing the intrinsic Josephson effect in BSCCO as a tunnel junction x-ray detector. However, it turned out that BSCCO was unrealistic for detectors because of the d-wave symmetry, unavoidable staking faults of different phases of Bi₂Sr₂CuO_x (2201), Bi₂Sr₂CaCu₂O_x (2212), Bi₂Sr₂Ca₂Cu₃O_x (2223), and Bi₂Sr₂Ca₃Cu₄O_x (2234),²¹ and long-range modulation superstructures.²² As a result, HTS-STJDs for single particles have never been realized up to now.

A successful usage of HTSs is seen in the SS type, which is essentially a particle counting device without measuring particle energies. The thermal noise is tolerable in SSPDs and, thus, an operating temperature of T_c/2 or higher is possible. One successful HTS-SSPD consists of MgB₂, in which the occurrence of superconductivity is explained within the s-wave framework with multiple energy gaps.^23,24 A cuprate superconductor LaSrCuO with the d-wave symmetry is close to single photon detection.²⁵ Several groups have been trying to employ YBCO for single-particle detection,^26–29 which may be operatable at 77 K. A preliminary result on single photon detection with YBCO is reported on dark count experiment.³⁰ However, single photon detection with YBCO remains to be realized in future. Most recently, an SSPD exfoliated from a bulk BSCCO single crystal succeeded in single photon detection at temperatures up to 25 K.³¹ 

The properties of the above-mentioned typical superconductors are summarized in Table I. Important parameters are listed for detector development. Among them, the Pearl length $( Λ = 2 λ 2 / t f)$³² that is important for SSPD operation may be unfamiliar to some readers. It is the penetration depth for a magnetic field perpendicular to film surfaces.

This chapter begins with the introduction to the detection principles on conventional quantum and thermal detectors, which helps us to understand why superconductivity plays a crucial role in achieving an ultimate energy resolution. The term energy resolution (ΔE) corresponds to the smallest detectable difference between two adjacent peaks for particles with different kinetic energies.⁴⁹ The ΔE values are often expressed by full width at half maximum (FWHM) of a peak for mono-energetic particles (E₀) (see  Appendix A for the details). A total absorption peak for mono-energetic x-ray photons is a good measure of the energy resolution. High resolution corresponds to a small ΔE value, which should not be confused with resolving power that is expressed by E₀/ΔE.

The statistical physics for semiconductors provides the primary basis for superconductor quantum detectors. In semiconductor detectors, particle incidence leads to electron excitation from the valence band to the conduction band and holes are left in the valence band, keeping charge neutrality or conserving the number of electrons. Those charge carries are collected by an electric field applied to the semiconductor, as illustrated in Fig. 3. Almost all charge carriers are collected in the modern Si detectors. The integration of the current pulse flowing R_L is equal to the number of the electron–hole (e–h) pairs, as well established in semiconductor detector physics. In other words, the voltage source applies a constant voltage to the detector only; the current injected to the detector is negligible even when an x ray is absorbed. In this condition, the circulating electron current through R_L annihilate the holes; therefore, the output charge number is half the total number of the quasiparticles created in the depletion layer. This is also important to consider the output charge number in superconductor detectors, which will be discussed in Secs. III B 2 and III B 3 b.

The Fano factor was devised to bridge a gap in variance between an experimental energy resolution of gaseous ionization detectors and Poisson distribution that should govern radiation counting.⁵¹ The experimental resolution of the gaseous detectors was significantly better than that of the Poisson distribution. The physical meaning of the Fano factor is still an open question; however, it is obvious that F is the variance ratio of an exact distribution function to the corresponding Poisson distribution having the same $N ¯$⁠. The Poisson statistics corresponds to the limit of n → ∞ and p → 0 in the binomial distribution, keeping a constant $np(= N ¯)$ value. The Fano factor is certainly a valuable indicator for a deviation from the Poisson distribution; a large deviation means a better energy resolution.

Instead of the analytical description, the Fano factor can be experimentally defined.⁵³ To be more practical, the ɛ value can be obtained by dividing E₀ by the number of detected charge carriers, and ΔE is substituted by the FWHM value of the observed total absorption peak. An empirical F value is obtained from Eq. (4). In this empirical definition, any combination of F and ɛ is possible without any constraints. This is a convenient way; however, this has no physical meaning. There must be a constraint condition between F and ɛ according to the definition of F and ɛ. At present, Monte Carlo simulations provide a reasonable measure of the statistical energy resolution limit.

The Fano factor is still an active research area even these days. Many models for ionization statistics have been proposed to explain F values that are significantly smaller than 1.0: statistical partitioning models including “crazy carpentry model”⁵⁴ in 1965, “phenomenological models of electron cascades”⁵⁵ in 2006, and “bathtub-whiskey-glass model”⁵⁶ in 2008. A recent theoretical study for a low energy region, in which only a few charge carriers are created, adopted a method with freedom on a $N ¯−F$ parameter space. The adopted probability function was the Conway–Maxwell–Poisson distribution, which was a member of the weighted Poisson distribution family covering both cases of F < 1 and F > 1.⁵⁷ More discussion on the Fano factor appears in that paper and references cited therein.

Superconductor detectors should follow Eqs. (2) and (4) in principle, since they are common to all quantum detectors having an energy gap. Although there are still uncertainties of F and ɛ, they signify that all quantum detectors should follow a dependence of $E 0$ regardless of semiconductors and superconductors. The lower the particle energy is, the better the energy resolution is.

Based on the statistical definition of F, very small F values mean that carrier creation trials along a particle trajectory are strongly dependent. In other words, in F ≪ 1.0, charge carriers are created efficiently, so that ɛ/E_g should approach 1.0. If all the particle energy is consumed to create e–h pairs and all the carriers are collected, there is no chance of statistical fluctuation: ɛ = E_g and F = ΔE = 0. In contrast, when ɛ/E_g is large, statistical fluctuation of the carrier number should be also large, approaching the Poisson statistics. Therefore, the empirical determination without any constraints is fraught of incorrect results due to experimental or simulation shortcomings. Another possible fluctuation or loss processes can be attributed to the binding energies of E_K, E_L, E_M, etc. A keV photoelectron or an Auger electron cannot carry the whole x-ray photon energy because of the biding energies of electron shells, that is, E₀-E_K, E₀-E_L, E₀-E_M, etc. When we subtract the biding energies from E₀, the relation of ɛ and F becomes more reasonable.⁵⁰ 

To estimate the ɛ–2Δ_g relation analytically, we can follow the Klein treatment⁵⁸ for semiconductors from the point of view of the density of the states (DOS). It calculates an average kinetic energy of excited electrons and holes in semiconductors within a free-particle approximation. Although the DOS profile of superconductors is completely different from that of semiconductors, it is worthwhile to apply the Klein treatment to superconductors. It provides a possible physical basis on the ɛ–2Δ_g relationship and is not inconsistent with experimental and Monte Carlo simulation data.⁵⁰ Although we may need to formulate a new model for the ɛ–2Δ_g or ɛ–E_g relation in superconductors as well as semiconductors, Owens and Peacock plotted energy resolutions expected for different superconductors as a function of 2Δ_g, extrapolating the data of semiconductors in a few eV toward a lower energy 2Δ_g region in a few meV.⁵⁹ The assumed Fano factors were 0.14 for semiconductors and 0.22 for superconductors. The expected ΔE values at 5.9 keV range from 9 eV of MgB₂ to 0.5 eV of Hf.

One concern unique to superconductors is the possible formation of normal cores or reduced Δ_g cores in several keV x-ray absorption^60,61 because of numerous quasiparticles in contrast to semiconductors that have a constant E_g value independent of e–h density. When it happens, quasiparticle loss occurs, resulting in a low charge output or a low energy resolution.

A schematic diagram of the thermal detection is shown in Fig. 4. The essence of cryogenic thermal detectors is that cryogenic low temperature T enables a small thermal energy fluctuation of an absorber with a small heat capacity C. In principle, there is no lower detection limit for photon energies or wavelengths because of no energy gap.

Superconductor thermal detectors should follow Eq. (6). Comparing Eqs. (4) and (6), it is apparent that ΔE of thermal detectors is independent of incoming particle energies, assuming that a temperature increase of the subsystem or absorber is negligibly small compared with the bath temperature.

Equation (6) is not a practical limit in modern thermal detectors. When the power spectrum of fluctuation is concerned, it spreads over a wide range of frequencies. Output signal pulses have a shape of exponential decay, so that it is possible to narrow a readout bandwidth. This is called optimal filtering. Accordingly, the energy fluctuation observed experimentally can be smaller than Eq. (6). Nevertheless, Eq. (6) is a primary guidance for detector design. The energy resolution becomes better or smaller when a thermal detector has a smaller C and operated at a lower T. In contrast to quantum detectors, of which energy resolution is proportional to $E 0$ and independent of T, thermal detectors have the energy resolution independent of E₀ and proportional to T. Thermal detectors can cover a wide particle-energy range at a constant energy resolution in principle. This difference suggests that thermal detectors are advantageous for high-energy particles, whereas quantum detectors perform well for low energy particles.

We can utilize a superconductor at one part or more for the three components: absorber, electric signal generation device, and readout circuit. In general, quantum detectors have a superconducting absorber, whereas thermal detectors have a superconductor thermometer. Superconductors have a smaller C value than normal metals because of the gap. Therefore, a superconductor can also be used as the absorber for thermal detectors. However, because of long-lived electronic excitations in superconductors, a normal metal absorber like a Bi/Au bilayer is a choice for modern superconductor thermal detectors.⁶³ 

Equation (2) indicates that an absorber with a small E_g value is beneficial to obtain a better energy resolution. Are there any solid-state materials that have a uniform small energy gap across a large volume enough for realizing a practical sensitive area or volume? The answer is to use superconductors, of which energy gaps are in a range of meV that is about 1000 times smaller than that of Si. The meV gaps are kept uniform across a macroscale without being influenced by local defects. Therefore, polycrystalline superconductors can be employed in contrast to the requirement for high quality single crystals for semiconductor detectors to avoid carrier trapping.

Provided that superconductor detectors have the same p value or Fano factor as semiconductor detectors, Eq. (2) indicates that superconductor detectors enable an approximately 30-times better energy resolution than that of semiconductor detectors: $E g / 2 Δ g≈30$⁠. It is surprising that the 30-times better energy resolution was predicted 58 years ago in the Wood's master’s thesis under the supervision of White in 1965.⁶⁴ In the 1960s, they faced practical problems of superconductor materials, fabrication, cryogenics, and readout electronics. In his master’s thesis on Al/AlO_x/Pb junctions, he wrote “The most formidable problem was degradation of Al/AlO_x/Pb junctions showing junction resistance increased with time after fabrication.” He also concluded “The experimental and theoretical study reported in this thesis has shown the plausibility of successfully developing a superconducting charged particle detector. There is, however, a considerable difference between a device working in theory and a device working in practice.” For the doctor’s thesis, they moved onto an Sn/SnO₂/Sn junction. They successfully observed output pulses from the Sn/SnO₂/Sn tunnel junction for 5.1-MeV α-particles, although no total absorption peak was observed in 1969.^65,66

Fifteen years after the work by Wood and White, an Sn–SnO₂–Sn junction was utilized for detecting α-particle by Kurakado and Mazaki in 1980.⁶⁷ The Sn-based junction successfully acquired pulse height spectra, which was the first spectroscopic measurement with STJDs; however, no distinct total absorption peak corresponding to 5.1-MeV α-particles was observed. A Monte Carlo simulation on ɛ and F values was performed for Sn in 1982.⁶⁸ The result on Sn was ɛ = 1.68Δ_g/e (=0.969 meV) and F = 0.195 for stimulation at 5.75 eV (=10⁴⋅ Δ_g/e), which is an important guideline for superconductor quantum detectors. This result is consistent with succeeding simulations for other superconductors.^69–71 The parameter values should be corrected to ɛ = 1.68 × 2Δ_g and F = 0.098 for quasiparticle pairs that are created simultaneously, as discussed in Section III B 2. In addition to α-particles, Twerenbold et al. utilized a Sn-based junction to detect 20-keV lysozyme protein ions (14 305 Da) for mass spectrometry (MS) in 1996.^72,73 Hilton improved an energy resolution for biomolecules by using a normal–insulator–superconductor (NIS) junction in 1998, enabling charge-state discrimination for bovine serum albumin (BSA, 66 430 Da).⁷⁴ Wenzel et al. realized detection of macromolecules, such as immunoglobulin M (IgM, 1 MDa) and von Willebrand factor (1.5 MDa), by using a 16-pixel Nb/Al-STJD in 2005.⁷⁵ 

The energy resolution limits expected by using the Monte Carlo simulation⁷¹ on Nb and Eq. (4) are 4.3 eV for 5895 eV (⁵⁵Fe x-ray source), 1.3 eV for 525 eV (O K-L₃), and 1.1 eV for 392 eV (N K-L₃), assuming that all the quasiparticles are collected once. They are far better than 119 eV for 5895 eV, 36 eV for 525 eV, and 31 eV for 392 eV by using a Monte Carlo simulation with ɛ = 3.70 eV and F = 0.117 for Si.⁷⁶ Experimentally, we recorded an intrinsic energy resolution of 3.6 eV at 400 eV with a Nb/Al-STJD in this article. A Ta/Al-STJD of Lawrence Livermore National Laboratory (LLNL) achieved 1.5 eV at 250 eV and 3.5 eV at 1 keV.⁷⁷ The STJ operation adds fluctuation due to imperfect charge collection, multiple quasiparticle tunneling, and spatial nonuniformity as discussed in Sec. III B 3.

