Effective Thermal Properties of a HTS Transition Edge Bolometer for High‐Flux Neutron Detection

Many applications require monitoring of increasingly high neutron flux levels. Pre‐neutron characterization is performed of a superconducting transition edge bolometric device, sensitive to neutrons by an enriched boron carbide (10B4C) layer. Heat from the absorption of neutrons is simulated using an attenuated laser, while the detector is cooled to the critical temperature, Tc. Frequency dependent (4–25 Hz) and constant input power measurements are performed on a 10B4C‐coated and a non‐coated pixel. Results are used to fit a two lumped element thermal model. The effective specific heat is highly dependent on the input pulse duration due to the interconnected stack structure. For modulated pulses, the active volume in the thermal circuit is the full depth of the substrate in the area under the pixel. For constant power input, a large portion of the substrate is heated and the interface conductivities to the cold finger are found to be important. The detectivity is D* = 3.5 × 108 cmHzW−1${{\rm cm}} \sqrt {{\rm Hz}}\,{{\rm W}^{-1}}$ , and is only slightly lower than similar detectors. Understanding the dynamics of the detector better and using neutron excitation, it is expected that optimizations can lead to an equally good detector for cold to thermal neutrons at flux levels >106 n cm−2s−1.


Introduction
High-temperature Superconducting (HTS) materials are used in several different fields as detectors, especially for infrared and X-ray radiation. [1,2]The simplest type of detector is the Transition Edge Sensor (TES), which utilizes the intrinsic superconducting-to-normal transition to convert minute changes in temperature to a measurable voltage signal using a bias current.As the sensor detects heat, it is often referred to as a Transition Edge Bolometer (TEB), which can be made sensitive to particles when using compatible absorbing layers. [3,4]n recent years, many applications require measurements and monitoring of increasingly high neutron flux levels.These include large-scale neutron science, fission and fusion reactors, and particle physics experiments.Simultaneously, 3 He, which is by far the most widespread material employed for detecting neutrons has an unstable supply chain with high fluctuations in price and availability, especially for large quantities. [5]Therefore alternatives, such as a neutron sensitive TEB based on other absorption materials and a robust technology are increasingly relevant.
Testing of high flux neutron detectors is usually required to take place at large-scale neutron facilities or test reactors with limited availability.In order to better understand the detector, a purely thermal test excitation system has been build, and the effective specific heat and thermal conductance of the prototype detector have been modeled based on the experimental data, in order to understand the thermal behavior of the detector.

Experimental Section
The TEB test detector was created as a stack consisting of a thermally and electrically conducting substrate of 50 μm thickness, a number of electrically insulating buffer layers, a superconducting meander structure of 1 μm thick YBa 2 Cu 3 O 6 −  (YBCO) and a neutron absorbing, amorphous, 10 B 4 C layer of 4 μm thickness.The bolometer was constructed as a solid substrate type with direct contact between superconductor, insulating layers, substrate, and cold finger.Due to the thermally interconnected nature of the stack, understanding the thermal transport dynamics and especially the interface conductances of the stack is important in order to optimize the detector properties toward specific applications.A sketch of the experimental setup can be seen in Figure 1, which also displays the stack structure.The fabrication of the TEB test detector is described in a previous publication by Brock et.al. [6] Two types of pixels were tested -one with the enriched boron carbide layer (P sc-b ) deposited on top of the superconductor pixel and one that was not coated (P sc ).

Cryogenic and Electrical Setup
The detector was cooled to the superconducting-to-normal transition temperature, T c , using an AIM SL400 linear Stirling cryo cooler.The detector was connected to the cold finger through a ceramic Aluminum Nitride (AlN) printed circuit board (PCB) using solid Al and Cu parts, while being kept in a vacuum of P =10 −4 mbar using a turbopump, which allowed a temperature stability within ±10 mK for the system.
Electrical connection to the test detector sample was established through Au wirebonds to the PCB, which was then soldered to the wires installed in the vacuum system.Electrical readout was performed by four-point measurements of every pixel individually using conventional source meters for bias current (Yokogawa 7651 and Keithley 2400) and a nanovoltmeter for voltage measurements (Keysight 34401A).
The noise level of the readout system for these measurements was defined by the integration time of the devices, which was then a compromise between a high signal-to-noise ratio and a high temporal resolution. [7]The fastest measurements were performed at 20 ms readout time, which corresponds to a Nyquist frequency of 25 Hz and constituted the lower resolvable noise level of the system within the level of thermal stability.The slowest measurements (continuous heating and cooling) were performed with an integration time of 2 s.

