# GHz Superconducting Single-Photon Detectors for Dark Matter Search

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}. In addition, the presence of ALPs created by the Primakoff process [7,8], consisting of photon-axion conversion in an external magnetic field, would alter stellar-evolution. Instead of using stellar energy losses to infer the axion exclusion limits, the flux of axions created by the sun can be detected through axion helioscopes, such as CAST [9] and IAXO [10] experiments. These experiments constantly point at the sun by means of a tracking system aiming at converting the solar axions into detectable X-ray photons through Primakoff effect. In microwave cavity experiments, such as ADMX [11] and QUAX [12], galactic halo axions may be detected by their resonant conversion into a quasi-monochromatic microwave signal in a high-quality-factor electromagnetic cavity permeated by a strong static magnetic field. The resonance frequency of the cavity is tuned to equalize the total axion energy. Interestingly, only these experiments are able to probe part of the quantum chromodynamics (QCD) Peccei-Quinn region [13].

## 2. Theoretical Modeling of a One-Dimensional Fully Superconducting Josephson Junction

- a thin insulating barrier forming a superconductor/insulator/superconductor SIS-JJ;
- a short section of normal metal creating superconductor/normal metal/superconductor SNS-JJ;
- a physical constriction in the superconductor producing an SsS-JJ (known as Dayem bridge);
- a short section of lower energy gap superconductor realizing a SS’S-JJ.

## 3. Experimental Demonstration of a 1DJ

#### 3.1. Density of States and One-Dimensionality

#### 3.2. Current Control of the R vs T

## 4. Operation Principle of the Nano-TES and JES

## 5. Single-Photon Detection Performance of the Nano-TES and the JES

#### 5.1. Modeling of the Nano-TES

#### 5.2. Modeling of the JES

#### 5.3. Performance Deduced from the Experimental Data

## 6. Materials and Methods

#### 6.1. Fabrication Procedure

_{2}thermally grown on an intrinsic silicon wafer. To obtain the resist suspended mask, a bilayer composed of a 950-nm-thick MMA (8.5) layer and a PMMA (A4, 950 k) film of thickness of about 300 nm was spin-coated on the substrate. The ratio between the electron irradiation doses to make the resists soluble is $DOS{E}_{MMA}:DOS{E}_{PMMA}\simeq 1:4$. The evaporations were performed in an ultra-high vacuum electron-beam evaporator with a base pressure of about ${10}^{-11}$ Torr by keeping the target substrate at room temperature. First, 13-nm-thick Al layer was evaporated at an angle of −40°. Second, the film was then oxidized by exposition to 200 mTorr of O

_{2}for 5 min to obtain the tunnel probes of the device devoted to the spectral and the thermal measurements. Third, the Al/Cu bilayer (${t}_{Al}=10.5$ nm and ${t}_{Cu}=15$ nm) forming the superconducting nanowire is evaporated at an angle of ${0}^{\circ}$. Fourth, a second 40-nm-thick Al film was evaporated at an angle of $+{40}^{\circ}$ to obtain the lateral electrodes completing the 1DJ. The angle resolution of each evaporation was $\sim {1}^{\circ}$. The average film thickness can be controlled during the evaporation process with the precision of 0.1 nm at the evaporation rate of about 1.5 angstrom/s.

#### 6.2. Measurement Setups

^{3}He-

^{4}He dilution refrigerator equipped with RC low-pass filters (cut-off frequency of about 800 Hz). The lowest electronic temperature obtained was 20 mK.

#### 6.3. Basic Properties of the Active Region

^{2}s

^{−1}for the Al thin film, and ${D}_{Cu}=8\times {10}^{-3}$ m

^{2}s

^{−1}for the Cu layer). The superconducting coherence length of the Al film is ${\xi}_{Al}=\sqrt{\hslash {D}_{Al}/{\Delta}_{Al}}\simeq 80$ nm, where ${\Delta}_{Al}\simeq 200\phantom{\rule{0.277778em}{0ex}}\mathsf{\mu}$eV is its measured superconducting energy gap. The normal-state coherence length of the Cu layer is ${\xi}_{Cu}=\sqrt{\hslash {D}_{Cu}/\left(2\pi {k}_{B}T\right)}\simeq 255$ nm, where $T=150$ mK is chosen in the worst possible working scenario (i.e., the critical temperature of A). The superconducting coherence length in A is given by ${\xi}_{A}=\sqrt{l\hslash /\left[({t}_{Al}{N}_{Al}+{t}_{Cu}{N}_{Cu}){R}_{N}{e}^{2}{\Delta}_{A,0}\right]}\simeq 220$ nm, where ${R}_{N}=80\phantom{\rule{0.277778em}{0ex}}\Omega $ is the nanowire normal-state resistance, and ${N}_{Al}=2.15\times {10}^{47}$ J