Another striking feature of the meV gap is to extend a range of single-photon energy measurement toward the lower energy side. Nb- and Ta-based STJDs marked a remarkable performance of ΔE/E₀ ≈ 10% near 1 eV in a range from ultraviolet to near-infrared photons.^78–80 This was the first single-photon energy measurement in an infrared–optical range. A similar measurement was also realized by thermal detection with TESDs.⁸¹ 

The other quantum detector type has a superconductor strip resonator structure to sense a surface impedance change in a microwave frequency range, based on Cooper pair breaking. It was recently invented in 2003: microwave kinetic inductance detector (MKID) for a large-scale array format for imaging.⁸² Quasiparticle excitation due to radiation induces a Q-factor change. Each pixel has its own resonance frequency so that it is possible to read out all pixels through one readout line.

In this section, we describe the e–h pair excitation in semiconductors first as a guidance on how to understand the quasiparticle excitation in superconductor quantum detectors. This approach also holds great significance as it addresses the crucial need for a proper understanding of the statistical treatment concerning the relationship between the number of quasiparticles in superconductor detectors and the output charge number.

There are two types of charge excitations in metals, semiconductors, insulators, and superconductors: electron-like excitation (quasielectron) and hole-like excitation (quasihole), which are collectively called quasiparticles. Energetic particles, such as photons, electrons, protons, and even atoms, create quasiparticles. Let us consider a case of x-ray photons, since all the particles end up the same excitation process as x rays except for initial interaction. The x-ray absorption can be transformed into an electron energy loss process after the emission of a keV photoelectron or an Auger electron from an atom. The highly excited electrons relax to the order of eV, which is equivalent to E_g for ∼0.1 ps, through inelastic scattering with electrons. They relax to the order of meV equivalent to Δ_g of superconductors in an elapsed time range of 1 ns–1 μs through electron–electron or electron–phonon scattering.⁸³ These collision cascades excite quasiparticles. Figure 6 covers the excitation energy range below 10 eV and depicts the DOS profiles of semiconductors and a superconductor.

In Si and GaP at 0 K, all states above E_c are vacant and all states below E_v are occupied by electrons. The Fermi level E_F of intrinsic semiconductors is placed at the middle of the energy gap. In a superconducting state, the superconducting energy gap 2Δ emerges at E_F according to Eq. (10). The E_F of Nb is 5.32 eV. Figure 6 clearly depicts the dissimilarity between the bandgap E_g and the superconducting gap 2Δ. Later, however, they are identically treated in statistical physics for the fluctuation of the number of charge carriers.

An excitation path in semiconductors is designated by the red near-horizontal arrow for Si in Fig. 6. An electron always accompanies a hole, forming the e–h pair according to charge neutrality, as illustrated in Fig. 3. When they meet, they recombine and annihilate. During electron energy relaxation, phonons are emitted. All the phonons cannot contribute to carrier creation in semiconductors because the maximum phonon energy or the Debye energy of Si is 55.6 meV, which is exceedingly smaller than an E_g value of 1.1 eV. In Nb, similar e–h pair creation beyond E_g initially occurs, regardless of whether Nb is in a superconducting state or not. In superconducting Nb, the electrons in a few eV range break Cooper pairs, exciting quasiparticles at levels far higher than Δ. Phonons are emitted during quasiparticle relaxation processes and on the occasion of recombination into Cooper pairs. The striking difference from semiconductors is that the Debye energy of 23.7 meV of Nb is higher than 2Δ/e (3.04 meV): therefore, most phonons can break Cooper pairs again, contributing to quasiparticle generation. High-energy quasiparticles finally relax to the edge of Δ since phonons below 2Δ cannot break Cooper pairs.

The above-mentioned quasiparticle excitation in superconductors naturally postulates that Cooper pairs exist implicitly. The processes of forming and breaking Cooper pairs are complex yet essential for superconductor quantum detectors. A simple and straightforward explanation of these processes for detector operation is rarely found in the available literature. Figure 6 provides a schematic illustration that sheds light on these processes. For forming a Cooper pair, it is necessary to add two electrons to a Fermi sea. When we consider an isolated superconductor, it corresponds to the electron excitation process leaving two holes below E_F as designated by the blue curved arrow near E_F(Nb), holding charge neutrality or conserving the electron number. This excitation requires an energy, but the energy reduction due to phonon-mediated attractive force, which forms the Cooper pair with the electrons near E_F, significantly exceeds the excitation energy. Almost no textbooks describe these holes accompanied. It can be understood by considering the gray dashed line for the electron occupation distribution $v k 2$ in the BCS ground state at T = 0; it is different from the Fermi step function in a normal-conducting state at T = 0. The holes left behind when forming the Cooper pair apply a hole-like character to quasiparticles when the two electrons in the Cooper pair breaks up. The excitation of quasiparticles requires the preservation of superconducting coherence, provided that the number of broken Cooper pairs is not enormous. When superconductivity is broken in a region beyond a coherence length, the two electrons simply revert back to the initial two electrons, following the reverse direction of the blue curved arrow, thereby precluding the generation of quasiparticles.

Returning to the coherent state, the red vertical arrow in Fig. 6 designates Cooper pair breaking. A photon or a phonon with an energy above 2Δ creates two quasiparticles by breaking a Cooper pair, obeying the laws of momentum and energy conservation. More precisely, we should consider the excitation on the momentum k space. When a Cooper pair $( k ↑ , − k ↓)$ is broken, a pair of excitations appears at $( k 1 ↑ , − k 1 ↓)$ and $( k 2 ↑ , − k 2 ↓)$⁠. The character of the two excitations can be electron-like or hole-like, depending on how momentum and energy are transferred to the two excitations or whether k₁ and k₂ are above or below k_F. On ensemble average, the quasiparticles have quantum superposition of 50%-electron and 50%-hole characters. In Fig. 6, the quasiparticles are represented by two ovals surrounding a filled circle and an open circle. When the excitation level is high enough well beyond Δ, the quasiparticles are close to either pure electron or pure hole. This is the case that we assume for the tunneling processes in Fig. 8 in Sec. III B 3.

The excitation in superconductors is called Bogoliubov quasiparticles or Bogoliubons. The original paper was published in 1958,⁸⁴ but it is still an active research area,⁸⁵ influencing many fields beyond superconductivity.⁸⁶ The character of Bogoliubov quasiparticles changes continuously between hole-like excitation and electron-like excitation on the single-particle picture; for example, 70% electron-like or 70% hole-like characters. This article follows a simple view for detectors: the absorption of an energy quantum above 2Δ create two quasiparticles, of which characters are either purely electron-like or hole-like, that is a quasielectron–quasihole pair; the superposition is an ensemble average. In this context, e–h pairs in semiconductors and quasiparticle pairs in superconductors can be treated similarly when we consider output current pulses flowing an external readout circuit in Fig. 3, although the origin of 2Δ in superconductors are totally different from E_g in semiconductors.

Up to here, we have described what happens when energetic particles are absorbed in superconductors. How can we read out an increase of the number of quasiparticle pairs and measure the incident particle energy E₀? Unlike semiconductor detectors in Fig. 3, it is impossible to collect carriers by applying a high electric field to superconductors. Fortunately, there are two readout methods: quantum tunneling in STJD biased in a subgap region and RF surface impedance change in MKID.

Figure 7 shows the top view and cross section of an Nb/Al-based STJD. One of the critical requirements for analytical instruments is a large sensitive area that should be enough to acquire signals such as characteristic x rays emitted from analytes. Detector junction sizes are usually between 100 and 200 μm² to avoid an unreasonably large junction capacitance and to ensure a pinhole free 1-nm thick tunnel barrier.

Detector junctions normally have the Al layers with a thickness of several 10 nm on both sides of the AlO_x tunnel barrier layer in expectation of quasiparticle trapping effect based on energy gap engineering proposed by Booth in 1987 and Kraus in 1989.^87,88 Most of the junctions have the polycrystalline structure for both top and bottom electrodes.⁸⁹ The bottom electrode can be made from a single crystalline superconductor film, which is advantageous to optical photon detection.^78,90 Higher Z superconductors such as Nb or Ta located at both ends of the layer structure serves as an x-ray absorber with a meaningful absorption efficiency below a few keV. Furthermore, the proximity bilayers maintaining T_c higher than 1.2 K of bulk Al are favorable for raising an operation temperature up to 0.5 K. The amorphous AlO_x tunnel barrier layer formed by oxidizing the Al surface has a thickness of about 1 nm without pinholes. It was reported that less than 10% of the junction area dominantly contributed to tunnel current because of a Gaussian variation of AlO_x thicknesses.⁹¹ In addition, the AlO_x barrier layer had an oxygen deficiency at the interface between the Al layer and the amorphous AlO_x layer.⁹² 

For connecting STJs and readout circuits, wiring leads are indispensable. The latest microfabrication technology for STJDs appears in Ref. 93, in which the junction and wiring layers are separated vertically by using a planarization process with chemical mechanical polishing (CMP). This fabrication technique has an advantage over large-scale close-packed array detectors (See Fig. 35).

When a particle impinges the STJD at <T_c/10, a similar I_sub current flows, although quasiparticle excitation distribution is different from the thermal equilibrium distribution in Fig. 8. The readout circuit for STJDs resembles that for semiconductor detectors in Fig. 3. Similarly, a bias source injects a negligibly small current to the junction biased in U_b< (Δ₁ + Δ₂)/2e when the DC Josephson current is suppressed by applying a parallel magnetic field. Therefore, on the analogy of semiconductor detectors, the experimental charge output value should correspond to the number of quasielectron-quasihole pairs rather than the number of total quasiparticles, holding charge neutrality or conserving the number of electrons. Statistical fluctuation should be also treated by the number of quasiparticle pairs because they are created simultaneously. In this context, E_g in semiconductors and 2Δ in superconductors is equivalent as the minimum energy gaps against electron–hole pair creation. We should pay attention that E_g is insensitive to the density of e–h pairs in semiconductors, whereas Δ is dependent on the quasiparticle density.