Laser Setup
The thermal excitation system consisted of a 5 mW, 633 nm laser, an optical chopper and four neutral density (ND) filters (10 −1 − 10 −4 ).The cross-section area of the laser beam was of a similar size as that of the pixel with radius approximately r = 1.2 mm estimated from an optical image of the laser beam on the pixel.The laser entered the detector environment through an optical window made of Kapton in the outer vacuum bell.The laser beam attenuation through the optical window was measured to be 50-59%, which was implemented for all data shown in this article.The entire setup can be seen in Figure 1.Neutral filters were used in order to lower the power to a level corresponding to a typical neutron flux found at a large scale facility.The reduced laser power at the detector position and the corresponding thermal neutron flux can be seen in Table 1.
In this study, the neutron flux equivalent, Φ n , to the laser power is calculated from the average energy deposited by the 10 B(n,) 7 Li absorption reaction, [8] the thermal parameters for the enriched boron carbide absorption layer, and the area of the pixel, P las is the incoming power from the laser, E n = 2.298 MeV n −1 is the average absorption energy per neutron, A las is the crosssection of the laser beam, and  is the absorption ratio of the Table 1.Power is measured after the attenuation of the filter and the optical window.The equivalent neutron flux is found using the average neutron energy of the 10 B(n,) 7 Li absorption reaction in Equation (1).
ND Filter 1) 10 −1 2) 10 −2 3) 10 −3 4) 10 thermal neutrons in the enriched boron carbide layer. [9]The absorption was calculated, using Beer-Lambert's law of attenuation and material parameters for the enriched boron carbide, and was found to be 20% [10] Neutrons were absorbed as single events, and as such the energy will be absorbed in small heat zones equal to the mean free path of the absorption products in the solid materials, which was a total of 10-12 μm.For large neutron fluxes of >10 6 n cm −2 s −1 , distributed homogeneously over the mm 2 -sized area of the pixel, the estimate of uniform heating by a laser was a good approximation.Each absorption interaction and the absorption of the resulting products lasts on the order of ps.The timescale at which this heat was transported through the absorbing layer and into the superconductor was at a much larger timescale of 10 −7 s, which smears that point-wise heat distribution given by the neutron absorption, thus imitating the constant heat of the laser excitation.The modeling of neutron absorption in a solid 10 B 4 C neutron conversion layer used as comparison for this work, has been modeled by Lundstedt et.al. for a boron carbide diode. [11]

Measurement Series
All measurements were performed with a bias current of 100 μA for each pixel.Measured voltages, V meas , were converted to a temperature difference, ΔT, using an experimentally determined superconducting-to-normal transition curve, R sc (T).Close to the superconducting transition temperature, T c ≈ 91 K, and well within the operating range of a few Ωs in excitation, the temperature change can be found by the linear slope of the transition, dR/dT, as ΔV meas I bias ⋅ ( dR dT where I bias is the bias current of the detector.Two series of thermal excitations were performed: Series A (frequency dependent) consists of a series of 100 s measurements performed at three different power inputs (see Table 1 ND 1-3) and at twelve frequencies for each power input (steps in the range 4-25 Hz).An example of data measured with the modulated signal and converted to temperature using Equation (2) can be seen in Figure 2. Series B (continuous power) consists of 100 s measurements performed without the chopper at four different power levels (see Table 1 ND 1-4).This yields a continuously increasing heating curve and a corresponding decreasing cooling curve of the pixels as the laser beam was turned off.

