1. Introduction
The transition-edge sensor (TES) microcalorimeter (μ-calorimeter) is a low-temperature photon detector [1]. Specifically, it is a thermal detector that transforms photon energy into heat through an absorber. The resistance of a superconducting film changes dramatically with the temperature of the film at the superconducting transition edge, and thus by measuring the variation in the current flowing through the film, one can deduce precisely the energy of the incident photons. In the soft X-ray band, the energy resolution of the TES μ-calorimeter is comparable to that of a grating or crystal spectrometer [2]. The TES μ-calorimeter can be placed close to a sample, and a large detection solid angle can be achieved especially if a detector array is used [2]. As an energy-dispersive detector, the TES μ-calorimeter does not require fine tuning of the optical path, and it is ideally suited to spectroscopic measurements of diffusive light glow [3], [4], [5] and liquid jet samples [6]. TES μ-calorimetry is a promising technique for chemical state analysis [7], [8], [9], astronomical X-ray observation [10], [11], [12] and radioactive isotope analysis [13], [14], [15].
The TES μ-calorimeter is usually operated at extremely low temperature to enhance its energy-resolving capability. The proximity effect is most often used to obtain superconducting films with a very low transition temperature. Material combinations such as Mo-Cu [16], [17], Mo-Au [18], [19] and Ti-Au [20], [21] have been successfully used to fabricate TES μ-calorimeters. These material choices lead to superconducting films with a low transition temperature, narrow transition width and low normal-state resistivity. In this paper, we report characterizations of the electric and thermal parameters of a TES μ-calorimeter made from proximately coupled Mo and Au films. The electrothermal parameters at the chosen working point can be considered as the fingerprint of a TES μ-calorimeter and they determine the performance of the μ-calorimeter. We estimate the instrumental energy resolution of the μ-calorimeter from the noise spectrum. The estimation is consistent with the energy resolution measured from the emission spectrum of a radioactive 55Fe X-ray source.
2. Detector fabrication
The μ-calorimeter was made of a Mo/Au/Au tri-layer film. The Mo film and the bottom layer of Au were deposited successively on a low-stress SiNx/SiOx/Si wafer through direct-current magnetron sputtering (CMS 18, Kurt J. Lesker, US). The purpose of the bottom Au layer was to prevent the oxidation of the Mo film. The MoAu film was then patterned through chemical wet etching using gold etchant (GE-8148). The surface of the etched Mo-Au structure was ion cleaned before the deposition of the top Au layer, which was fabricated by adopting a lift-off process. The top Au layer was thermally evaporated to reduce the resistivity of the gold film. The width of the top Au structure was slightly larger than that of the patterned Mo-Au structure to prevent the formation of superconducting shorts along the edges of the bottom Mo-Au structure.
The absorber of the μ-calorimeter had an overhanging design. The absorber was supported by five Ti/Au pillars. One was located at the center of the μ-calorimeter and the other four were located on the four corners of the SiNx membrane. The supporting pillars and absorber were fabricated by gold electroplating and the seed layer for the electroplating process was created through direct-current magnetron sputtering.
After the fabrication of the absorber, the backside of the Si substrate was etched adopting a deep silicon-etching process to reduce the thermal conductance between the μ-calorimeter and heat bath. A thin silicon nitride membrane substrate was left to support the μ-calorimeter.
Fig. 1. (a) Optical image of a 4 × 4 TES μ-calorimeter array (b) Design layout of a single TES pixel (c) The cutaway view of a single TES pixel. |
Table 1. Geometrical parameters of the μ-calorimeter pixel. |
| TES detector | Absorber | SiNx membrane | Supporting pillars | |
|---|---|---|---|---|
| Dimensions | 150 μm × 150 μm | 190 μm × 190 μm | 210 μm × 210 μm | 5 μm × 5 μm (small pillar) 20 μm × 20 μm (center pillar) |
| Thickness | 425 nm (Mo45/Au30/Au350) | 2.5 μm | 500 nm | 1.8 μm |
3. Electrothermal parameters of the μ-calorimeter
The electrothermal parameters are important for understanding the performance of a μ-calorimeter. We conducted a series of measurements to deduce these parameters. Details of the measurements are given below.