STJ output pulse and sub-gap structure: For proper detector operation, junctions are cooled below ∼T_c/10, eliminating thermal excitation. Particle incidence induces energy-mode imbalance.⁹⁶ A possibility of charge-mode imbalance will be discussed separately.^50,97 The increase of the quasiparticles generates a current pulse through the tunnel paths when a junction is biased in a subgap region. The typical bias point for particle detection in our experiments is just below Δ_g/e or just after the beginning of a rapid current increase, as indicated by the vertical dashed line on the I–U curve in Fig. 9(a). Although it exceeds the optimum bias point for maximum pulse height, this bias point is effective in avoiding bias-point jumps due to Fiske steps.⁹⁸ It is also beneficial for operating many junctions simultaneously and stably.

When a particle incident event creates quasiparticles, the bias point shifts from almost zero current to a high current according to the 210-Ω load line (dotted arrow) in Fig. 9(a). A cryostat wiring lead resistance of 160 Ω and a readout circuit input resistance of 50 Ω determine the load line. This voltage shift may affect quasiparticle tunneling processes slightly, but it is negligible in an x-ray energy range below 1 keV. Figure 9(b) shows the current pulse for a 6-keV x-ray photon from a ⁵⁵Fe source. The rise time is determined by a readout bandwidth, whereas the fall time corresponds to the effective quasiparticle lifetime τ_eff. The single exponential decay curve (solid line) has a τ_eff value of 1.42 μs, which is significantly longer than a material-dependent intrinsic lifetime because of the trapping effect of phonons created upon quasiparticle recombination into Cooper pairs.⁹⁹ In addition, phonon escape to a substrate should be considered.¹⁰⁰ The integration of the current pulse equals to the detected charge corresponding to the number of the created quasiparticle pairs or the particle kinetic energy E₀. The charge pulse for 6 keV has ∼10⁶ electrons, which is surely higher than ∼1600 electrons in Si. Even though no strong charge collection mechanism exists in superconductors, the extremely high output-charge ensures a high signal/noise (S/N) ratio and a high-energy resolution statistically.

In Fig. 9(a), the subgap current begins to increase considerably at Δ_g/e, which deviates from the BCS-DOS prediction. The current increase at 2Δ_g/ne, where n is integral, was attributed to multiple Andreev reflection (MAR) before,^101–103 although other mechanisms were also proposed. It is called the subharmonic gap structure. On the MAR model, multiple charges of ne are transferred at the bias voltage 2Δ_g/ne. It was believed that probable pinholes in an AlO_x barrier layer caused MAR.¹⁰³ However, the gap structure may also exist in pinhole-free junctions in a variety of barrier transparency.¹⁰⁴ Furthermore, no pinholes were found in a 1-nm thick AlO_x layer.⁹¹ The junctions for analytical instruments in this article are classified as a low transparency category with j_c = ∼200 A/cm² to obtain high resistance in a subgap region. Pinholes are unlikely at this low j_c, but the anomaly clearly appears at n = 2, although higher-order subharmonic gap structure is obscure in the inset of Fig. 9(a). The subharmonic structure without pinholes can be explained by multiple cycles of quasiparticle sequential tunneling (MQT) proposed by Kozorezov et al.¹⁰⁵ The quasiparticles that tunnel via path 1 and path 2 in Fig. 8 acquire an energy of eU_b. The forward-and-back multiple tunneling causes energy accumulation. This MQT process leads to current jumps at 2Δ_g/e n, where n is the number of tunneling times. Although the MQT model is most promising, the origin of the subharmonic gap structure is still a long-standing open question.¹⁰⁴ 

Pulse height spectrum: Figure 10 shows a histogram of output charge values for the x-ray photons of 6.00 keV from a synchrotron radiation source. The x rays irradiated an area with a diameter of 5 μm at the center of the 146-μm square Nb/Al junction with 30-nm-thick Al layers.¹⁰⁶ The histogram of photon counts per a small energy bin as a function of detected energies is termed the pulse height spectrum. The energy resolution ΔE value of 92 eV for the bottom electrode exceeds a typical ΔE value of 120 eV for Si detectors. However, Fig. 10 exhibits two disadvantages of STJDs in this x-ray energy range.

The first disadvantage is that 90% of the incoming 6-keV photons pass through the whole junction layer structure, which produces a double peak due to the absorption events in either the top electrode or the bottom one.^107,108 The double peak phenomenon is primarily due to a difference in τ_eff between the top and bottom electrodes. The top Nb electrode is covered by a few nm thick NbO_x metallic layer, which acts as a quasiparticle trap, leading to a short τ_eff.^107,109 Because of the low absorption efficiency of the Nb electrodes, the Si substrate absorbs 90% of the incoming photons, which induced phonon excitation. The junction detects these high-energy phonons, appearing as a large number of phonon events (or substrate events) below the channel number 60 in Fig. 10. The second disadvantage is that the peak positions of the Si K-L_2,3 (Kα_1,2) events at 1.74 keV and the Nb-L escape events at 3.84 keV in Fig. 10 are not proportional to E₀. This is termed energy nonlinearity, which disturbs the assignment of peaks appearing on pulse height spectra in material analysis. The section describes the energy nonlinearity or linearity and how to solve the problem.

Energy linearity: We measured the energy linearity of the 200-μm square junction with the 70-nm thick Al layers in detail. A synchrotron radiation microbeam with a diameter of 10 μm irradiated the center of the junction at 6, 7, 8, 9, and 10 keV. Figure 11 shows pulse heights as a faction of photon energies with the data of the Si K-L_2,3 and Nb L escape peaks. The solid line of the junction with the 70-nm-thick Al layers has an extremely better linearity up to 8 keV than the dashed line of the junction with the 30-nm Al layers in Fig. 10. Furthermore, the 70-nm STJD covers up to ∼30 keV with a tolerable energy nonlinearity as displayed in the lower right inset, which plots the data for photons and multiply charged lysozyme ions (14 305 Da) accelerated at 7 kV.¹¹⁰ Beyond ∼30 keV, a strong nonlinearity may prevent proper particle energy measurement. The upper-left inset shows the full illumination peak data at energies less than 1 keV. The full illumination energy resolution at 400 eV was 28 eV with an electrical noise of 25 eV, which corresponds to an intrinsic energy resolution of 13 eV.

Energy resolution: To increase the absorption efficiency and expand the covering energy range, the thickness of the top Nb layer was increased to 300 nm. Figure 12 shows a pulse height spectrum of 400-eV full illumination for a 100-μm square asymmetric junction having a layer structure of Nb(300)/Al(70)/AlO_x/Al(70)/Nb(100)/Si in nm. The absorption coefficients for the 300-nm-thick Nb top film are 80% at 200 eV, 99.7% at 400 eV, and 92.2% at 700 eV;¹¹¹ the effect of the double peak is negligible below ∼1 keV. The FWHM value at 400 eV is 4.1 eV, which is 10 times better than 45 eV of the latest 10 mm² silicon drift detector (SDD).¹¹² After subtracting an electrical noise of 2.0 eV, the full illumination intrinsic energy resolution is 3.6 eV. A better energy resolution in this soft x-ray range was obtained with a Ta/Al-based junction developed by Friedrich et al. primarily because α-Ta has a lower Δ_g/e of 0.7 meV than 1.52 meV of Nb. The Ta/Al junction marked 2.5 eV at 250 eV and 4 eV at 1 keV including an electrical noise of 2 eV.⁷⁷ 

The energy resolution of the Nb/Al-STJD exceeds the typical natural linewidths of elements embedded in a matrix. For example, the carbon atoms in CaCO₃ have a K-line width of ∼14 eV (Sec. I A, Fig. 2). In addition, the Zn L₃-M_4,5 (Zn L) line and other lines in a dentin sample has a linewidth of ∼15 eV (Section V B 1 d, Fig. 44). Broadening of characteristic lines exceeds the level widths of electronic shells because of chemical bonds to other elements or electronic band structure. Therefore, no further energy resolution improvement is necessary for elemental mapping analysis in a range below ∼2 keV.

Magnetic environments in the course of the cooling process considerably affect output charge values. We observed that the output charge is significantly reduced after cooling in the geomagnetic field. This reduction stems from Abrikosov vortices (AVs) vertically piercing the junction layer structure. Since polycrystalline Nb films normally behave as a type II superconductor, AVs can be trapped while crossing T_c.^113,114 The effects of AV trapping were studied by low temperature scanning electron microscope (LTSEM) at Tübingen.^83,115 The reduction of output signal amplitude and quasiparticle effective lifetime as a function of perpendicular magnetic field strengths were observed at 1.6 K.¹¹³ Moreover, LTSEM signals indirectly visualized each AV trapped in junctions with electron-beam stimulated signals.¹¹⁶ 

We directly visualized AVs by measuring magnetic field distribution with a SQUID microscope.¹¹⁷ To investigate AV trapping patterns, the 200- or 400-μm square junctions were cooled in a perpendicular magnetic field between 0 and 7.3 μT. Figure 13 shows the dependence of AV trapping patterns on the perpendicular magnetic field strengths during cooling. The black dots in Fig. 13(a) denote the AVs carrying one magnetic flux quantum Φ₀ of 2.067 × 10^–15 Wb. The number of AVs increases with increasing the magnetic fields from 1.3 to 7.3 μT. Above ∼10 μT, the SQUID microscope was unable to distinguish individual AVs because the AV density exceeds a spatial resolution of ∼20 μm. We need to shield a geomagnetic field of 35.4–35.9 μT at AIST-Tsukuba¹¹⁸ for proper STJ operation.

One of the striking features in Fig. 13(a) is that no AVs are trapped near the junction edges. The width of the AV-free edge region decreases with increasing magnetic field. This phenomenon is explained by vortex nucleation under a surface barrier of superconductor films, based on the classical Bean and Livingstone surface barrier model.¹¹⁷ The AV densities are plotted against magnetic fields in Fig. 13(b). The lower critical field H_c1 values are 0.68 μT for the 200-μm junction and 0.95 μT for the 400-μm junction: the large junction has the higher H_c1. In addition, the straight line fitted to the data of the 400-μm junction has a lower gradient than that of the 200-μm junction. It signifies that the large junction size is advantageous to exclude the magnetic flux.

AVs prevent quasiparticle generation, turning Cooper pairs into normal electrons along the reverse direction of the blue curved arrow Fig. 6. Moreover, they cause quasiparticle lifetime shortening and thus output charge reduction.^113,114 The quasiparticle lifetime depends on AV density and excited quasiparticle density. Figure 14 exhibits these two loss processes. The absorption events for 5.9-keV x rays are plotted according to output charge values and the pulse rise times of the charge-sensitive preamplifier. The rise times correspond to the τ_eff values. The 50-μm square junction with the 40-nm thick Al layers was cooled with and without the magnetic shield having an attenuation factor of well over 1000. The x-ray events form the total absorption group at 8.2 × 10⁶ electrons at ∼0 μT and 3.6 × 10⁶ electrons at ∼35 μT. The events due to photoelectron escape, Si K-L_2,3, and Nb L escape continuously connect the total absorption event group and the substrate event group.

The quasiparticle loss rate is given by 1/τ_eff= 1/τ_r+ 1/τ_AV, where τ_r is the quasiparticle lifetime against recombination into Cooper pairs without AV trapping and τ_AV is the effective quasiparticle loss time due to AVs. The τ_r values at ∼0 μT were 2.2 μs for the low-energy substrate events and 1.0 μs for the total absorption events of the 6-keV x rays. The continuous τ_r reduction with increasing output charge at 0 μT represents self-recombination into Cooper pairs.⁹⁰ The higher the density of the excited quasiparticles is, the shorter the τ_r is in a condition that thermal excitation is negligible at 0.3 K. τ_r was independent of photon energies below ∼2 keV, in which STJDs excellently operate in a linear regime.