Two-Lump Thermal Model
The model is a transient lump-based model with either one or two heat capacities, and one to two thermal bridges.These connect the two lumps (for the two-lump model), and the lump with the thermal bath.The two-lump model can be seen in Figure 3.The lumps are described by their temperatures T i (t), and all temperatures are defined relative to T sub = T 0 .The two lumps are described by the incoming and outgoing heat for each lump: where T i and C i is the temperature and heat capacity of lump i, respectively, and G i is the thermal conductivity of the bridge between lumps as corresponding to Figure 3.The temperature is assumed to be uniform throughout a single lump, at all times, and are therefore not described by the physical dimensions.The thermal capacity, C i , is related to the specific heat, c i , by the amount of material given by the mass, m i , such that, where A i and t i is the area and thickness of the individual layers, and  mat is the density of the layer material.The relation between thermal conductivity G i and the thermal conductance, k i , is described similar to Equation (5).Equation ( 3) describes the dynamics between incoming power and thermal conduction to the lower model layer 2, while Equation (4) describes the transport of heat, from the first lump, transported to the cold finger underneath thus acting as heat sink.
For the single lump model the lump is an average across all layers in the stacked pixel.In the two lump model, the upper layer (2) describes the superconductor and interface layers, while the intermediate layer ( 1) describes the substrate and interface conductance to the cold finger.All interface conductances are included as an averaged parameter in the modelled G i parameters of the lumps.
For Series A, with a modulated input signal, the input power is modeled using a periodic signal based on a harmonic wave, which is numerically easy to solve, and provides equal results to using square input signal, with frequency  as the modulation frequency of the chopper and the amplitude P las .The data measured at each frequency in series A, as seen in Figure 2, show a direct response without saturation, that resembles a harmonic signal.Therefore the average response, ΔT A , is calculated as, where n is the number of peaks in a series, and T A, i, max and T A, i, min , are the maximum and minimum measured temperature of each peak, respectively.These are plotted as the datapoints in Figure 4 for all measured frequencies.The parameters C i and G i are found by fitting these combined plots to the single lump model using a least squares curvefit.The results of this fit is shown in Figure 4. Due to the fast, direct response without  saturation, a single lump thermal model is sufficient to describe the behavior.
For Series B, the model parameters were fitted directly to the measured data, which can be seen in Figure 5.A two-lump model is fitted for these measurements, as two lumps with separate thermal properties are needed in order to satisfy the saturation behavior for longer thermal excitations.For power P ⩾ 26 μW the noise level is comparably low and the data can be fitted well using a two-lump model.For P =2.6 μW a significant uncertainty is observed due to the lower power providing a smaller signal and thus a lower signal-to-noise ratio.Due to the large difference in noise, the weighted averages of the effective specific heat for the two-lump model fitted for all energies are calculated and these are shown in Table 2.

Results and Discussion
Thermal signals plotted in Figures 4 and 5 demonstrate a high noise level at the ND3 power input of P in = 2.6 μW.A good fit was not possible to achieve at ND4 equivalent to an input power of 0.7 μW.Evidently, this may be interpreted as a lower bound for the neutron flux sensitivity equal to 10 6 n cm −2 s −1 for the present design.This neutron flux is comparable to many present imaging and other high-flux neutron beamlines at large-scale facilities. [12]ble 2. Weighted average of the effective specific heat of the test detector system found by fitting of the one-lump model to frequency dependent data (Series A), and the two-lump model to continuous heating data (Series B).

Response Time and Sensitivity
From the results in Table 2 response times can be calculated for each pixel and each model as  = C i /G i . [13]For the single lump model fitted to Series A this gives  A-high =0.2-0.3 s for the highest power at ND1 and  A-low =0.02-0.06s for ND2 and ND3.These values are similar to the duration of the modulated signal pulses ranging from 0.25 to 0.04 s.The pixel with the enriched boron carbide coating showed response times ≈30% higher than those for the non-coated pixel.
For the tests in Series B the time constants are  B, 2 = 0.6 s for the top lump (2) of enriched boron carbide and superconductor, and  B,1 = 5-9 s for the bottom lump (1).A general trend of lower time constant values for heating compared to cooling was observed and lower time constant values for lower input powers were also recorded.Heating rates are expected to be higher than the cooling rates, as cooling is controlled mainly by the diffusion through the insulation layers and substrate to the cold finger. [14]he intrinsic response time for the ideal detector, is given as  int = C sc G sub = 6 ms from the material parameters at T c , where C sc is the thermal capacity of the superconducting layer, and G sub is the thermal conductivity of the substrate. [15]The relation that  B ≫  A ≫  int reveals that the limits in the response time is not given by the intrinsic material properties of the test detector, but rather the limits of the signal provided by the optical chopper.Using a modulated beam with a higher frequency is thus expected to decrease the response time to a value close to the intrinsic response time.Similar response characteristics for a comparable system have been observed by Gerbaldo et.al. [16]