^{−1}m

^{−3}and ${N}_{Cu}=1.56\times {10}^{47}$ J

^{−1}m

^{−3}are the density of states at the Fermi level of Al and Cu, respectively. The London penetration depth for the magnetic field of A takes the form ${\lambda}_{L,A}=\sqrt{(\hslash w{t}_{A}{R}_{N})/\left(\pi {\mu}_{0}l{\Delta}_{A,0}\right)}\simeq 970$ nm, where ${\mathsf{\mu}}_{0}$ is the magnetic permeability of vacuum. The magnetic field generated at the wire surface in correspondence of the maximum bias current is ${B}_{I,max}={\mathsf{\mu}}_{0}{I}_{max}/\left(2\pi t\right)\simeq 4.7$$\mathsf{\mu}$T, where ${I}_{max}=370$ nA, and ${\mathsf{\mu}}_{0}$ is the vacuum magnetic permeability.

## 7. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Laboratory axion search. (

**a**) Photon-axion coupling (g) versus axion mass, where axions origins are indicated. The diagonal band shows the parameter space consistent with the quantum chromodynamics (QCD) axion from the Peccei-Quinn theory. The grey area depicts the operating range of the detectors presented in this review. (

**b**) Conceptual representation of a light-shining-through-wall (LSW) experiment. A laser (violet) feeds photons in a Fabry-Perot cavity immersed in a constant magnetic field. An axion is generated by the conversion of a photon through Primakoff effect and passes through the wall (grey). The axions converts back in a photon of same energy in the second magn etic field area and is revealed by a single-photon detector (red).

**Figure 2.**Structure and transport properties of a one-dimensional fully superconducting Josephson junction (1DJ). (

**a**) Top: Scheme of the structure a 1DJ. Two superconducting electrodes (S, blue) are separated by a weak link consisting of a superconducting wire (A, bronze). Its width (w) and thickness (t) are indicated. The two superconducting energy gaps follow ${\Delta}_{A}\ll {\Delta}_{S}$. The current (I) flowing along the 1DJ is shown. Bottom: resistively shunted junction (RSJ) model of a 1DJ, where I is the bias current, J is the junction, and ${R}_{N}$ is the shunt resistor. (

**b**) Tilted washboard potential of a 1DJ calculated for different values of I. The energy barrier ($\delta U$) for the escape of the phase particle from the washboard potential (WP) decreases by rising the bias current; thus, the probability of the transition of the 1DJ to the normal-state increases with I. The phase particle in the WP [34] is indicated by means of the black dot. (

**c**) Energy barrier ($\delta U$) normalized with respect to the zero-temperature Josephson energy (${E}_{J,0}$) calculated by varying the critical current (top) and the bias current (bottom), respectively. (

**d**) temperature dependence of the normalized resistance ($R/{R}_{N}$) calculated for selected values of I. ${R}_{N}$ is the normal-state resistance of the 1DJ. (

**e**) Temperature derivative of the resistance ($\mathrm{d}R/\mathrm{d}T$) calculated for the same values of I in panel (

**d**).

**Figure 3.**Definition of escape and critical temperature. (

**a**) Calculated normalized resistance ($R/{R}_{N}$) versus temperature calculated for a 1DJ within the RSJ model. The escape temperature (${T}_{e}$) and the critical temperature (${T}_{C}$) are indicated. (

**b**) Critical temperature (${T}_{C}$, blue) and escape temperature (${T}_{e}$, orange) as a function of the bias current (I) calculated by means of the RSJ model of a 1DJ.

**Figure 4.**Measurement of the density of states in a 1DJ. (

**a**) False-color scanning electron micrograph of a device used to measure the DOS of a 1DJ. The 1DJ is made of an Al/Cu bilayer nanowire (yellow) interrupting two Al electrodes (blue). The Al probes (red) allow to perform tunnel spectroscopy. In the experiment, a voltage (V) is applied between A and one probe while recording the current (I). (

**b**) Tunneling current (I) as a function of voltage (V) characteristics recorded at ${T}_{bath}=20$ mK (blue) and ${T}_{bath}=250$ mK (yellow). (

**c**) Zoom of the $IV$ characteristics in correspondence of the transition to the normal-state. It is possible to extract ${\Delta}_{A,0}\simeq 23\phantom{\rule{0.277778em}{0ex}}\mu $eV and ${\Delta}_{P,0}\simeq 200\phantom{\rule{0.277778em}{0ex}}\mathsf{\mu}$eV as the crossing between the black dotted lines and $I=0$.

