Highly multiplexible thermal kinetic inductance detectors for x-ray imaging spectroscopy

Thermal Kinetic Inductance Detectors (TKIDs) are recently developed superconducting X-ray detectors.^1,2 They have the potential to achieve excellent time and energy resolution and, most importantly, can be scaled up to kilo- or even mega-pixel arrays.³ TKIDs promise enormous advancement in X-ray imaging spectroscopy, a powerful and versatile tool to study scientific questions as diverse as the astrophysical processes around black holes,⁴ understanding of semiconductor fabrication irregularities,⁵ and elemental identification in biological samples.⁶ Many modern applications of X-ray imaging spectroscopy require both high energy and spatial resolution over a large field of view. Excellent energy resolution (down to 0.5 eV at 5.9 keV⁷) can be achieved with diffractometers, but they have to be scanned mechanically over their energy range creating an impractical time penalty, and they are not applicable to spatially extended X-ray sources such as supernova remnants. For these reasons, Transition Edge Sensor (TES) arrays are currently the dominant spatially resolving X-ray detectors. They still resolve an impressive 2.4 eV at 5.9 keV,⁸ and arrays with more than 1000 pixels are under development. However, scaling TESs up to larger pixel counts is very challenging. As they operate below 100 mK, it is impossible to connect several thousand TESs with three wires each to room temperature, forcing larger TES arrays to use complex multiplexing schemes. Impressively, elaborate approaches^9,10 have been demonstrated for TES multiplexing, but the sheer complexity necessary makes more than several kilopixels impracticable. TKIDs offer a straightforward and feasible route to kilo- or even mega-pixel arrays. Further development could improve TKID energy resolution to rival TESs, while future TES multiplexing is unlikely to scale to the pixel counts TKIDs can reach.

TKIDs are a variation of Microwave Kinetic Inductance Detectors (MKIDs^11–14). Every TKID pixel is formed by combining a superconducting capacitor and an inductor into an LC resonator circuit with a unique, lithographically defined resonance frequency, coupled to a microwave feed line. More than 2000 resonators can be coupled to a single feed line and amplifier, allowing them to be read out simultaneously with just two microwave lines (signal in/signal out), connecting the sensors from around 100 mK to room temperature. Working MKID arrays with 2024 pixels have already been demonstrated,¹⁵ and significant further array growth is restrained more by the cost of room temperature readout electronics (about $10 per pixel¹⁶ at the moment) than by fundamental scientific challenges.

Our TKIDs are fabricated from a 200 nm thick sub-stoichiometric TiN_x film,¹⁷ a Nb ground plane and feed line, and a 500 nm thick superconducting Ta absorber that will absorb about 26% of all passing Mn K-α X-ray photons,¹⁸ see Fig. 1. A free-standing Si₃N₄ membrane is used to thermally isolate the absorber and a part of the TKID inductor (for further details please see the supplementary material¹⁹). We use sub-stoichiometric TiN_x (x < 1)²⁰ in order to obtain a 1.6 K critical temperature (T_c). We operate our TKIDs between 75 mK and 200 mK, well below T_c, and expect the membrane temperature to rise by about 200–300 mK when a 6 keV photon is absorbed. In future development, we intend to use mushroom shaped absorbers^21,22 to cover the entire resonator pixel, eliminating X-ray photon hits in undesired locations and increasing the pixel fill factor. However, to demonstrate the device, we have started with a simpler design that still allows many photons to hit the Si substrate. These produce pulse shapes clearly different from hits on the Ta absorber, allowing us to filter out these substrate hits during data analysis.

TKID resonators are read out by driving them at their respective resonance frequencies and monitoring the phase between excitation and oscillation.^14,23 They generate a signal pulse for each individual photon absorbed (Fig. 2 (Ref. 19)) and operate as microcalorimeters:²⁴ when a photon hits the absorber, the temperature of the free-standing island increases. This temperature rise breaks Cooper pairs and creates unpaired single electron excitations (so called quasiparticles) in the superconducting TiN_x inductor on the island. As Cooper pair density and kinetic inductance are coupled, the kinetic inductance and along with it the resonator's total inductance increases. This causes its resonance frequency to decrease (Fig. 3(c)) and, as the drive frequency stays constant, the phase between oscillation and excitation to shift (Fig. 3(a)). We plot this phase shift over time and get a unique pulse for every absorbed photon (Fig. 2(b)), allowing us not only to determine photon number and arrival times but also to determine the energy of every single photon from the maximum island temperature it provoked (see below). As is conventional with MKIDs,²³ we measure phase not with respect to the origin of the IQ plane but to the center of the resonance loop (Fig. 2(a)). Thus, the resonator phase shifts from π to −π during a frequency sweep.