Specific Heat and Thermal Conductance
Modeling of the difference in thermal response between the enriched boron carbide coated pixel P sc-b and the non-coated pixel P sc for Series B measurements showed that the enriched boron carbide coated pixel had higher thermal capacity as well as higher thermal conductance in the bridges.This results in a calculated increased thermal mass (+18%) and time constant (+13%) for the enriched boron carbide coated pixel.This indicates that the enriched boron carbide coated pixel may be characterized as less sensitive to thermal variations and have a higher cooling time constant than the non-coated pixel.A reduction in the superconducting transition temperature, and an increase of the transition width, was previously demonstrated to be an indirect result of the boron carbide layer deposition process. [6]A widening of the tran-sition curve leads to lower dR/dT values and thereby decreased sensitivity.Absorption of the laser power directly in the superconducting layers rather than in an absorption layer, can also cause a difference in response.The superconducting structure is expected to absorb >90% at a wavelength of 635 nm, which is a similar value to that of the boron carbide layer. [17,18]An increased reflectance of the substrate surrounding the superconducting meander structure is expected as the surface appears shiny, which is expected to cause a decrease in the absorption directly in the substrate between meander paths, and thereby a small decrease in the total measured signal.
The values for specific heat found with both models show values much higher than the sum of the specific heat of the enriched boron carbide, superconducting meander and insulation layers for a volume defined by the area of the pixel and thickness of each layer.This indicates that a volume of the substrate, which is larger than the pixel area, is an active part of the thermal circuit and contributes to the response of the detector.This is consistent with previous observations of similar TEB structures, where a large active area compared to the thickness of the substrate has lead to a large thermally active volume in the substrate. [19,20]This indicates that a 1D model of the detector is not sufficient to fully described the thermal response of the pixels.
For the frequency modulated power input in test series A, the effective specific heat is found to be of an equal area of incidence as the pixel above it with a depth equal to the substrate thickness.For the longer power input with constant power (series B), the substrate volume that is part of the bolometer response is on average C 2 /C times larger equal to 40 times larger than that defined by the area of the pixel and the thickness of the substrate.This is visualized in Figure 6, where the dark red area defines the volume needed for the frequency dependent measurements in Series A, while the transparent red and the striped area constitute the active volume for the long constant power measurements in Series B.
A similar relation was found by a 3D model of a solid substrate type YBCO bolometer created by Ansari et.al.For  < 4 Hz, a significant volume of substrate, larger than that solely underneath the superconducting structure, is heated during each pulse. [21]he layout of the bolometer tested in this study is also that of a solid substrate type with direct contact between layers in the stack.For this geometry the interface thermal conductivity between substrate and cold finger is often seen to be the dominating factor in the thermal dynamics.Therefore, the discrepancy between physical volumetric material constants and modeled parameters is also expected to be caused by these not well known interface properties.

Efficiency
The amplitude of the frequency dependent signal and the incoming power from the laser is compared in order to estimate the thermal efficiency of the test detector in the setup.The efficiency in this work is defined as the ratio between the thermal power associated with the temperature increase and the laser power.This can be related to the materials properties of the superconductor and the laser by, where ΔT describes the change of temperature in the superconductor during a pulse, V sc ,  sc , c sc are the volume, density and specific heat of the superconductor at T c , respectively, and P laser is the integrated power of the laser during the rise of one pulse.g is a geometric factor that accounts for >99% of the laser power being absorbed at the top of the enriched boron carbide layer and the spread that is caused by the heat diffusing through the layer to the neighboring superconducting layer underneath. [18]he transport of heat via radiation from the enriched boron carbide surface is two orders of magnitude smaller than the transport governed by the thermal conductivity within the material and therefore this is disregarded in the analysis.The enriched boron carbide is deposited on an area (2.4 × 2.2 mm 2 ) not only covering the superconducting meander (2.0 × 2.0 mm 2 ), but also a small area around it.The meander has a filling factor 50%, which means that heat is also expected to be transported directly from the enriched boron carbide layer to the substrate in the areas of direct contact.Considering the additional enriched boron carbide coverage on the sides of the pixels and in these intermediate areas the geometric factor is set to g = 0.4.Using the above numbers yields an efficiency of 7% at 4 Hz.
In the 7% efficiency, the continuous transport of heat through the superconductor to the substrate is not taken into account.Measuring with a low sample rate of tens of ms the transient energy transport taking place will show as a loss, artificially reducing the efficiency.Furthermore the laser was not focused by any additional optical components and the laser beam crosssection was slightly larger than the pixel, which with a Gaussian distribution of the power in the laser beam accounts for a loss of 7% compared to the input power.