The trapped AVs lead to τ_eff reduction with a factor of 4 and output charge reduction with a factor of 2.3. The AV quasiparticle loss rate 1/τ_AV is given by 1.82n_vπR_eff²/τ₀, where n_v is the AV density, R_eff is the effective vortex radius, and τ₀ is the material dependent characteristic time.^113,114,119 τ_eff for the low-energy events is shortened by the AVs from 2.2 to 0.7 μs, which gives τ_AV= 1.0 μs. The τ_eff for the total absorption events are shortened from 1.0 to 0.5 μs, which also gives τ_AV= 1.0 μs. Therefore, the τ_AV estimation is consistent. Considering n_v= 1.7 × 10¹⁰ m⁻², τ₀= 0.149 ns,⁹⁹ and ξ = 23 nm³³ or 10 nm¹⁰⁸ for the polycrystalline Nb films, the τ_AV value leads to R_eff ≈ 1.5ξ or 3ξ.¹¹⁴ This R_eff range is consistent with the values in the literature.^113,119

The proper detector operation requires no AVs. Complete exclusion of AVs requires a magnetic shield with an attenuation factor of over 40–50. We are fortunate in not installing a massive magnetic shield for SQUID-MEG but using a simple shield. Nevertheless, electric parts near the detector chip often include ferromagnetic materials, which degrades the detector performance as shown in Fig. 14. Therefore, careful selection of the electric parts is necessary.

In this section, we selectively use Δ for the order parameter and Δ_g for the bottom edge of a DOS profile. In other words, the order parameter Δ describes the Cooper pair potential, whereas the gap Δ_g is the minimum excitation energy appearing on the quasiparticle DOS profiles. Gap engineering involves two superconductors, of which Δ_g values are different. A combination of Al and either Nb or Ta is common for STJDs. In the “trapping model” of gap engineering, quasiparticles excited inside an Nb layer with an intact Δ_g,Nb/e value of 1.52 meV are trapped inside an Al layer with a lower Δ_g/e value, which varies between 1.52 meV and the intact Δ_g,Al/e value of 0.17 meV depending on the Al thicknesses. The trapping model is certainly valid and effective to collect quasiparticles when a tunnel junction is attached to a very thick bulk superconductor with a higher gap or when two junctions are attached at both ends of a-few-10-μm long superconductor film for imaging detection as proposed by N. Booth⁸⁷ and H. Kraus.^88,120,121 However, for direct detection with a thin-film tunnel junction geometry, this trapping model should be reconsidered and modified according to the theoretical studies on the proximity effect.^33,122–124 The “gap-reduction model” is proposed in this section below.

Table II lists the parameters based on the trapping and gap-reduction models for the STJs with 200-nm thick Nb layers and different Al thicknesses. We obtained the gap Δ_g, effective quasiparticle lifetime τ_eff, and output-charge number Q_c/e from experimental data. The excited quasiparticle number Q₀/e(=E₀/ɛ) in the Nb absorber is calculated based on the Monte Carlo result;⁷¹ it should be noted that the used relation is not ɛ = 1.72Δ_g for creating one quasiparticle but ɛ = 3.44Δ or ɛ = 1.72 × 2Δ_g for one quasielectron–quasihole pair, as discussed in Sec. III B 3 b. The ratio of the observed charge number Q_c/e to Q₀/e gives the average tunneling times $N ¯ tun$⁠. The tunneling probability P_tun values are valid for electrodes with a uniform Δ_g value: they are listed only for the gap-reduction model to discuss the multiple quasiparticle tunneling model in Sec. III B 3 f.

On the trapping model, the Q₀/e values for quasiparticle-pairs are calculated by E₀/(3.44 × 1.52 meV) for the intact Nb absorber. On the other hand, the gap-reduction model predicts that Nb/Al electrodes have a constant Δ_g value throughout the bilayer. The Δ_g/e values of 0.44–1.39 meV in Table II are evaluated for the different Al thicknesses by extrapolating I-U curve slopes at 2Δ_g/e to zero current. The Q₀/e values on the gap-reduction model are calculated by E₀/(3.44×Δ_g) with reduced Δ_g values inside the Nb layer, provided that the DOS has sufficient room for the 5.9-keV photon absorption.

Figure 15(a) illustrates the trapping model for the proximity tunnel junction with a symmetric layer structure of Nb(200)/Al(30)/AlO_x/Al(30)/Nb(200), of which I–U curve and output pulse are shown in Fig. 9. The order parameter Δ and the gap Δ_g are plotted as a function of the distances from the Al/AlO_x interface. Δ and Δ_g are equal or different depending on the circumstances. In an ideal BCS-DOS profile, which has a singularity at Δ_g on the trapping model, we can treat it with Δ = Δ_g. Because of the Nb/Al proximity effect, the Δ_g (0 K)/e value of 0.832 meV at the Al/AlO_x interface is considerably higher than the intact Al value of Δ_g,Al/e = 0.17 meV. On the trapping model, Δ_g inside the Nb layers is equal to the Nb intact value (Δ_g,Nb/e = 1.52 meV). This surely occurs when Nb/Al interfaces are dirty and have a low transparency. Figure 15(a) illustrates a case that an x-ray photon or a particle is absorbed inside the Nb layer on the left side. After the initial photoionization and electron energy loss, the quasiparticles are excited to high levels. They relax close to the edge of Δ_g,Nb afterward. Once quasiparticles are trapped in the left Al layer, they cannot enter the Nb layers again because of the higher Δ_g wall at the Nb/Al interfaces. The quasiparticles confined within the two Al layers can tunnel forth and back before they recombine into Cooper pairs. Since the Al layers are usually several times thinner than Nb absorber layers, the tunneling probability that is inversely proportional to the film thicknesses becomes high. Consequently, the quasiparticles created inside the Nb layer efficiently contribute to the output charge. The tunneling time enhancement is significant for the junctions with the Al layers of 30-nm or thicker in Table II, leading to gains in the ratio of Q_c/e to Q₀/e or $N ¯ tun$⁠, as depicted in Fig. 16 later. One can explain the experimental data with the trapping model; however, $N ¯ tun$ tends to be unreasonably high in some cases. A more plausible interpretation is possible by the gap-reduction model described below.

Figure 15(b) illustrates the gap-reduction model, in which non-negligible DOS exists below the intact Δ_g,Nb/e of 1.52 meV in the Nb layer with a thickness scale of a few 100 nm in accordance with the Usadal equations.¹²⁵ The schematic DOS profiles at different locations are based on the calculation results.^33,122–124 The Δ_g/e value of 0.832 meV obtained from the I–U curve holds constant at any location throughout the Nb/Al bilayer. The high-energy quasiparticles relax to the edge of Δ_g/e = 0.832 meV inside the Nb layer through a possible short trap at the DOS peak energy corresponding to the intact Δ_g,Nb/e of 1.52 meV.¹²² The relaxed quasiparticles tunnel to the counter electrode and spread inside the counter Nb/Al bilayer because of the constant Δ_g: there is no trapping effect. As an example, an Nb(100)/Al(90) bilayer keeping a constant Δ_g/e of 0.5 meV has a DOS value of 1.43 × 10⁶ states/meV/μm³ at a kinetic energy of 0.3 × Δ_g,Nb and at the Nb free surface.³³ This DOS value is large enough for accepting the quasiparticles excited by a 6-keV x-ray photon; an excited quasiparticle cloud of ∼10⁶ can be accommodated within an energy bin of 0.1 meV inside a presumed Nb volume of 20 × 20 × 0.2 μm³. Since the quasiparticles spread throughout the whole electrode volume, the quasiparticle density is considerably reduced by a factor of ∼100 and thus far below the allowable DOS. For x rays below ∼1 keV, the Nb layer has more DOS room at 0.832 meV. Consequently, we can treat the Nb/Al bilayers as uniform superconductors. This enables a simple treatment for multiple quasiparticle tunneling in Sec. III B 3 f. If the DOS at the Nb free surface is insufficient, quasiparticle densities vary along the horizontal direction in Fig. 15(b), and nonlinear effects such as DOS vanishing can occur below the intact Δ_g,Nb.

The exact quasiparticle distribution in the proximity bilayer depends on many factors such as particle energies, relaxation processes from a-few-eV electrons to the edge of Δ_g, Andreev reflection, quasiparticle reflection at the Nb free surface, and dynamic processes concerning the depth dependence of the DOS profiles. At this moment, there is no exact analytical solution of this quasiparticle dynamics. An intermediate situation between (a) and (b) can arise depending on film thicknesses, Nb/Al interface transparency, and particle energies.

Let us compare the two models with respect to a junction with very thick Al layers: Nb(150)/Al(200)/AlO_x/Al(200)/Nb(240) in nm fabricated by Mears at al.¹²⁶ They reported that the 2Δ_g/e of the Al layers was 0.34 meV at 0.95 K. The Δ_g/e value of 0.17 meV is equal to that of the intact Al. This is understandable from the extremely thick Al layers: the proximity effect due to the Nb layer is negligible. The trapping model was used to explain the experimental data: the $N ¯ tun$ value becomes 10 by dividing Q_c/e (=2.3 × 10⁷ electrons) by E₀/ɛ (=2.3 × 10⁶ electrons), assuming that x rays are absorbed in the top Nb layer with the intact Δ_g/e value of 1.52 meV. Since the E₀/ɛ value should be the number of created quasiparticle pairs, the corrected $N ¯ tun$ value is 20. One should not anticipate this significantly high $N ¯ tun$ value for the 200-nm thick Al layers because of the inverse relationship between the quasiparticle tunneling probability and film thicknesses. Alternatively, based on the gap-reduction model, the Δ_g/e values are constant at 0.17 meV throughout the bilayers with enough DOS. The initial quasiparticle-pair number inside the 150-nm thick top Nb layer is evaluated at E₀/ɛ = 1.0 × 10⁷ electrons, which gives $N ¯ tun=2.3$⁠. This value on the gap-reduction model is more plausible than 20.

Consequently, the gap-reduction model predicts that the most striking feature of gap engineering is not the strong quasiparticle trapping, but primarily the Δ_g reduction in the high Z absorber like Nb. The Δ_g reduction inside the Nb layer leads to the increase of the initially excited quasiparticles. However, most papers up to now have followed the trapping model for thin-film junction detectors. In addition, the transparency at Nb/Al interfaces and the strength of the proximity effect affect the exact DOS profiles inside Nb or Ta layers significantly,¹²⁷ so that experimental results should vary depending on thin-film deposition conditions. Hopefully, MKIDs with a bilayer geometry may play a crucial role to verify the gap-reduction model better than STJDs. MKIDs simply utilize surface impedance change due to quasiparticle excitation in superconductor strips as described in Sec. III B 4. In addition, when we fabricate a RF filter with a bilayer geometry, the gap-reduction model should be applied for a low power range with a small number of quasiparticles. The gap-reduction model in Fig. 15(b) is proposed in this article for the first time; therefore, further experimental and theoretical studies are required.

We need to optimize the Al layer thickness for analytical instruments within the framework of the gap-reduction model. Table II indicates that the τ_eff values become longer monotonically with increasing Al thickness, which is beneficial to collecting quasiparticles. This is consistent because Al has an intrinsic quasiparticle lifetime extremely longer than that of Nb: τ₀(Nb) = 0.149 ns and τ₀(Al) = 438 ns.⁹⁹  Figure 16 shows the $N ¯ tun$ curves for 6-keV x-ray absorption as a function of the Al thicknesses. The blue solid curve for P_tun on the gap-reduction model is discussed in Sec. III B 3 f. The red dashed $N ¯ tun$ curve on the trapping model is plotted for comparison. The red solid $N ¯ tun$ curve for the gap-reduction model exhibits the maximum value of 3.1 at 40 nm. At 40 nm, the output charge is considerably enhanced by the two mechanisms: lowering Δ_g in the Nb layers and keeping a reasonably high $N ¯ tun$ of 3.1. At 70 nm, $N ¯ tun$ slightly decreases to 2.2. Nevertheless, the energy linearity is the best at 70 nm in Fig. 11. The Al thickness of 70 nm is also beneficial to a spatial uniformity in Sec. III B 3 g. Therefore, we adopt 70 nm to realize high-energy resolution XRS.