A photon's energy is coded into the maximum of its phase pulse as that corresponds to the maximum island temperature reached. However, since every pulse has a slightly different rise time (likely caused by varying absorption locations) and thermal energy is lost to the bath even during thermalization, analyzing just the maximum pulse height is insufficient (Fig. 2(c)). Instead, we fit every pulse in order to reconstruct the maximum signal at the moment of photon absorption. To better illustrate, Fig. 3(a) shows four frequency sweeps of the same resonator at different temperatures. Photon absorption increases the island temperature and thus causes this frequency response curve to shift to one of the higher temperature values and then relax back. A phase pulse, as shown in Fig. 2(b), is generated by monitoring the shifting frequency response at a constant measurement frequency. A steeper frequency response curve thus results in higher resonator sensitivities (but also reduces the dynamic range). The slope of the response curve is dominated by the resonator's measured quality factor, Q_m, which consists of the internal quality factor, Q_i, and the coupling quality factor, Q_c (⁠$1/Qm=1/Qi+1/Qc$⁠). Therefore, we require a superconductor with high Q_i to achieve high sensitivity.¹⁹ 

Fig. 3(c) shows the dependence of our TKIDs' resonance frequency, f_r, on temperature as well as a fit^23,24 (based on Mattis-Bardeen theory) for a T_c of 1.6 K. In the temperature range our TKIDs operate in, this f_r(T) behavior can be approximated as two linear regions with different slopes. Since after absorbing an X-ray photon the island cools exponentially, these two slopes cause the phase pulse to exhibit two different exponential decay times. Therefore, we use a double exponential decay to fit TKID phase pulses. We found that if the two decay times and the ratio between the amplitudes are fixed at the same values for all photons, this simplified double-exponential model still replicates the complex pulse shapes sufficiently. We use it to fit every phase pulse to obtain the phase shift at the moment of absorption, which is proportional to the photon energy. We expect to be able to further improve performance by developing a fit that exploits the unique shapes of both the resonance frequency versus temperature dependence as well as the frequency response. We design our TKIDs with long signal decay times even though those reduce the maximum possible count rate—if a second photon hits on the decaying tail of a previous one, reconstructing its energy becomes difficult. However, long decay times also reduce phonon noise,²⁵ help to average down HEMT and TLS noise,²⁶ and allow more accurate fits to the pulses, increasing the accuracy of photon energy reconstruction. It is straightforward to engineer decay times to a desired maximum count rate by varying the thermal link between island and temperature bath.

The final energy resolution of one of our prototype TKIDs is demonstrated in Fig. 4, showing 4970 photons collected at a base temperature of 170 mK. We use a Fe⁵⁵ X-ray source and, fitting the spectrum with three normal distributions centered at the Mn K-α and K-β lines, we calculate an energy resolution of 75 eV at 5.9 keV.

The operating temperature T₀ can be used to customize a TKID for desired sensitivity and photon energy. As our TKIDs use superconducting Ta absorbers, their heat capacities are small but rise quickly with membrane temperature (C ∼ T³ as we operate well below the Ta T_c of about 4.5 K). Furthermore, the membrane temperature rise we expect is large compared to our operating temperature, thus, modest changes in T₀ have little effect on the maximum island temperature after photon absorption. Hence, lowering T₀ results in bigger pulses (see Fig. 3(c)), and increases sensitivity and signal to noise ratio. Higher T₀ decreases pulse heights but allows for measurement of higher photon energies. However, if T₀ is not sufficiently below the T_c of TiN_x, Q_i decreases and noise increases.

A 6 keV photon increases our TKID island temperature by about 200–300 mK, resulting in a pulse height of more than 150°. As the slope of the frequency response curve gets smaller above phase shifts of about 100° (see Fig. 3(a)), this reduces our energy resolution. We reduce this saturation effect by working with a TiN_x T_c of 1.6 K and operating at 170 mK, but we cannot eliminate it completely as the decline of Q_i starts to dominate for higher T₀ and a higher TiN_x T_c reduces the island's heat capacity too much. This means our prototype TKIDs are still too sensitive, and we should be able to increase energy resolution by optimizing their thermal design. The straightforward way to reduce the sensitivity of a TKID is to increase its absorber's heat capacity, but first experiments with Au absorbers failed. As TKIDs are driven at resonance, the current density in the inductor is high and can induce substantial eddy currents in the nearby absorber. We think these eddy currents heat a non-superconducting absorber significantly and are the reason our initial Au absorber devices did not work. Finding the optimal tradeoff between TiN_x T_c, sensitivity, saturation, and noise is work in progress. Preliminary calculations show that a device with an optimized thermal design utilizing the steepest part of the frequency response curves of Fig. 3(a) should be able to resolve less than 10 eV at 5.9 keV.¹⁹ Further optimization of sensor fabrication and thermal design should thus bring TKIDs into the energy resolution range currently achieved by TES microcalorimeters, facilitating significant progress in both astronomical X-ray observations with next generation telescopes, as well as high resolution X-ray spectroscopy at synchrotron light sources.

The authors would like to thank the NASA Space Technology Research Fellowship (NSTRF) program (Grant No. NNX13AL70H) for funding graduate student P.S. The work was funded by the NASA ROSES-APRA detectors program Grant Nos. NNX13AH34G and NNX14AI79G. Devices were made at the UC Santa Barbara Nanofabrication Facility; a part of the NSF funded National Nanotechnology Infrastructure Network.