Responsivity and Detectivity
For a model where the area of the superconductor is much larger than the thickness of the substrate, then G and C can be assumed to be independent of frequency for lower frequency regimes. [22]he frequency dependent responsivity can then be found as: where μ = g is the optical absorption of the sample based on the geometric factor previously described, I bias is the bias current of the superconductor during measurements, dR dT is the slope of the superconducting-to-normal transition, f is the modulation fre- quency, and G bo and C bo are the thermal conductivity and the thermal capacity of the bolometer, respectively.For the case of the frequency dependent measurements the modeled material parameters from the single lump Series A model are used.A plot of the absolute values of the responsivity (|S(f)|) can be seen in Figure 7 with a maximum value of 18 VW −1 at 1 Hz.
As described in Equation ( 9) the responsivity can be increased by a higher bias current.The bias current is however limited by thermal runaway, caused by excessive joule heating in the system.The thermal runaway factor is described by Fardmanesh et.al. [19] : Using the parameters from the measurements in this study, performed at I bias = 100 μA and with substrate material parameters at T c , the thermal runaway is calculated to be  = 0.06, which is a stable configuration ( < 1) according to Fardmanesh et.al. [19] The limit of the bias current is found at I bias = 400 μA using the material parameters of the superconductor and Equation ( 10), but using G sub = G 1 as measured in this study for Series B, the result becomes up to an order of magnitude higher.This is expected as the pure substrate thermal conductance is lower than what is measured for the stacked substrate system.The measured values are found to be 500-1000 μA.
The Noise Equivalent Power (NEP) is the detection power limit of the bolometer.For a HTS bolometer, this consists of three main elements: The Johnson noise (NEP john ), caused by the thermal fluctuations of the finite electrical resistance of the bolometer, R, the phonon noise (NEP phon ), which describes the noise from random phonon coupling through the substrate, and the low frequency noise (NEP 1/f ) described by the empirical Hooge relationship, where k B is Boltzmann's constant, T o = T c is the operating temperature, R = R(T c ) is the resistance of the bolometer at the operating temperature, S(f) is the responsivity described in Equation (9), μ = g is the absorption coefficient, n is the charge carrier density of the superconducting structure, and  H is the Hooge noise parameter. [23]The carrier density is assumed to be 2.5 × 10 21 cm −3 based on previously measured values. [24]The noise parameter has not been measured for this sample, but the value  H =0.2, was found for a similar bolometer on a solid SrTiO 3 substrate. [23]The noise parameters in YBCO scales with the temperature such that  H (T)∝R X (T), where X is an experimentally determined parameter.Operating at the transition temperature will increase the noise parameter, and therefore it is assumed slightly higher at  H =10, which is still within the range found for HTS thin films. [25]he total noise of the detector can then be found as the squared sum of all the contributions, Detectivity describes the detectors ability to convert the incoming signal to a measured output.The specific detectivity, D*, describes the frequency-dependent signal-to-noise, per 1 Hz bandwidth and with an irradiation of 1 W on a 1 cm 2 detector.It is found as the inverse of the noise, normalized with the active area of the pixel, A pix , A plot of the frequency dependent detectivity, D*(f), along with the responsivity, S(f), can be seen in Figure 7.
From Equation ( 11)-( 14) it is seen that four main parameters are important for the detectivity of the detector: the active area of the pixel, A pix , the absorption coefficient , the thermal conductivities G, and the thermal capacities, C, of the device.As seen in Figure 7, the noise of HTS-TES is dominated by the 1/f-noise, which is again dominated by the material parameters G and C of the detector.This relation also means that noise measured for the low-frequency regime is observed to be much higher than for the same detector at higher frequency corresponding to the inverse relation to frequency. [26,27]For the system measured here, this also implies that values for G i and C i modeled for low frequency data is not expected to be correct for higher frequency regimes.
The detectivity as function of frequency is similar to that measured for a similarly large-area, but free-standing patterned HTS bolometer in the low frequency regime as well as a semifreestanding YBCO-bolometer on a thinned substrate. [28]A welldefined knee-frequency is not visible in Figure 7, but a change in frequency dependence at 4-8 Hz is noticeable.This interval corresponds to that of f K = 3 Hz for a thick substrate sample without a separate absorber layer. [29]e responsivity and detectivity shown in Figure 7 is similar for the pixel with and without enriched boron carbide, which further supports that the dominating parameters in the detector are connected to the thermal conductivity of the interface between layers as well as the specific heat of the substrate rather than parameters of the absorbing or superconducting layers.
The knee frequency describes at which frequency the penetration depth of the heat is equal to the stack thickness.At midrange frequencies (Series A) the dynamics is described by the knee frequency, whereas for low frequencies (Series B) the thermal contact between substrate and heat sink also contributes significantly.At high frequencies the dynamics is governed by the thermal interface conductivity between superconductor, insulating layers and substrate, since the penetration depth of the heat is similar to the thickness of the superconductor itself.Therefore, to investigate the influence of the thermal interface resistance between substrate and cold finger, the knee frequency is calculated, where D sub is the thermal diffusivity given as the ratio between the thermal conductance k sub , and the specific heat c sub and density  sub of the substrate, and h sub is the thickness of the substrate. [22]All material parameters are used for T = T c .Using the material parameters, the knee frequency f K = 500 Hz, is found, which is far from the range of 4-8 Hz estimated from the local minima of d|S(f)|/df on the responsivity plot in Figure 7.
To obtain an estimated thermal interface conductivity, Equation ( 16) is solved for k sub and f K =6 Hz.The specific heat of the substrate used to calculate the new thermal conductance k meas is similar to that found in the model and shown in Table 2.This yields a thermal conductance of k meas = 80 mW m −1 K −1 , which is two orders of magnitude lower than that found for a similar volume of substrate material using standard material parameters.This suggests that there is an additional thermal interface resistance (R int ) in series with the substrate, such that with R meas being the total thermal resistance as given by the observed knee frequency of 6 Hz, and R sub being the thermal resistance of the substrate.The thermal resistances can be found from the inverse thermal conductances of the material along with the dimensions, R i = t i /(k i • A i ), where t i is the thickness of the layer and A i is the area.As the area is similar in all cases, the thermal interface resistance of the stack can be described as, where t total ≈ t sub are the thicknesses of the entire stack of substrate and interface and the thickness of the substrate, which is by far dominated by the substrate thickness. [30]The interface resistance is found to be R int =0.6 mK m 2 W −1 , which is similar to that of a partly oxidized steel/ceramic interface. [30]The large thermal contact resistance is expected to be caused by the setup, since the sample and the PCB were not pressed together directly underneath the pixels and therefore did not have a good thermal contact.The thermal contact resistance can be decreased for future measurements by introducing a thermally conducting interface layer, such as a thermally conductive grease in between the sample and ceramic PCB leading to the cold finger.