3.1. Transition temperature (TC)
A Lakeshore 372 AC resistance bridge was used to measure the resistance of the μ-calorimeter. The excitation current of the resistance bridge was set at 10 μA. By slowly increasing the bath temperature of a cryo-cooler, the curve of the resistance versus temperature was recorded as shown in Fig. 2a. The normal-state resistance of the μ-calorimeter (Rn) was found to be 17.5 mΩ. Additionally, the onset temperature of the superconducting transition was approximately 91.6 mK, which was determined from the crossing point of two linear fits to the data above TC and to the transition part, as shown in the inset of Fig. 2a. The bath temperature was swept in 1-mK step, and the transition width of the device shown in Fig. 2a is approximately 3.2 mK.
Fig. 2. (a) R-T curve of the μ-calorimeter under a small excitation current of 10μA, TC(onset) is approximately 91.6mK. (b) I-V curves of the μ-calorimeter at various bath temperature. |
3.2. I-V characteristics
The I-V curves of the μ-calorimeter were measured using a two-stage superconducting quantum interference device (SQUID) amplifier system provided by Star Cryo-electronics. The total bias current Ibias was ramped from 0 to 3 mA in steps of 1 μA. The current flowing through the μ-calorimeter ITES was calculated using Eq. (3.1), in which Vout is the SQUID output voltage in flux-locked mode, V0 is the offset voltage, and Gsquid is the trans-impedance gain of the SQUID amplifier system. The voltage drop across the μ-calorimeter was calculated using Eq. (3.2), in which Rsh is the shunt resistance. We measured the I-V curves as shown in Fig. 2b by varying the bath temperature. The bath temperature was increased from 55 to 90 mK in step of 5 mK.
$I_{T E S}=\frac{V_{\text {out }}-V_{0}}{G_{\text {SQUTD }}}$
$V_{T E S}=\left(I_{\text {bias }}-I_{T E S}\right) R_{s h}$
3.3. Thermal conductance (G) and heat capacity (C)
We extracted the thermal conductance G from the I-V curves. The thermal conductance is defined by Eq. (3.3), in which Pbath is the power flow to the heat bath, K is a prefactor, and n is the exponent of the power flow. The resistance of a μ-calorimeter depends on both the temperature and current. However, empirical experience suggests that the resistance of a μ-calorimeter is only dependent on temperature when it exceeds 0.8Rn [22]. We denote by T0.8 the temperature at which the resistance of the μ-calorimeter is equal to 0.8Rn. The thermal conductance can be rewritten in the form of Eq. (3.4). By integrating the thermal conductance, the Joule power PTES generated by the current flowing through the μ-calorimeter is expressed as Eq. (3.5), from which the thermal conductance G(T) can be fitted.
We plot PTES versus the bath temperature in Fig. 3a. The PTES values are calculated from the working points marked with red stars in Fig. 2b. At these working points, the resistance of the μ-calorimeter is equal to 0.8 Rn. By fitting the data points shown in Fig. 3a using Eq. (3.5), the three fitting parameters, namely n, T0.8 and the thermal conductance at T0.8, were deduced. These parameters are given in Table 2. The thermal conductance at other temperatures can be calculated using Eq. (3.4).
