A method to measure superconducting transition temperature of microwave kinetic inductance detector by changing power of readout microwaves

A superconducting detector is a sensitive device because its gap energy is much smaller than that of a semiconductor detector. A microwave kinetic inductance detector (MKID)¹ is a superconducting microwave resonator. This resonant circuit is fabricated of a thin superconductor film on a silicon or sapphire substrate. The resonant frequency is tuned by the total length of the resonator. Therefore, numerous detectors can be read by multi-frequency tones using the same readout line. This advantage allows the production of a large MKID array, such as one containing 100–1000 detectors per single readout line. This has resulted in the rapid progress of radio and infrared astronomical observations.^2–5 

An MKID has a simple detection mechanism. Radiation entering the detector breaks a Cooper pair in the resonator when the radiation energy is larger than twice the gap energy of the superconducting film. A broken Cooper pair yields two additional quasiparticles. The inductance of the MKID circuit varies with the number of quasiparticles, and the resonant condition of this MKID changes with the inductance. We can measure the incident radiation energy using the variation in the resonant phase and amplitude.

The superconducting transition temperature (T_c) of the MKID is an important parameter. This is because various MKID characterization parameters depend on T_c. In this paper, we propose a method to measure the T_c of the MKID by changing the readout power rapidly.

According to the BCS theory,⁶ the relation between the gap energy (Δ) and T_c is given by the following formula:

where k_B is the Boltzmann constant. A coefficient in this relation is confirmed with 1% precision for aluminum.⁷ It slightly varies for different materials. Using Δ, the number of quasiparticles (N_qp) under a low-temperature condition (i.e., T ≪ T_c, where T is the device temperature) is obtained by the following formula:⁶ 

where N₀ is the single-spin density of states at the Fermi level and V is the volume of the resonator. It is known that N₀ = 1.74 × 10¹⁰ eV⁻¹μm⁻³ for aluminum,⁸ and its deviation from the other studies is less than 3%.^9,10 The intrinsic quasiparticle lifetime, (⁠$τqpi$⁠), is obtained by the following formula:¹¹ 

where τ₀ is the electron–phonon interaction time (τ₀ = 458 ± 10 ns for an aluminum MKID¹²). The noise equivalent power derived from the generation and recombination of the quasiparticles (NEP_gr) is obtained by the following formula:¹³ 

where η_pb is the pair breaking efficiency¹⁴ (η_pb = 0.57 for an aluminum¹⁵). Figure 1 presents the parameters of an aluminum MKID as a function of T_c, where we display the plots of the device temperature at ∼0.3 K. They are sensitive to T_c. Various previous studies have used different T_c, with the deviation being ∼10%.^12,16,17 Understanding the reason of this difference is a recent research topic, e.g., Fyhrie et al.¹⁶ discussed the relation between T_c and film thickness.

Measuring T_c using four DC probes is a well-established method. However, its setup is different from the MKID readout via microwave lines. It is worthwhile to have methods for measuring T_c using the readout microwaves. Monitoring the transmittance of the readout microwaves with the device temperature variation is a conventional method to measure the T_c of an MKID.^18,19 This method is referred to as the “S21 method” in this paper. Note that this method is not applicable for a hybrid-type MKID.^20,21

Another method estimates T_c using the power spectrum density (S_x), which is modeled by the following formula:²² 

where subscript x denotes the phase or amplitude response, τ_qp is the quasiparticle lifetime under the measurement condition, f is the frequency of the detector response, and X_system is the noise of the readout system. τ_res is the resonator ring time given by τ_res = Q_r/πf_r (where Q_r is the quality factor of the resonance and f_r is the resonant frequency). We extract τ_qp by fitting the power spectrum density with the above formula. Under the assumption of $τqpi=τqp$⁠, we obtain T_c using Eq. (3). This method requires that X_system is lower than the contribution of the first term in Eq. (5). Moreover, another contribution due to a two level system noise²³ should be low enough in the case of the phase response.