Cross-Talk
spread of the heat to a volume of substrate much larger than the pixel size shown in the model, can result in significant crosstalk between pixels.Measured temperature response for the noncoated pixel, as illustrated in Figure 1, while exciting the noncoated and coated pixel with the laser, separately, can be seen in Figure 8.
For the continuous heating in Series B, shown in Figure 8, a signal 60% the size of the direct signal is measured in the neighboring pixel.Considering the geometric factor g described previously only ≈40% of the signal would be seen as the direct signal.If the heated volume is considered as the entire depth of the substrate and it spreads out isotropically in the plane of the substrate with a Gaussian profile, this would yield an expected signal of approximately 17% of the entire signal in the neighboring pixel.This corresponds well with the measured relation between the direct and indirect signal of ≈60% as shown in Figure 8.
Cross-talk measurements were also performed for the frequency-dependent Series A, which showed a signal approximately 100 times smaller for the measured cross-talk signal compared to the direct measured signal.This corresponds well with the smaller thermal capacity for Series A than Series B visualized in Figure 6, leading to a smaller active volume.As the heated substrate volume is similar in area to the pixel itself and the thickness of the substrate, heat does not diffuse far out in the substrate under neighboring pixels for these fast measurements.
The cross-talk measured does not show any significant delay compared to the direct signal.This is expected to be due to the very low frequency regime, and the close proximity of the pixels on the substrate.For higher frequency signals and longer inter-pixel distances, the delay caused by the thermal capacity of the substrate would be visible.This was shown for HTS bolometer devices with feature sizes of 40-170 μm that showed significant cross-talk up to a cut-off frequency of 1-15 kHz depending on geometry. [31]or future designs it would be recommendable to make geometric adjustments to the design, where the superconducting meander has a higher filling factor to lower the effect of g.With the current fabrication methods a filling factor of at least 70% is possible.Adjusting the interface thermal conductivities between pixel, substrate and cold finger by adjusting the stack geometry and fabrication process would make it possible to optimize the detector for specific applications.Both are expected to significantly increase the efficiency, decrease the cross-talk, and increase the detectivity. [32,33]