**Figure 5.**Current control of the resistance versus temperature characteristics in a 1DJ. (

**a**) False-colored scanning electron micrograph of a 1DJ. The nanowire is made of an Al/Cu bilayer (yellow) separating two thick Al electrodes (blue). The 1DJ is AC current-biased (amplitude I), while the voltage drop across the wire (${V}_{out}$) is measured through a lock-in amplifier. The load resistor (${R}_{L}\gg {R}_{N}$) guarantees constant bias current while transitioning to the normal-state. (

**b**) Selected normalized resistance (R) versus temperature (T) characteristics recorded for different values of bias current (I). (

**c**) Dependence of the escape temperature (${T}_{e}$) on bias current normalized with respect to the zero-temperature 1DJ critical current ($I/{I}_{C}$) for two different samples. (

**d**) Width of the phase transition ($\delta {T}_{C}$) versus I for two different 1DJs.

**Figure 6.**General thermal and electrical model of the nano-transition edge sensor (TES) and Josephson escape sensor (JES). (

**a**) Schematic representation of a typical biasing circuit for the nano-TES and JES. The parallel connection of the sensor (of variable resistance R) and the shunt resistor (${R}_{S}$) is biased by the current ${I}_{Bias}$. The role of ${R}_{S}$ is to limit the Joule overheating of A when transitioning to the normal-state. The variations of the current (I) flowing through the sensing element are measured by means of the inductance L. For instance, an inductively-coupled SQUID amplifier could serve as read-out element. (

**b**) Thermal model of the nano-sensors, where the main thermal exchange channels are shown. ${P}_{in}$ is the power coming from the incoming radiation, and ${P}_{e-ph}$ is the heat exchanged between electrons in the active region (yellow) at ${T}_{A}$ and lattice phonons (grey) at ${T}_{bath}$, while ${P}_{A-B}$ is the heat current flowing towards the superconducting electrodes (blue) residing at ${T}_{bath}$.

**Figure 7.**Performance calculated for the JES. (

**a**) Frequency resolution $\delta {\nu}_{JES}$ as a function of I for sample 1 (blue triangles) and 2 (yellow squares). The dashed lines indicate the frequency resolutions of the corresponding nano-TESs. (

**b**) Resolving power ($\nu /\delta {\nu}_{JES}$) as a function of the frequency ($\nu $) of the incident single-photons calculated for sample 1. The dashed lines indicate the resolving power of the corresponding nano-TES. (

**c**) Time constant versus bias current for sample 1 (blue triangles) and 2 (yellow squares). The dashed lines indicate the time constants of the corresponding nano-TESs.

**Table 1.**Main figures of merit deduced for the nano-TES. The time constant $\tau $, the pulse recovery time ${\tau}_{eff}$, the frequency resolution $\delta \nu $, and the resolving power $\nu /\delta \nu $ (at 100 and 300 GHz) are reported for the two fabricated nano-TESs.

Sample | ${\mathbf{T}}_{\mathbf{c}}$ | $\mathit{\tau}$ | ${\mathit{\tau}}_{\mathit{eff}}$ | $\mathit{\delta}\mathit{\nu}$ | $\mathit{\nu}/\mathit{\delta}\mathit{\nu}$ |
---|---|---|---|---|---|

(mK) | ($\mathit{\mu}\mathit{s}$) | ($\mathit{\mu}\mathit{s}$) | (GHz) | 100 GHz 300 GHz | |

1 | 128 | 6 | 0.01 | 100 | 1 3 |

2 | 139 | 5 | 0.2 | 540 | 0.18 0.55 |

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Paolucci, F.; Giazotto, F. GHz Superconducting Single-Photon Detectors for Dark Matter Search. *Instruments* **2021**, *5*, 14.
https://doi.org/10.3390/instruments5020014

**AMA Style**

Paolucci F, Giazotto F. GHz Superconducting Single-Photon Detectors for Dark Matter Search. *Instruments*. 2021; 5(2):14.
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**Chicago/Turabian Style**

Paolucci, Federico, and Francesco Giazotto. 2021. "GHz Superconducting Single-Photon Detectors for Dark Matter Search" *Instruments* 5, no. 2: 14.
https://doi.org/10.3390/instruments5020014