We propose the third method to measure T_c, which uses a loss of the readout microwaves in the MKID. The quasiparticle lifetime (τ_qp) decreases with the increase in the number of additional quasiparticles (⁠$Nqp′$⁠) produced by the readout power loss,^22,24–27

Because the phase response (θ) of the MKID is proportional to the number of additional quasiparticles,²⁸ the above formula is rewritten as

where αθ = $Nqp′/Nqp$⁠. This suggests that $τqpi$ can be estimated from the relation between τ_qp and θ.^27,28 This relation is easily measured using our previous method to measure the phase responsivity.²⁹ A small fraction of the readout power is deposited in the MKID, and the response of the MKID increases with this power. In this paper, we demonstrate the measurement of this relation for an aluminum MKID. Subsequently, we obtain T_c with Eq. (3) and estimate $τqpi$⁠.

Figure 2 presents the diagram of the MKID readout. Our cryostat (Niki Glass Co., Ltd.) consists of 4 K and 40 K thermal shields from inside to outside. They are insulated from the room temperature (300 K) in a vacuum chamber and are cooled using a pulse tube refrigerator (PT407RM, Cryomech Co., Ltd.). The 4 K thermal shield also acts as a magnetic shield (A4K, Amuneal Co., Ltd.) for mitigating the effects of geomagnetism. The MKID device is set in a light-tight copper box. The box is cooled in a ³He-sorption refrigerator, and it is maintained at T = 311 mK with an accuracy of 6 mK.

The MKID device is fabricated at RIKEN. This device consists of a quarter-wave coplanar waveguide resonator and a feedline (there is no coupling with any antenna). The width of the center strip and gap of the resonator (the feedline) are 4 μm (12 μm) and 1.5 μm (8 μm), respectively. All the circuits patterns are formed using an aluminum film on a silicon wafer. The volume of the resonator is 2600 μm³ (the width is 4 μm, the length is 6500 μm, and the thickness is 100 nm). Its resonant frequency and quality factor are f_r = 4.30 GHz and Q_r = 2.61 × 10⁴, respectively. Our readout system^30,31 measures the response based on a direct down-conversion logic with a 200 MHz sampling speed, and the data are down-sampled to a 1 MHz step. The power of the readout microwave is controlled by a variable attenuator (LDA-602, Vaunix Co., Ltd.). It requires a few microseconds to change the attenuation value.

We use eight attenuation setups to change the readout power from high power (P_H) to low power (P_L), as summarized in Table I. The readout power into the feedline is approximately −65 dBm at P_L. We use the same treatment for the effects of a cable delay and linearity correction, as described in Ref. 29. Figure 3 shows the phase response as a function of time. We reset the attenuation value at t = 100 µs. We fit the data to Eq. (5) in Ref. 29, and we obtain the phase response (θ) and the quasiparticle lifetime (τ_qp) for each setup using Eq. (7). We mask the data in the short period, t = 95 µs–105 µs, because of the uncontrolled state of attenuation soon after the reset. We measure 50 samples for each set of power change. Table I summarizes the fitted results for each setup. Figure 4 displays the relation between τ_qp and θ. We obtain $τqpi=31.3±0.2μs$ from the fit with Eq. (7). Subsequently, we obtain T_c = 1.278 ± 0.001 K using Eq. (3), where we only estimate the statistical error. We estimate the systematic uncertainties for T_c, device temperature (0.025 K), time for changing the attenuation value (0.014 K), and electron–phonon interaction time (0.004 K). We finally obtain T_c = 1.28 ± 0.03 K, including the systematic error.

For comparison, we also measure T_c by the S21 method. Figure 5 presents the S21 intensity results at 4.35 GHz as a function of the device temperature. We determine T_c as a temperature at the middle of the transition, T_c = 1.27 ± 0.04 K. Here, the error includes the difference from the onset of the superconducting transition and the uncertainty of the thermometer. We confirm the consistency in the results from the two methods.

In summary, we propose a method to measure the T_c of an MKID device by changing the power of the readout microwaves. In this method, we obtain the intrinsic quasiparticle lifetime using the device temperature. Subsequently, we estimate T_c using them. We demonstrate our method using an aluminum MKID maintained at 311 mK. The results are consistent with those obtained by the conventional method: monitoring the microwave transmittance by changing the device temperature. The method proposed in this paper is applicable to other types, such as a hybrid-type MKID.