Conclusion
Thermal excitations using a laser in the visible spectrum of a magnitude corresponding to that of a high neutron flux 10 6 -10 9 n cm −2 s −1 , has been measured in the High-T c TEB for two pixels with and without a deposited enriched boron carbide neutron absorption later.The tests show that it is possible to optimize a bolometric neutron detector based on non-neutron-based test sources.A lumped transient model is fitted to the measured response of both a chopped, and constant signal and the effective thermal capacity found by the model suggests that a large volume of the substrate extending further than the pixel area is part of the responsive thermal circuit of the bolometer detector.This large active volume is greatly reduced for measurements performed in the low-mid range frequency spectrum, and is expected to decrease further in the high-frequency regime, thereby providing a higher time resolution of the detector.
The comparison between model and signal suggests that interface conductivities have a noticeable effect on the dynamics, and changing the cooling interface might reduce the contact resistance by up to two orders of magnitude.Responsivity and detectivity as a function of frequency, show that the noise at low frequencies is dominated by 1/f-noise, and exhibit D* = 3 × 10 8 cm √ Hz W −1 .Significant thermal cross-talk through the substrate has been measured for low-frequency signals, but these are minimized when moving toward the mid-range frequencies.Detection limits for neutron equivalent fluxes show a lower limit of 10 6 n cm −2 s −1 and no upper limit of saturation, which suggests that the detector is highly applicable for monitoring in high neutron flux environments.

Figure 1 .
Figure 1.Conceptual sketch of the thermal excitation setup based on laser pulses created by a chopper and sent into a cryo cooler holding the TEB detector behind an optical window.The detector consist of two pixels with a size of 2 mm × 2 mm, which are connected in a four probe measurement setup as indicated in the inset.The cross-sectional architecture of a pure superconducting pixel (P sc ) and a pixel also holding a neutron absorbing enriched boron carbide layer (P sc-b ) is shown at the bottom.

Figure 2 .
Figure 2. Example of response data measured at a modulated laser signal at a frequency of 4 Hz for Series A at power input P in =250 μW.Voltage to temperature conversion was determined using Equation (2).

Figure 3 .
Figure 3. Sketch of the two-lump transient model with the top layer representing a neutron absorbing layer and a superconducting layer, the middle layer a metal substrate and the bottom layer is the cold finger functioning as a thermal bath.Each lump is described by its temperature T i (t) and thermal capacity C i , and each bridge by its thermal conductivity, G i .For the single lump model, only C 1 and G 1 are used as fitting parameters and are therefore averages of the thermal parameters of absorbing layer, superconductor, and substrate.

Figure 4 .
Figure 4. Single lump model fitted (solid lines) to the response amplitude of the measured modulated signals, ΔT A , (data points, see Equation (7)) of pixel P sc without absorption layer, and P sc-b with an enriched boron carbide absorption layer at three different powers.

Figure 5 .
Figure 5. Two lump model fitted (lines) to measurements of ΔT (points) for continuous (30 s) heating and cooling at three power levels.Data from test Series B.

Figure 6 .
Figure 6.Visualization of the approximate relation between the magnitudes of effective specific heat for the frequency modulated power input in series A (single lump model) and the constant power input of series B (two-lump model).The two-lump C 1 in the substrate extends beyond the boundaries of the figure.

Figure 7 .
Figure 7. Top) Responsivity (|S(f)|) and specific detectivity (D*) for the bolometric pixel with (P sc-b ) and without (P sc ) boron carbide.Bottom) Noise equivalent power for the pixel without boron carbide divided into main contributions: phonon noise (NEP phon ), Johnson noise (NEP john ), and low frequency noise (NEP 1/f ).

Figure 8 .
Figure 8. Cross-talk curve: measured temperature response of non-coated pixel (P sc ), while the laser is directed toward the pixel itself (P sc ) or its neighboring boron carbide coated pixel (P sc-b ).Data shown for continuous heating, Series B at P in =250 μW.