The junction with the 70-nm thick Al layers in Table II has the parameters of P_tun= 0.69 (G = 1.4). Prediction of the theoretical ΔE limit requires a F value. The Fano factor is the variance ratio of a Monte Carlo simulation peak to the Poisson distribution with the same average number $N ¯= 1 n ∑ i = 1 n⁡ N i$⁠, where N_i is the carrier number and n is the simulation times. It gives $F= 1 n ∑ i = 1 n⁡ ( N i − N ¯ ) 2/ N ¯$⁠. When we take the Monte Carlo simulation result of F = 0.214 for all quasiparticles in Nb,⁷¹ it should be modified for quasiparticle pairs. Two quasiparticles are formed simultaneously when a Cooper pair is broken; therefore, the statistical fluctuation should be treated by the number of quasiparticle pairs. The $N ¯$ and N_i values become half for the quasiparticle pairs, which gives F = 0.107. Therefore, Eq. (16) suggests that the most fluctuation originates from the tunneling noise rather than the Fano fluctuation. In contrast, the charge collection in semiconductor detectors with a strong electric field is equivalent to $N ¯ tun=1$⁠, P_t= 1, and P_b= 0 and then G = 0. The high G value of STJDs is one of the disadvantages. Nonetheless, the effect of the extremely small Δ_g exceeds the extra tunneling noise: STJDs outperform semiconductor detectors by a factor of ∼10 experimentally.

Let us compare the experimental data on the energy resolution with the theoretical calculation based on the gap-reduction model in Table II. Figure 17 shows experimental intrinsic ΔE values as a function of photon energies below 1400 eV for the 100-μm square junctions with the Al layer thicknesses of 30 and 70 nm.¹³⁰ The monochromatic photons from synchrotron radiation uniformly irradiated the whole junction area. The electrical noises of 6.5 eV for 30 nm and 7.0 eV for 70 nm were subtracted from the measured ΔE values. The red and blue solid lines depict the theoretical curves for 30 and 70 nm, respectively. For the theoretical energy resolutions, we used the Δ_g and P_tun(=1/G) values in Table II and ɛ = 1.72 × 2Δ_g and F = 0.107 for quasiparticle pairs, G = 1.5 at 30 nm, and G = 1.4 at 70 nm.

The ΔE values of the junction with 30 nm deviate from the $E 0$ dependence above 500 eV abruptly, exceeding the ΔE of SDDs above ∼800 eV. This abrupt ΔE degradation is due to spatial nonuniformity as is reported in Ref. 90, which is the absorption-position dependence of output charge values. This is the topic of Sec. III B 3 g. The ΔE degradation in the high-energy range is typical for junctions with thin Al layers. Nevertheless, it is intriguing that below ∼400 eV the junction with 30 nm has a similarly small ΔE to that of the junction with 70 nm. This implies that the spatial nonuniformity is small even at 30 nm in that energy range. Direct measurement results on the Al thickness dependence on spatial nonuniformity will be shown in Fig. 21 in Sec. III B 3 g.

The junction with the 70-nm thick Al layers follows the dashed curve, which is the theoretical curve multiplied by 7. The best experimental intrinsic ΔE value for 70 nm is 9.7 eV at 500 eV, which is worse than the theoretical limit of 2.5 eV by $ΔE( eV)=2.355 E 0 ε ( F + G )=0.112 E 0$⁠. One of the possible reasons for the factor 7 at most energies is explained by the dip structure of the blue dots between 400 and 600 eV. The dip structure stems from the x-ray transmission of the top 200-nm-thick Nb near the Nb M-edge. The green solid line depicts the absorption coefficients of the top Nb layer.¹¹¹ In the energy range between 400 and 600 eV, almost all x-ray photons are absorbed by the top Nb layer. On the other hand, at the data points below 400 eV and above 800 eV, 30% or more of the photons transmit the top Nb layer and are absorbed in the bottom Nb layer. The absorption events in the bottom Nb layer produce slightly higher output pulses. This causes the wider FWHM values on the occasion of Gaussian fitting to the experimental peaks. Another possibility is a spatial nonuniformity that slightly remains event at 70 nm as shown in Fig. 21 in Sec. III B 3 g.

Additionally, we should consider the fact that the Monte Carlo simulations were performed by assuming imparted energies of 5.75–56 eV.^68,71 These energies are extremely smaller than those of keV x rays by a factor of 10–1000. In the x-ray energy rage, we should consider a cascade of radiative or nonradiative transition processes related to the K, L, and M shells as mentioned in  Appendix B. The number of the e–h pairs fluctuates before considering quasiparticle pair generation. This causes additional broadening of the experimental peaks. Consequently, the possible origins for the factor 7 are the double peak formation, the spatial nonuniformity, and the e–h pair fluctuation.

The black square at 400 eV in Fig. 17 plots the intrinsic ΔE value of 3.6 eV for the asymmetric junction in Fig. 12. This ΔE value is closer to the theoretical value of 2.2 eV because the top thick Nb layer reduces the double peak occurrence. Up to now, this junction has the best ΔE value at AIST.

The spatial nonuniformity occasionally exceeds other noise sources; for example, the junction with the 30-nm thick Al layers in Fig. 17. The linear ΔE degradation results in the significant deviation from the statistical fluctuation proportional to $E 0$⁠. To investigate spatial nonuniformity directly, we first employed the low temperature scanning electron microscope (LTSEM) at the University of Tübingen.^83,115 The base temperature of LTSEM was ∼1.5 K at the lowest by pumping ⁴He with an open hole for an electron beam. Despite the high base temperature, LTSEM has a high spatial resolution of well better than 0.1 μm that enables fine imaging analysis of STJDs. The LTSEM measurement revealed that the bottom electrode of a Nb/Al junction with the 10-nm thick Al layer normally produces a higher signal than the top electrode; it is explained by a difference in τ_eff and oxidation of the top Nb surface.¹⁰⁷ Furthermore, for a geometrical design guideline on wiring leads and junction shapes, we analyzed the junction shown in Fig. 18: wide wiring leads, which is necessary for measuring full I–U curves beyond I_c, and the round junction with a diameter of 113 μm, of which sensitive area is equivalent to 100-μm square. The layer structure was Nb(200)/Al(20)/AlO_x/Al(20)/Nb(200)/Si in nm for the operation at ∼1.5 K. The electron-pulse beam enabled to measure 2D profiles of signal amplitudes and fall-time constants.

The 2D image in Fig. 18(a) visualizes that the amplitudes near the junction rim are lower than the junction center, which is commonly observed for any junction shape. This leads to the low energy tail of the x-ray total absorption peak as observed in Fig. 12. To eliminate the low energy tail, one solution is to fabricate a shadow mask in front of junctions so that the junction edges are covered. The white solid line exhibits the position dependence of signal amplitudes along the green arrow. It visualizes the response of the contact hole and the wiring leads. The pulse fall-time constants in Fig. 18(a) are 150 ns in the top electrode and 210 ns in the bottom electrode or bottom lead just near the junction, which causes the double peak phenomenon.^107,108 In the contact hole, the fall time of 190 ns is longer than 150 ns on the top electrode. This can be attributed to quasiparticle diffusion time to the junction area in the thick Nb as shown in Fig. 18(b). The LTSEM observation suggests that contact holes must be small and wiring leads must be narrow as much as possible to achieve a high-energy resolution.

It is necessary for analyzing STJDs at a normal operation temperature of ∼0.3 K. We developed the imaging instrument using an x-ray microbeam with a diameter of 5–10 μm from synchrotron radiation: low temperature scanning synchrotron microscope (LTSSM) proposed by Pressler.^106,131,132 The LTSSM employs ³He sorption cryostat and x-ray windows that block infrared radiation, which ensures ∼0.3 K. Before and after the LTSSM studies, location-selective x-ray irradiation experiments were performed in Refs. 133–135. However, precise full 2D x-ray imaging analysis was realized only by LTSSM.

The diffusion model was applied to the 146-μm square junction with a τ_eff value of 1.4 μs and a l value of 10 nm in Fig. 10.¹⁰⁶ The fitting parameters are R, D, and γ in Eq. (18). Figure 20 shows the comparison between the model calculation and the LTSSM experimental data of the bottom electrode events at bias points of 0.15 and 0.30 mV, both of which are well below Δ_g/e = 0.841 meV. The 5-keV x-ray microbeam scanned the square junction diagonally. The best fit to the experimental profiles was obtained at D = 1.8 cm²/s, R = 0.999952, γ = 1/0.594 μs⁻¹ for 0.15 mV and D = 4.5 cm²/s, R = 0.999916, γ = 1/0.343 μs⁻¹ for 0.30 mV. The R values are slightly less than 1.0, which implies that the edge loss results in the convex dome shape.

It is reasonable that the γ values change with the bias points, since the net tunneling probability increases as the bias voltage increases in a range below the maximum tunnel current according to the BCS-DOS profile in Fig. 8. A diffusion constant of 1.0 cm²/s for polycrystalline Nb in the literature¹⁰⁹ is not inconsistent with the present value of 1.8 cm²/s at 0.15 mV. However, the D value of 4.5 cm²/s at 0.30 mV is unreasonable. Since the quasiparticle diffusion constant should be essentially unaffected by the bias points. Therefore, we should consider factors other than quasiparticle diffusion.

It is surprising that, in a photon energy range below 300 eV, the junction with the 30-nm thick Al layers has almost the same energy resolution as the junction with the 70-nm Al layers, as shown in Fig. 17. This signifies that the abnormally large spatial nonuniformity emerges in a high photon energy range like 5 keV. The LTSSM scan at 300 eV failed because of the optics problem; however, it is apparent that the spatial uniformity rapidly degrades above ∼500 eV for the 30-nm thick Al layers. The concrete explanation of this behavior is difficult at this moment. Here, it is only pointed out that for practical usage of STJDs, we should pay attention to photon energies and Al thicknesses. Figure 21 shows the LTSSM scans at 6 keV for the junction with different Al thicknesses:¹³⁸ the spatial uniformity is improved by increasing Al thickness significantly. One possibility of this behavior is that the thick Al layers lead to a large quasiparticle diffusion due to the long τ_eff, and, thus, the spatial nonuniformity becomes insignificant. It is also probably related to the gap-reduction model in Fig. 15. The thicker the Al layers are, the more the DOS below the intact Δ_g,Nb exists in the Nb layer. This leads to a linear response even to the 6-keV photons. For the 30-nm thick Al layers, the DOS below Δ_g,Nb is less, but still enough for the quasiparticle density at photon energies below 300 eV. The possible mechanism of the spatial uniformity degradation at the high-energy range will be discussed considering the MQT process and bias points separately.⁵⁰ 

One of the important performance figures is the count rate (count per second: cps). For analytical instruments with imaging capability, a high count rate is crucial for a tolerable measurement time to scan an area to be analyzed. Figure 22(a) shows the energy resolution of the 100-μm square symmetric junction with the 70-nm thick Al layers as a function of photon count rates at 400 eV. Because of a large electrical noise of 17 eV in this experimental run at a synchrotron radiation beamline, the ΔE value was 20 eV instead of 12 eV in Fig. 17. No energy resolution degradation was observed up to ∼6 kcps that is the maximum photon flux at the bending magnet beamline. A single junction can be operatable up to ∼50 kcps without significant degradation of the energy resolution.^139,140 However, this count rate is still 10 times lower than a typical count rate of several 100 kcps of SDDs.¹¹² Therefore, we need to array STJ pixels and readout circuits simultaneously to realize a comparable count rate. It is possible to produce a detector chip with 1024 pixels,¹⁴¹ but a typical complete system with readout circuits has 100-to-200 channels.

Figure 22(b) shows the count-rate performance of a 100-pixel array STJD with a ΔE value of 6 eV in Fig. 12. Briefly, the signal readout circuit for the 100 pixels consists of arrays with charge-sensitive preamplifiers, flash ADCs, and digital signal processing (DSP) circuits that have pulse shaping processors and multi-channel analyzer memories for the 100 pixels. The details of the readout circuits are described in Sec. IV B. For operating the 100 pixels simultaneously and stably, the bias point was just after the rapid subgap-current increase near Δ_g/e as usual. The 100 pixels had very similar I–U curves, but each pixel had own gain with a typical relative standard deviation of 4% (standard deviation divided by average peak position on pulse height spectra). Therefore, before the count-rate measurement, energy calibration of each pixel was performed by using known characteristic x-ray peaks. A control software automatically processes this gain calibration for the 100 pixels. When precise gain calibration is necessary for measuring ∼1-eV peak shift in chemical analysis, as shown in Fig. 47 later, non-linear off-line calibration should be performed by using a few characteristic peaks. For this count-rate measurement, the linear calibration using one peak was enough, because the natural K-line width of the carbon atoms in the carbon tape was 11.0 eV after subtracting the STJD energy resolution from the measured 12.5 eV.