$\mathrm{G}(\mathrm{T})=\frac{\mathrm{dP}_{\text {bath }}}{\mathrm{dT}}=\mathrm{nKT}^{\mathrm{n}-1}$
Fig. 3. (a) Joule Power of the Mo/Au/Au μ-calorimeter with R0 = 0.8 Rn versus bath temperature. (b) The signal pulse of the μ-calorimeter measured at a bath temperature close to the superconducting transition temperature of the detector. |
Table 2. Summary of the electro-thermal parameters. |
| Tc(onset) = 91.6mK | Rn = 17.5mΩ | C = 0.737pJ/K |
|---|---|---|
| n = 3.05 | T0.8 = 90.3mK | G(T0.8) = 319pW/K |
| Parameters | Mathematical Expression | Value |
| I0 | N/A | 32.9 μA |
| R0 | N/A | 6.26 mΩ |
| $T_0$ | N/A | 86.5 mK |
| $G(T_0)$ | $\begin{array}{l} \mathrm{G}\left(\mathrm{T}_{0}\right)=\frac{\mathrm{dP}_{\text {both }}}{\mathrm{dT}}=\mathrm{nKT} \mathrm{T}^{\mathrm{n}-1}= \\ \mathrm{G}\left(\mathrm{T}_{0.8}\right) *\left(\frac{\mathrm{T}}{\mathrm{T}_{0.8}}\right)^{\mathrm{n}-1} \end{array}$ | 292 pW/K |
| $α_I$ | $\alpha_{I}=\left.\frac{T_{0}}{R_{0}} \frac{\partial R}{\partial T}\right|_{I_{0}}$ | 47.2 |
| $β_I$ | $\beta_{I}=\left.\frac{I_{0}}{R_{0}} \frac{\partial R}{\partial I}\right|_{T_{0}}$ | 1.15 |
| τ | $\tau=\frac{\mathrm{C}}{\mathrm{G}}$ | 2.31 ms |
| $L_I$ | $\mathscr{L}_{I}=\frac{\alpha_{I} P_{0}}{G\left(T_{0}\right) T_{0}}$ | 11.62 |
| $τ_I$ | $\tau_{I}=\frac{\tau}{1-\mathscr{L}_{I}}$ | −217 μs |
| $τ_+$ | $\tau_{+}=\tau_{e l} \approx \frac{L}{R_{0}\left(1+\beta_{I}\right)}$ | 4.59 μs |
| $τ_-$ | $\tau_{-}=\tau_{e f f} \approx \tau \frac{1+\beta_{I}}{1+\beta_{I}+\mathscr{L}_{I}}$ | 361 μs |
$Letting \mathrm{G}\left(\mathrm{T}_{0.8}\right)=\mathrm{nKT}_{0.8}^{\mathrm{n}-1} \mathrm{G}(\mathrm{T})=\mathrm{G}\left(\mathrm{T}_{0.8}\right) *\left(\frac{\mathrm{T}}{\mathrm{T}_{0. \mathrm{e}}}\right)^{\mathrm{n}-1}$
$P_{T E S}=\int_{T_{\text {bath }}}^{T_{0}} G(T) d T=\frac{\mathrm{G}\left(\mathrm{T}_{0.8}^{0.-1}\right.}{n \mathrm{~T}_{0.8}^{0-1}}\left(T_{0}^{n}-T_{b a t h}^{n}\right)$
The heat capacity of the μ-calorimeter is another important thermal parameter. We treated the heat capacity of the gold absorber and that of the MoAu detector as a single entity for simplicity. To deduce the heat capacity, we set the bath temperature close to the transition temperature TC of the MoAu detector and measured the μ-calorimeter response to incident X-ray photons. At such a high bath temperature, the negative electrothermal feedback effect was greatly reduced and the falling-edge time constant of the signal pulses approached the natural time constant τ = C/G. A pulse measured under this condition is shown in Fig. 3b. The fitted falling-edge time constant was 2.31 ms. Using a thermal conductance G(TC) of 319 pW/K, we found that the heat capacity C was approximately 0.737 pJ/K.
3.4. Temperature sensitivity (α) and current sensitivity (β)
The temperature sensitivity and current sensitivity describe how the resistance of the detector evolves with the detector temperature and the current flow through the detector. Both parameters affect the energy resolution and time constants of the μ-calorimeter. They can be deduced by densely measuring the I-V curve around a certain bath temperature and then calculating the derivative of the detector resistance against the detector temperature or detector current along iso-current or iso-thermal lines. Fig. 4a and 4b presents the temperature and current sensitivity versus detector resistance, with the measurements made at a bath temperature of approximately 55 mK. Both parameters decreased with increasing detector resistance. The current sensitivity dropped to less than 0.1 when the detector resistance exceeded 0.8Rn. The results are consistent with our assumption that the detector resistance is nearly current independent above 0.8Rn. The evolution of α and β shown in Fig. 4 agrees qualitatively with the results of previous studies on TES μ-calorimeters made from MoAu [18], [23] or MoAuAu [19] films.