For the 100-pixel STJ array detector in Fig. 22(b), we employed an x-ray source using a carbon nanotube electron gun¹⁴² that irradiated a conductive carbon tape on an Al plate, producing the characteristic x rays of the C K, O K, Al L-lines, and Bremsstrahlung. This count-rate test reproduces an actual material analysis condition like SEM. Real-time acquisition of pulse height spectra was performed without off-line data processing. Figure 22(b) shows the FWHM values for the C K-line as a function of the photon count rates including all characteristic x rays and Bremsstrahlung. The FWHM values including an electrical noise of 2 eV begin to degrade rapidly above 200 kcps, which is the upper limit for x-ray material analysis. Insufficient bias current supply to each junction at a high count rate limits the count rate. The count rate of 200 kcps is the same as that of the 25-mm² SDD in Fig. 2: this value is the highest record among superconductor detector systems. STJ arrays with 200 or more pixels will achieve a higher count rate and a larger sensitive area in future.⁹³ 

Considering superconductors in an RF range, kinetic inductance due to Cooper pairs have a non-negligible contribution to the total inductance in most cases. The complex surface impedance is expressed by Z_s= R_s+ iωL_s, where R_s is the surface resistance and ω is the angular frequency. At temperatures lower than ∼T_c/10, a condition of R_s ≪ L_s is fulfilled since almost no quasiparticles are thermally excited. The L_s has two components of energy storage: magnetic field within a penetration depth and Cooper pair inertia. The sum of the magnetic inductance and the kinetic inductance is roughly proportional to μ₀λ, where μ₀ is the vacuum permeability and λ is the magnetic penetration depth. Particle incidence results in Cooper pair breaking and produces quasiparticles. The decrease of the Cooper pair density leads to the increase in λ and thus in L_s since the London penetration depth is inversely proportional to the square root of the density of Cooper pairs. In addition to the change in L_s, R_s increases because of quasiparticles dissipating an energy. The increase of L_s and R_s induces the resonance curve change shown in Fig. 23(c). The absorbed energy is determined by measuring the resonance amplitude change or the degree of phase change.

An advantage of the MKI type is that a single microwave cable is enough to read out thousands of pixels. The covering energy range is between infrared light and soft x rays for the single-particle detection mode without additional absorber structure. Particle flux measurement is also possible. The MKID type is suitable in a range of near-infrared to ultraviolet for astrophysics,^143,144 and frequently used for cosmic microwave background (CMB) radiation.^145,146 The performance, physics, and applications appear in the latest LTD workshop proceedings.¹⁴⁷ 

For optical/IR astronomy, the figures of merit are compared for MKID, TESD, and STJD in Ref. 148. They concluded that the MKI type is the most promising because of a tremendous number of pixels and a large sensitive area. This is a reasonable conclusion in the field of optical/IR astronomy. One of the latest MKID applications is the 20 000-pixel camera for the 8-m Subaru telescope at Mauna Kea.¹⁴⁹ The MKID camera has a photon detection capability of arrival time, location, and photon energy, which cannot be realized by charge coupled device (CCD) cameras in a near-infrared range of 800–1400 nm.

The MKI type can play a crucial role in investigating the relation between ɛ and Δ_g in superconductors, since output signals reflect the number of broken Cooper pairs in a simple superconductor strip form unlike the complex signal generation processes in STJDs. In addition, the gap-reduction model for superconducting gap engineering in Fig. 15(b) can probably be verified by using proximity bilayer films.

Superconductor thermal detectors, such as TESD and SSPD, are also installed into analytical instruments for synchrotron radiation x-ray analysis, mass spectrometry, infrared spectroscopy, and visible-infrared spectroscopy.^150–154 In this article, SSPD is categorized as thermal detection, since it utilizes S-N transition accompanied with heating and cooling processes due to ETF.^155–157 The history of thermal detector development is to realize a smaller C value and a higher temperature-sensitivity dR/dT value for TESDs or to induce S-N transition with ETF in SSPDs.

For calorimeters, single-particle events must produce a reasonably large temperature rise to be measured with a thermometer.¹⁵⁸ Since the heat capacity of solids decreases as temperature decreases, it is effective to cool down an absorber at a cryogenic temperature. In a cryogenic temperature range, however, there were no proper thermometers in the 1930s. After the discovery of superconductivity in mercury in 1911, the possible use of sharp S-N transition as a thermometer was proposed in 1939–1941.^159,160 The first successful superconducting phase transition (SPT) bolometer was reported by Andrews in 1942.¹⁶¹ This was also the first practical application of superconductivity in all fields, which was 13 years earlier than a superconductor electromagnet in 1955.¹⁶² An SPT thermometer using a Ta wire coil was attached to an Al sheet absorber with a thickness of 80 μm and an area of 8 × 10 mm². The Al absorber sheet was supported by silk threads for thermal isolation. He observed a detector response that was consistent with the Stefan–Boltzmann law for blackbody radiation between 24 and 55 K. Figure 24 depicts the superconducting transition of the Ta wire coil in (a), the equivalent circuit of the SPT bolometer in (b), and the temperature balance diagram in (c). The thermal circuit was designed such that the heat escape to a heat bath balanced with the Joule self-heating of the Ta coil thermometer that is in the middle of S-N transition.

When the switch in Fig. 24(b) is turned off, it operates at the balance point between the blue line and the black line: G • ΔT = I₁²R(T), where G is the heat conductance between the Al absorber and the heat bath. However, this balance point is unstable because I₁²R is steeper than G • ΔT. When the switch is turned on, the balance stability is improved: G • ΔT = I₂²R(T) provides the stable balance. By choosing a suitable R_s value, the temperature of the Ta coils is automatically regulated within the middle of the S-N transition. For radiation input, I₂ decreases because of the increase of R, which provides a weak negative ETF. The details of the ETF operation of the first SPT bolometer will be described elsewhere.⁵⁰ 

It is surprising that the first superconductor bolometer had the self-regulation circuit to automatically hold the temperature within the S-N transition width. However, the potentiometer–galvanometer combination in 1942 required a sufficient voltage change across the Ta coil, which precludes the strong negative ETF operation with constant voltage biasing. The modern ETF operation was realized by a SQUID amplifier based on a superconductor galvanometer (SLUG-SQUID) conceived in 1966.¹⁶⁴ The modern TES type proposed by Irwin in 1995 utilizes a condition of R_s ≪ R and R_w= 0 with SQUID readout circuitry;^165,166 therefore, constant U_T can be applied to an SPT or TES thermometer and current change can be measured. The self-heating power of the TES thermometer is expressed by U_T²/R, which leads to a negative slope. This bias scheme clockwise rotates the green I₂²R line to the dotted orange U_T²/R line in (c). As temperature increases or R increases because of the heat input J, the self-heating power is reduced according to U_T²/R, which is called strong negative ETF.¹⁶⁵ 

A Mo/Au-TESD marked an energy resolution of 1.58 eV for 5.9-keV x-ray photons with a fall time of probably ∼1 ms in 2012.^167,168 Conventional calorimeters with a semiconductor thermistor thermometer had not reached to this impressive ΔE value. Besides an x-ray range, TESDs perform well for photons in an infrared-to-optical range. In addition to STJDs, a W-TESD achieved an energy resolution of 0.15 eV and a fall time of 60 μs for photons above 0.3 eV by Cabrera et al. in 1998.⁸¹ The fall time was improved to 200 ns in 2009 by Fukuda et al.¹⁶⁹ Furthermore, an Au/Ti TESD by Hattori et al. marked a ΔE value of 0.067 eV and a fall time of 4.6 μs in 2022.¹⁷⁰ 

Magnetic calorimetry using Er-doped garnets was proposed and demonstrated for 5.5 MeV α-particles by Bühler and Umlauf in 1988.^171,172 The metallic magnetic calorimeter using a metallic host, which has a stronger spin-phonon interaction than dielectric garnet hosts, was proposed by Bandler et al. in 1993.¹⁷³ Enss et al. demonstrated spectroscopy for x rays and γ rays in 2000.¹⁷⁴ The MMCD marked an energy resolution of 1.58 eV and a fall time of ∼1 ms for 5.9-keV x-ray photons.¹⁷⁵ 

An unconventional thermal detector type utilized a simple superconductor strip (SS type or SSPD), which appeared 38 years later after the discovery of superconductivity. A superconductor NbN strip of 6-μm thick and 0.4 × 3.5-mm square detected α-particles in 1949, which was reported by Andrews,¹⁷⁶ who was also the author of the SPT bolometer.¹⁶¹ This was the first SSPD for single-particle counting, although the term of bolometer was used. The NbN strip was firmly mounted on a copper base for a good thermal contact unlike conventional bolometers or calorimeters. Voltage output pulses were produced by a resistive change due to α-particles under constant current bias. When it was held in the middle of S-N transition just like TES, the best counting performance was obtained.

A different operation mode for superconductor strips maintained at a temperature well below T_c was proposed many years later in 1962.¹⁷⁷ The strip was supposed to be biased near a half of critical current (I_c). The presumed detection mechanism for 5-MeV α-particles or 60-MeV fission fragments was an S-N transition of a Sn strip of 100-nm thick and 10-μm wide. No growth of the normal region with Joule heating was expected in this simulation, which was different from modern SSPDs with ETF heating. An experimental paper on this detection scheme was reported in 1965.¹⁷⁸ The SS type was also capable of detecting electrons from a beta source with an NbN strip of 0.36-μm thick, 4-μm wide, and 2-mm long in 1990.¹⁷⁹ It was also possible to detect 6-keV x rays with a W strip of 0.24-μm thick, 1.8-μm wide, and 0.55-mm long in 1992.¹⁸⁰ The W strip by Gabutti et al. operated in either a high bias current region for hot spot formation and subsequent Joule heating or a low bias region for normal region formation bridging the strip width without Joule heating. The recovery times were in a range of 10–50 ns, which were ∼10 times longer than modern SSPDs; nonetheless, the former detection mode is the same as the one proposed for the detection of infrared-to-optical photons.

The infrared photon detection succeeded with a nanomized NbN nanostrip of 5-nm thick, 200-nm wide, and 1-μm long by Gol'tsman et al. in 2001.¹⁸¹ The proposed operation mode was the same as the W strip, that is, Joule-heating-assisted signal creation after initial hot-spot formation or recently proposed vortex motion. The photon detection in a telecommunication wavelength region with SSPDs covers applications such as quantum cryptography, quantum computing, and time-of-flight (TOF) ranging using single photon detection capability.¹⁸² Furthermore, possible application of SSPD to ion detection in TOF MS was suggested by Frank et al. in a review paper on superconductor detectors for MS in 1999.¹⁸³ A possibility of kinetic inductance operation of SSPDs for MS was reported in 2005.¹⁸⁴ Experimentally, the first successful acquisition of MS spectra for a biomolecule of bovine serum albumin (BSA) was reported by Suzuki et al. in 2008.¹⁸⁵ Neutrons were also detected with a MgB₂ strip with ¹⁰B by Ishida et al. in 2008.¹⁸⁶ 

It should be noted that the pioneers conceived the ideas using superconductivity for thermal detection in the 1940s. In particular, we have reverence for Andrews, who is the pioneer of both TES type in 1942 and SS type in 1949.^161,176 The modern TESDs and SSPDs have marked far better performance and far wider particle-energy coverage compared with the early years.

The TES type usually consists of an absorber, a superconductor film thermometer, and a SQUID current–voltage conversion amplifier. Figure 25 depicts the equivalent circuit, detector structure, and I–U characteristic. The remarkable change from the first SPT bolometer¹⁶¹ in Fig. 24 is that the potentiometer–galvanometer combination located at room temperature is replaced by the SQUID amplifier at a cryogenic temperature. It is also noticeable that R_s is repositioned from room temperature to the vicinity of TES to hold a constant bias voltage, providing R_s ≪ R_T(T).

In Fig. 25(a), the DC SQUID ring is magnetically coupled with the coil L connected in series with the TES–R_s circuit. The TES current is converted to magnetic flux and then voltage U_S. The circle with two Josephson junction symbols represents a DC SQUID with the small cross attribute for magnetic dependence.¹⁶³ Although only one DC SQUID is present in (a), a SQUID series array is normally employed to obtain output voltage sufficient for signal processing at room temperature.

In the ETF operation below 1.2 μV, I_T changes such that a relation of U_TI_T= 9.35 pW is satisfied according to Eq. (21). R_T(T) decreases rapidly below the onset temperature of the superconducting transition, following U_T²/9.35 pW. As a result, the temperature of the Ir/Au TESD is automatically fixed within the S-N transition width. The thermal escape of 9.35 pW is maintained by the SiN_x membrane. The membrane structure in Fig. 25(b) is fragile because the backside is fully open. We conceived a novel durable structure by using a silicon-on-insulator (SOI) wafer, reinforcing the backside of the membrane with the Si substrate.¹⁸⁸ 

The ETF improves the energy resolution beyond the limit of conventional calorimeters by a factor of 1/α^0.5. A TESD specialized for a soft x-ray range marked an energy resolution of 1.1–1.5 eV for 500 eV x-ray photons.^150,151 which is 2–3 times better than STJDs. For 6-keV photons, the best ΔE value is 1.58 eV of a Mo/Au TES with an Au or Bi/Au absorber,^167,168 which outperforms the energy resolution of 29 eV of the Nb/Al-STJD with a SQUID amplifier.¹⁹¹ The TES type covers infrared to γ rays, designing the necessary thickness and heat capacity for absorbers. However, the best TESD energy resolution is normally obtained at a slow pulse fall time of ∼1 ms. The details of the TESD-wide applications are referred to Refs. 5 and 150.

Spatial nonuniformity is also a problem of the TES type, since it has a dimension larger than λ or λ_L. Figure 26 shows LTSSM images of a 50-nm thick and 500-μm square Ir-TES on a SiN_x membrane.¹³² The pure Ir film has T_c= 134.5 mK, ΔT_c= 1.3 mK, and R_n= 2.15 Ω just above T_c. While scanning the Ir-TESD biased at 3.61 μV with a 3-keV x-ray microbeam having a diameter of 20 μm at the ETL-TERAS synchrotron radiation facility, the current pulse parameters, such as τ_rise, τ_fall, and pulse height, were recorded according to the x-ray microbeam coordinates by Pressler et al.¹³² The Ir film geometry in (a) is superimposed with the white line on the LTSSM signals toned by the rainbow color scale from black to white in (b)–(d). In (e), it is noticeable that the pulse height near the upper edge indicated by the red spot in (b) is three times higher than that near the TES center indicated by the light green spot. The images of τ_rise and τ_fall also show the large nonuniformities. This behavior can be explained by phase separation between a superconducting region and a normal-conducting region below.

Figure 27 shows the LTSSM 1D pulse-height profiles at different bias points.¹³² The black arrow in the 90°-rotated optical image in (a) indicates the x-ray microbeam scan direction. As the bias voltage increases in (b), the maximum pulse heights decrease and simultaneously the maximum positions move from right to left. The inset in (b) shows the measured Ir-TES resistance as a function of the distances between the film right edge and the open arrow positions. These observations indicate that as the bias voltage increases, the S-N boundary moves from right to left, expanding the normal region maintained by self-heating. In the course of this S-N boundary movement, the maximum pulse height values decrease and τ_fall values increase, which represents a reduction of α in Eq. (22). The α value monotonically decreases as U_T approaches to 8.5 μV, at which the whole film becomes normal-conducting. The predicted temperature variation profiles are shown in Fig. 27(c), providing a thermal healing length of ∼700 μm. The estimated peak-to-peak temperature variation of ±8 mK exceeds ΔT_c. This is caused by a low thermal conductivity of the pure Ir film. Intriguingly, the superconducting region and the normal one occasionally swapped each other by a single photon event as exhibited by the dashed line in (b). This indicates that the side of the normal region is independent of current directions. Furthermore, the photon absorption events in the superconducting region produced spike pulses that are the same mechanism as SSPDs.¹⁹² This indicates that the operation of the TES and SS types are close, which will be discussed in detail elsewhere.⁵⁰ 

The spatial nonuniformity should be reduced to achieve a superior energy resolution. Proximity bilayer structure is crucial for increasing thermal conductivity as well as adjusting T_c. We designed the bilayer structure such that the temperature variation would be less than ∼1 mK. Figure 28 shows the LTSSM scan data of the 200-μm square Ir(100)/Au(25) bilayer in nm. The photon event number image for 6-keV x-ray microbeam in (a) visualizes the Ir/Au film geometry and the Nb leads. The LTSSM 1D pulse-height profiles in (b) are flat except the Nb lead regions, which signifies that no S-N phase separation occurs. Interestingly, the τ_rise image in Fig. 28(c) still exhibits a small nonuniformity between 5.0 and 6.4 μs, which conceivably represents a slight nonuniformity of bias current like Fig. 26(a). Nevertheless, the ⁵⁵Fe full-illumination energy resolution was 9.4 eV. This energy resolution was not the best in 2003 but exceeded ∼120 eV of SDDs by a factor of more than 10.

For the simple square geometry, an unexpected excess noise emerges.¹⁹³ The excess noise was explained by the RSJ model for Josephson weak links, for example.^194,195 Additionally, it is also reasonable that the bias-current nonuniformity, which can occur in films with a dimension larger than λ, causes extra peak broadening due to pulse-height spatial nonuniformity. Experimentally, the excess noise can be reduced by arranging Cu normal metal bars on bilayer films or applying a perpendicular magnetic field.¹⁶⁷ In fact, a Mo/Au bilayer film with the normal metal feature and the Au absorber marked a superior energy resolution of 1.58 eV at 5.9 keV.¹⁶⁷ 

The MMC type is one of the important thermal detector types using a SQUID readout circuit. The details on the MMC operation appear in Ref. 175, for example. The MMC type is based on thermal detection with a paramagnetic temperature sensor. Figure 29 shows a schematic illustration of MMCD, which consists of the paramagnetic material M(T) and the SQUID magnetometer. The paramagnetic temperature sensor is attached to an absorber, which is often Au. A high M(T) coefficient is realized by a diluted alloy, of which temperature dependence is depicted in (b). The block of the absorber and the paramagnetic sensor is thermally connected to a low temperature heat bath via a weak link having a small heat conductance G. An example of the diluted alloys is Au:Er with an Er concentration of ∼300 ppm.¹⁷⁴ When a particle impinges the absorber, M(T) decreases as indicated by the arrow in (b). The magnetic flux in the paramagnetic temperature sensor changes under a bias magnetic field H. The flux change is converted to a voltage pulse by the SQUID amplifier.

The fundamental energy resolution of the MMC type is expressed by Eq. (6) for thermal detection, limited by the energy fluctuation of a canonical ensemble. The time response or τ_fall is simply given by C/G. More detailed expressions appear in Ref. 175. Experimentally, an impressive energy resolution of 1.58 eV at 6 keV, which is exactly the same as the Mo/Cu TESD,¹⁶⁸ was achieved at a base temperature of 30 mK in 2018.¹⁷⁵ The best resolving powers are ∼4000 for x rays and ∼6000 for γ rays, which correspond to ∼10 eV in a wide energy range up to 60 keV.¹⁹⁶ In addition to the TES type, the MMC type is utilized for XRS, γ-ray spectroscopy, neutrinoless double beta decay for Majorana nature of neutrinos,¹⁹⁷ neutrino mass measurement with electron capture in Ho,¹⁹⁸ MS for neutral fragments in ion storage rings,^199,200 and dark matter search.⁷ The MMC type has no function for shortening τ_fall of output pulses in a range of ∼ms. Nevertheless, the MMC type with the strong spin-phonon interaction has the fastest τ_rise value of 70 ns among calorimeters,²⁰¹ which is advantageous to the timing measurement in imaging MS.^199,200

The SS type usually has a meander geometry of a superconductor nanostrip, which has typically a few nm thickness, a few 100 nm width, and ∼mm length. This nanoscale dimensions enable to count infrared-optical single photons near ∼1 eV that is 10⁶ times smaller than MeV α-particles in the early years of microscale SSPDs. In addition to photons near 1 eV, one can detect electrons,¹⁷⁹ x-ray photons,^180,202,203 and macromolecules that generate phonons upon impact.^185,204,205 Recently, ultrawide strips with $w≫λ$ also enable photon detection near 1 eV,^206,207 which will be separately described in Sec. III C 4 f. SSPDs normally produce pulses with a constant height independent of particle types and energies. Although energy discrimination was possible by adjusting bias current¹⁵² or single-photon energy measurement was reported in a range between 0.4 and 2.0 eV,²⁰⁸ SSPDs cannot be generally employed for high resolution spectroscopy or EDS of single particles.

For detector physics, 2D approximation is appropriate in most cases²⁰⁹ because of the nanostrip dimensions: $t f≪ξ≪w$ and $w ≲λ$⁠. Because of the 2D nature of SSPD operation, this article follows the term used by the pioneers in the early years and the IEC terminology:^6,210 “strip” or “nanostrip” by contrast to “nanowire” or “microwire.” The term of “wire” represents 1D physics, disregarding lateral extent.

Initial perturbation models for a photon energy range between ∼0.5 and 3 eV, including a telecommunication wavelength (e.g., 1.55 μm or 0.8 eV) remains controversial: bias-current assisted normal-region bridging (hot spot) proposed initially in 2001,^181,211 single vortex crossing (SVC),^212,213 vortex-antivortex departing (VAP),^214,215 and phase slip center (PSC).^209,216 The hot spot model is based on normal core formation with a temperature of higher than T_c or hot-electron core formation. Supercurrent detours around the hot spot, leading to bridging the strip width with a normal domain, inducing a voltage drop. In SVC, a single vortex crosses a strip because of Lorenz force. In VAP, Cooper pair breaking causes vortex-antivortex pair creation, depairing, and subsequent movement to the opposite directions. The vortex motion due to Lorentz force causes a voltage drop. Almost all materials for SSPDs are type-II superconductors, which allows vortex nucleation. The initial response to photons well below 1 eV can be either SVC or VAP. The SVC-VAP mixture means that both can occur at similar probabilities. In addition, hot-spot-induced VAP mechanism was also proposed.²¹⁷ The SVC model is valid only for a case that w is considerably smaller than Pearl length values in Table I: $ξ≪w≪ Λ$⁠.^212,218 The phase slip mechanism is inappropriate for most SSPDs because of the 2D nature except relatively narrow NbN strips with w ≈ 100 nm.²¹⁹ The detection principles can change depending on superconductor materials as well as the strip dimensions.