Fig. 4. (a) Temperature sensitivity, (b) current sensitivity of the Mo/Au/Au μ-calorimeter measured at various working points. |
3.5. Summary of electrothermal parameters
We summarize the electrical and thermal parameters of the TES μ-calorimeter in Table 2. The working point of the μ-calorimeter was set as R0 = 0.35Rn and the bath temperature was set at 55 mK.
4. Noise analysis and estimation of the energy resolution
The main noise components [1] of the μ-calorimeter are the Johnson noise of the TES and its shunt resistor, the phonon noise due to thermal fluctuations (TFN) of the thermal conductance G and the SQUID readout noise. F(T0, Tbath) is a unitless function of the temperature of the TES. It depends on the thermal conductance exponent and on whether phonon reflection from the boundaries is specular or diffuse. F(T0, Tbath) typically lies between 0.5 and 1 [1]. Additionally, there is a noise component referred to as excess noise, whose mechanism remains an open question [24], [25], [26]. We usually assume that excess noise [27] is proportional to the detector Johnson noise (Eq. (4.1)), with the prefactor M referred to as the excess noise factor. We used an audio analyzer to measure the total current noise (Eq. (4.2)) of TES at the working point (R0 = 0.35Rn, Tbath = 55mK). By subtracting the contributions of the Johnson noise, TFN noise and SQUID amplifier noise, the excess noise was deduced and we obtained an excess noise factor M of ∼ 2.48. Fig. 5a shows the current noise spectrum measured at the working point (R0 = 0.35Rn) at a bath temperature of 55 mK. Using the electric and the thermal parameters deduced above, we calculated the power-to-current responsivity function (sI(ω)) as shown in Fig. 5b. The power-to-current responsivity was used to convert all the current noise terms into power noise. The noise equivalent power is defined in Eq. (4.3).
$S_{I E X C E S S}=M^{2} * S_{I T E S}$
$S_{\text {ITOT }}=S_{\text {ITES }}+S_{\text {ITFN }}+S_{I L}+S_{\text {ISQUID }}+S_{\text {IEXCESS }}$
$N E P_{T O T}^{2}=\frac{S_{I T O T}}{\left|s_{I}(\omega)^{2}\right|}$
Fig. 5. (a) Current noise spectrum of the μ-calorimeter. (b) the amplitude of power to current responsivity versus frequency. |
We can use the measured total current noise to estimate the energy resolution of the μ-calorimeter. At the small-signal limit, the energy resolution of the μ-calorimeter is given by Eq. (4.4). Using this equation, we estimated the energy resolution of the μ-calorimeter to be 4.01 eV.
$\delta \mathrm{E}_{\mathrm{FWHM}}=2 \sqrt{2 \ln 2}\left(\int_{0}^{\infty} \frac{4}{N E P_{T O T}^{2}} \mathrm{df}\right)^{-1 / 2}$
5. μ-calorimeter response to X-ray photons
We investigated the actual energy-resolving capability of the μ-calorimeter using a radioactive 55Fe X-ray source. The 55Fe nucleus undergoes an electron capture process and decays into 55Mn. The decaying nucleus emits X-ray photons and Auger electrons [28], [29]. The two main X-ray emission lines are the Kα1 and Kα2 lines, whose central energy is 5898.80 and 5887.59 eV, respectively. The emission yields are 16.6% for the Kα1 line and 8.4% for the Kα2 line. The 55Fe source was mounted on the 3 K radiation shield of the cryo-cooler, and the distance between the source and detector was 11 mm, as shown in Fig. 6. We measured the average number of photons that arrived at the detector per second, which is 2.8 cps.
Fig. 6. Schematic view of the X-ray response measurement setup. |
Using the setup described above, we measured the μ-calorimeter response to incident X-ray photons. The bath temperature was set at 55 mK. A typical signal pulse (R0 = 0.35Rn) is shown in Fig. 7a. The pulse was fitted by $\delta V(t)=A(E) *\left(e^{-\frac{t-t_{0}}{\tau_{+}}}-e^{-\frac{t-t_{0}}{\tau_{-}}}\right)$. The time constant of the falling edge was found to be 342 μs, which is close to the calculated τeff given in Table II.
Fig. 7. (a) A typical current pulse of the TES μ-calorimeter. The red line was the double exponential fitting of the raw pulse. The inset is the calibration curve of TES for to translate the pulse amplitude into photon energy. (b)the measured energy spectrum of Mn Kα by Mo/Au/Au TES. The red solid line is used to fit the data with Eq. (5.3). The blue dash line represents the natural line shape of Mn Kα lines. The inset gives the energy resolution measured at six different working points. |
We used the following data analysis pipeline to deduce the instrumental energy resolution of the μ-calorimeter. We first excluded low-quality signal pulses that had an abnormal time constant or showed a pulse pile up. The raw pulses were then digitally filtered adopting principal component analysis to reduce the noise, and the amplitude of each pulse was extracted. We then calibrated the photon energy and pulse amplitude. The inset in Fig. 7a shows the calibration curve, which is a polynomial fitting to the data points. The photon energy of the characteristic X-ray emission lines of the 55Fe source is known and the corresponding pulse amplitude voltage was estimated through Gaussian fitting of each individual peak in the pulse amplitude histogram.
Using the calibration curve, we plotted the photon energy histogram as shown in Fig. 7b. To deduce the instrumental energy resolution of the μ-calorimeter, we subtracted the natural spectral line broadening [30] of the 55Fe emission lines, which was considered as the sum of several Lorentzian functions as shown in Eq. (5.1).
$L(\mathrm{E})=\sum_{i j} \frac{I_{i j}}{1+\left(\left(\mathrm{E}-\mathrm{E}_{i j}\right) / \gamma_{i j}\right)^{2}}$
Here, Iij is the amplitude of each Lorentzian function, E' is the central energy and γ is the half width at half maximum. The index i indicates the emission lines Kα1 and Kα2 and the index j indicates the sub-lines of the energy peak. i.e.,
The μ-calorimeter was assumed to have a Gaussian response function that describes the probability that the detector identifies an incident photon with energy E′ as a photon with energy E. Additionally, σ is the standard deviation of the Gaussian function.
$\mathrm{G}(\mathrm{E})=\frac{1}{\sigma \sqrt{2 * \pi}} * \exp \left(-\frac{\left(\mathrm{E}-\mathrm{E}^{2}\right)^{2}}{2 * \sigma^{2}}\right)$
Here, we convolve Eqs. (5.1), (5.2) and obtain Eq. (5.3), which was used to fit the measured histogram.
$S(E)=\int_{-\infty}^{\infty} \mathrm{AG}\left(\mathrm{E}-\mathrm{E}^{\prime}\right) \mathrm{L}\left(\mathrm{E}^{\prime}\right) \mathrm{dE}^{\prime}$
Here, A is a fitting parameter. From the fitting, we found the instrumental energy resolution of the μ-calorimeter, defined as the full width at half maximum of the Gaussian function, $\Delta \mathrm{E}=2 \sqrt{\ln (2)} * \sigma$. The best energy resolution 4.13 eV was achieved at the working point R0 = 0.35Rn as shown in the inset of Fig. 7b. This result agrees qualitatively with the estimation from the noise spectrum. We note that this result comes from a single pixel in this 16-pixel array. The energy resolution of a randomly selected 2nd pixel in this array shows an energy resolution of 6.83 eV. More measurements are on the way and we are investigating the cause of the difference in the energy resolution from pixel to pixel.
6. Conclusion
We successfully fabricated a TES μ-calorimeter using proximately coupled Mo and Au films. We characterized the thermal and electric parameters of the μ-calorimeter and measured the noise spectrum. We estimated the energy resolution from the measured noise spectrum. The estimation gave an energy resolution of 4.01 eV, which agrees qualitatively with the instrumental energy resolution of 4.13 eV deduced from the spectral histogram of a 55Fe X-ray source. We expect that the energy resolution can be further improved by reducing the excess noise factor. Measurements on the devices with metal structures to restrain the excess noise are ongoing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.s
Acknowledgments
The project is supported by China National Space Administration (CNSA) under grant No. D050104 and by the grant for low energy gamma-ray detection research based on SQUID technique. The nanofabrication work is supported by the Superconducting Electronics Facility (SELF) of Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences.