According to the physical parameters in Table I, we can assign SVC, VAP, or SVC-VAP mixture to the listed superconductor materials. For t_f= ξ and w = 200 nm, the possible response to photons near 1 eV is SVC for NbN and YBCO, VAP for Nb, and SVC-VAP mixture for MgB₂. As of now, none of the proposed detection mechanisms can fully explain all the experimental observations. A possibility is that the different mechanisms are engaged depending on particle energies, nanostrip dimensions, elapsed time after particle incidence events, distance from a particle incident point, and operation conditions. Further theoretical and experimental studies are required for detection mechanisms.^220–222 In this article, we assume hot spot formation for keV ion detection²⁰⁴ and vortex motion due to either SVC or VAP for photon detection below ∼1 eV. The boundary between hot spot formation and vortex motion is at ∼1 eV for a 4-μm square NbN SSPD with t_f= 4.5 nm and w = 100 nm.²⁰⁸ 

Compared with extremely wide inhomogeneous strips for LTSEM experiments,⁸³ SSPDs consist of high-quality uniform narrow strips. Therefore, the bias current distribution along the lateral direction should be uniform. Even in this condition, the 2D nature was observed by analyzing position dependent local detection efficiency and detection threshold current for ∼1-eV photons at a spatial resolution of 10 nm.²²³ An ingenious method of quantum detector tomography achieved a surprisingly high spatial resolution. They observed that the local detection threshold bias-currents for an NbN film with t_f= 5 nm, w = 150 nm, and l = 100 nm were slightly nonuniform: 27.5 μA at the strip center and 25.3 μA near the edge. This behavior was explained by an SVC model.²²³ In addition, it conceived that the threshold bias-currents are 27.5 μA for VAP and 25.3 μA for SVC. Therefore, it is concluded that most SSPDs have a 2D nature even in a strip width close to the ISO nanoscale definition of smaller than ∼100 nm.²²⁴ 

Figure 30(a) shows a laser scanning microscope image of the 200-μm square SSPD with a single-crystalline NbN on a MgO substrate, which was fabricated at NICT by Miki et al.²²⁵ The 200-μm square NbN-SSPD in (a) has a meander nanostrip with t_f= 10 nm, w = 800 nm, and l = 40 mm, and a space of 200 nm.^152, Figure 30(b) shows a typical output pulse from an analogous 50-μm square NbN SSPD with t_f= 7 nm, w = 300 nm, l = 4.15 mm, a space of 300 nm, and T_c= 14 K.²⁰⁵ Angiotensin-I biomolecules accelerated at 17.5 kV impinged on the 50-μm square SSPD at a bias current of 170 μA, that is, 94% of I_c and 4.2 K. Angiotensin-I is a peptide biomolecule with a molecular weight (MW) value of 1296. The output pulse parameters are τ_rise= 360 ps and τ_fall= 9 ns in Fig. 30(b). Output pulse shapes were independent of MW or molecular species: biomolecules such as bovine serum albumin (BSA) of 66.4 kDa and Immunoglobulin G (IgG) of 146 kDa. Furthermore, the 50-μm square NbN-SSPD generated exactly the same output pulse shape even for 0.8-eV photons.²²⁵ The ETF process described in Sec. III B 4 c governs the output pulse shape regardless of photons and molecules at this SSPD dimensions.

The SSPD operation resembles avalanche ionization under a high voltage in Geiger–Müller counters or avalanche photodiodes (APDs) to improve quantum efficiency. The constant pulse height implies that, on the hot spot model, an initial resistive band formed across the strip width provokes ETF due to Joule self-heating, leading to expansion and contraction of the subsequent resistive domain. In SVC and VAP, the initial resistive band formation can be replaced by local voltage drop induced by photon-assisted vortex motion.²¹⁵  Figure 31 illustrates the SSPD-ETF operation assuming the uniform bias-current distribution. A photon for direct Cooper pair breaking can replace the IgG molecule ion for phonon-mediated Cooper pair breaking in (a).

As an example, the observed τ_rise was 640 ps (∼500 ps in Fig. 31) for the 50-μm square NbN SSPD. The corresponding R_k is evaluated at 1.7 kΩ, which is equivalent to a normal-domain length of 1.5 μm.²⁰⁵ Because of I_L diverted to R_L, the current flowing the strip decreases to j_n(x) in (b). The heat sustaining the normal domain rapidly diffuses to the substrate unlike TESDs. When the strip temperature decreases below T_c with a typical time scale of ∼500 ps, the normal current j_n is transformed to the light red supercurrent j_s in Fig. 31(c). The temporal change of resistive domain expansion was measured for parallel NbN nanostrips with t_f= 9 nm and w = 100 nm by Ejrnaes et al.²²⁸ The expansion lengths ranged from 0.59 to 3.5 μm depending on parallel configuration. These lengths are consistent with 1.5 μm, although we cannot directly compare the parallel and meander SSPDs.

When the duration of output I_L pulses is longer than the normal domain cooling time, which is primarily determined by the inelastic electron–phonon scattering time τ_e-p in Table I, I_k flowing the strip immediately after events is efficiently reduced.^155–157 In this negative ETF, the strip returns to a superconducting state without latching in a normal state. The negative ETF strength order is TESD > SSPD > Ta-SPT. No latching in SSPDs occurs when the ratio R_L/R_K is well less than ∼10. On the other hand, when τ_fall is shorter than the cooling time, for example, a large R_L, the Joule self-heating continues on and the resistive domain expands to the whole strip length, which is the positive ETF leading to latching.^155–157 It is necessary for using a small R_L; however, this leads to slowing down both τ_rise and τ_fall. Therefore, the time constant of the SSPD readout circuit should be designed such that continuous particle counting is as fast as possible without latching.

It is worth considering superconductor materials for fast response. The τ_e-p values determine the upper limit for high frequency applications for optical or quantum communication. Aluminum in Table I is an essential material for the STJ type, but the extremely long τ_e-p 395 μs is unsuitable for fast response. Niobium with 330 ps is acceptable, and NbN with 88 ps is favorable. Other LTS materials are also employed. Since the SS type simply utilizes the S-N transition, high temperature superconductors are promising for operation at a higher temperature regardless of multiple superconducting energy gaps or d-wave symmetry. The compounds MgB₂ and YBCO with a few ps response are attractive for ultra-high-speed applications. Single photon detection with MgB₂-SSPDs is reviewed by Shibata in Ref. 232. Although it is difficult to prepare uniform YBCO films, it is close to single photon detection.³⁰ We fabricated a meander YBCO-SSPD with t_f= 200 nm, w = 1.4 μm, T_c= 77 K; however, 17.5-keV molecule detection was unsuccessful because of film inhomogeneity that causes a poor I–U characteristics.²⁶ A BSCCO-SSPD exfoliated from a bulk single crystal has recently succeeded in detecting single photons.³¹ 

We experimentally validated the above-mentioned hot-spot model for keV ion bombardment.^204,233 Threshold bias current I_th values for ion detection were measured as a function of the kinetic energies E₀ of Ar ions that irradiated a 200-μm square single-crystalline NbN SSPD with t_f= 10 nm, w = 800 nm, and l = 40 mm. The Ar ions were charged singly, doubly, or triply by electron impact (EI) and accelerated at different voltages. Unlike biomolecules, the Ar ions exhibit no fission upon impact; therefore, fluctuation of energies imparted to the NbN film can be minimized. The I_th values were determined at the level of 1% of the saturated detection efficiency in Fig. 32(a). Figure 32(b) is the plot of the I_th values normalized with I_c as a function of E₀ values. We assume that the hot spot region has a cylindrical shape, and its diameter is expressed by d = C × E₀^0.5, where C is determined by DOS at the Fermi level, T_c, and the efficiency of quasiparticles creation.²²⁶ The dashed line in (b) depicts a calculation curve with C = 3.5 × 10⁻⁹ eV^−0.5, following Eq. (24) by substituting I_th for I_b. The hot spot model curve well explains the experimental data of the Ar ions in an energy range between 0.6 and 9 keV. The hot spot diameter d is estimated at 110 nm for the 1-keV Ar ions.

To track the output pulse generation process in Fig. 31 more precisely, Zen and Mawatari solved a 2D time-dependent Ginzburg–Landau (TDGL) equation coupled with a thermal diffusion equation for a MgB₂-SSPD with t_f= 10 nm, w = 250 nm, and l = 200 μm,²⁴ following the method in Ref. 215. The calculation results were compared with the experimental result on 20-keV angiotensin-I ions. Figure 33 exhibits the temporal changes of the normal domain in (a), superfluid electron density at the different locations in (b), and output voltage across R_L in (c). The molecule ion impact was simulated by the power input Q_ion depicted by the red dashed-dotted line in (b). Figure 33(b) reveals that the superfluid density becomes zero after a few ps at the incidence location. The normal domain appears and expands to a length of ∼1 μm after 16 ps because of self-heating due to the bias current, and then the MgB₂ strip returns to a superconducting state after 400 ps. This time scale is in the same order as the 50-μm square NbN SSPD in Fig. 31. The highest temperature of the resistive domain was 48 K. The superfluid density curves at different positions marked by A, B, C, and D in (a) show an intriguing behavior. Curves A and B near the molecule impact location follow the smooth temporal changes without oscillation, which means that the detection mechanism is the normal-core hot-spot formation. On the other hand, at the place apart from the hot spot region, curve C exhibits a clear oscillation after 10 ps, which signifies the vortex creation and motion. Although the primary behavior is the hot spot formation, VAP depairing can occur depending on the elapsed time and locations.

Similar simulations were performed for an Nb SSPD. Although the material parameters of Nb are significantly different from MgB₂ in Table I, the calculation result was similar to Fig. 33. This suggests that the keV particle detection is rather insensitive to material parameters. The time evolution for keV ion detection is predominantly governed by normal domain growth and ETF. Contrarily, it is anticipated that the ∼1-eV photon detection is influenced by the material parameters, strip dimensions, bias currents, temperatures, and substrate types.

The co-existence of the different excitation phenomena explains experimental results on particle energy dependent pulse heights. Semenov et al. reported photon energy measurement in a range between 0.4 and ∼2 eV with a 4-μm square NbN SSPD with t_f= 4.5 nm and w = 100 nm.²⁰⁸ The best energy resolution was 0.55 eV for 0.8–1.2 eV. Below ∼0.3 eV and above ∼2 eV, no pulse height variation was observed, which corresponds to either model of dark counts due to vortex motion or the hot spot formation with ETF. In the intermediate energy range, the pulse amplitude almost linearly decreased with increasing the photon energy. Although the photon-energy dependence of pulse height is opposite from the common belief, it implies a transition between the vortex motion and the hot spot formation.

Detection of photons in a telecommunication wavelength normally requires superconductor strips with a thickness of several nm and a width of less than a few 100 nm. Even for keV ions, the empirical upper limit of the strip width is ∼1 μm. However, it has recently been proposed that several-μm wide strips still enable to detect photons near 1 eV. The single photon detection was confirmed by using an NbN SSPD with a thickness of 5.8 nm and widths up to 5.15 μm.^206,207 The key factors for the photon detection with the ultrawide strips are a ratio of electron-specific heat capacity to phonon one, which should be larger than 1, and a critical current, which should be larger than 70% of depairing current (critical pair-breaking current).²⁰⁶ Recently, new amorphous superconductor Nb_0.15Re_0.85 with a thickness of 4 nm and a width of up to 2.5 μm enabled 0.8-eV single photon detection at a low bias current of 21% of departing current, which is common to amorphous superconductors.²³⁴ The ultrawide microstrips may overcome the inherent disadvantage, that is, a small sensitive area. Nanostrips are difficult to cover a meaningfully large sensitive area. On the other hand, ultrawide microstrips enable a practically large sensitive area more easily; for example, WSi- and MoSi-SSPDs covering a 1-mm² area with 8 pixels.²³⁵ The ultrawide SSPD development is new trend for practical use.

Table III lists the important performance figures to clarify the strengths and weaknesses of the different detector types. Energy resolutions are paramount in the LTD community, employing a ⁵⁵Fe x-ray source at 5.9 keV. However, direct energy resolution comparison among different detector types is often purposeless, since suitable energy coverage ranges are different from detector to detector. Table III lists the ΔE values followed by photon energies after “@.” Analytical requirements add the second paramount figure that is the pulse fall time τ_fall, which is related to the photon count rate. When a particle arrives in the period of τ_fall for a preceding particle, it is called tail pileup that results in energy resolution degradation on pulse height spectra. The second type of pileup is called peak pileup that produces sum peaks. The probability of the second type also depends on τ_fall. Considering the spatial resolutions and the operating temperatures as well as ΔE and τ_fall, one can conceive suitable application fields for each detector type.