GHz Superconducting Single-Photon Detectors for Dark Matter Search
AArticle
Sensitive Superconducting Calorimeters for DarkMatter Search
Federico Paolucci and Francesco Giazotto NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, Italy† [email protected]‡ [email protected]: date; Accepted: date; Published: date
Abstract:
The composition of dark matter is one of the puzzling topics in astrophysics. Since, theexistence of axions would fill this gap of knowledge, several experiments for the search of axions havebeen designed in the last twenty years. Among all the others, light shining through walls experimentspromise to push the exclusion limits to lower energies. To this end, effort is put for the development ofsingle-photon detectors operating at frequencies <
100 Ghz. Here, we review recent advancements insuperconducting single-photon detection. In particular, we present two sensors based on one-dimensionalJosephson junctions with the capability to be in situ tuned by simple current bias: the nanoscale transitionedge sensor (nano-TES) and the Josephson escape sensor (JES). These two sensors seem to be the perfectcandidates for the realization of microwave light shining through walls (LSW) experiments, since theyshow unprecedented frequency resolutions of about 100 GHz and 2 GHz for the nano-TES and JES,respectively.
Keywords: axion; single-photon detectors; superconducting detectors
1. Introduction
Axions and weakly interacting massive particles (WIMPs) are expected to be possible componentsof the cold dark matter. Furthermore, axions and axion-like particles (ALPs) are proposed to solve thecharge-conjugation parity (CP) problem in quantum chromodynamics (QCD) by means of the Peccei-Quinnmechanism [1–3]. Up to now, the experimental searches focusing on axions or ALPS produced null resultswith corresponding excluded regions in the coupling constant ( g ) versus mass parameter space shown inFig. 1(a). Two classes of experiments are performed: astrophysical experiments observing astrophysicalphenomena or attempting to detect cosmic axions, and laboratory-based experiments which aim todemonstrate the existence of axions in strictly controlled settings [4].The observation of solar axions is strongly affected by the limits of the solar models. Indeed, thecoupling of low-mass weakly interacting particles produced in the sun with normal matter is boundedby the observations of stellar lifetimes and energy loss rates. Solar models together with measurementof neutrino fluxes imply limits on the magnitude of the coupling constant to g ≤ × − GeV − . Inaddition, the presence of ALPs created by the Primakoff process [5,6], that is photon-axion conversionin an external magnetic field, would alter stellar-evolution. Thus, many exclusion limits on the axionmass have been argued. Instead of using stellar energy losses to infer the axion exclusion limits, the fluxof axions created by the sun can be detected through an axion helioscope, such as CAST [7] and IAXO[8] experiments. These experiments constantly point at the sun by means of a tracking system aiming toconvert the solar axions into detectable X-ray photons through Primakoff effect. Instead, in microwavecavity experiments, such as ADMX [11] and QUAX [10], galactic halo axions may be detected by theirresonant conversion into a quasi-monochromatic microwave signal in a high-quality-factor electromagnetic a r X i v : . [ phy s i c s . i n s - d e t ] J a n of 15 cavity permeated by a strong static magnetic field. The resonance frequency of the cavity is tuned toequalize the total axion energy. Interestingly, only these experiments are able to to probe part of the QCDPeccei-Quinn region.Light shining through walls (LSW) experiments, such as ALPS [11] and STAX [12], are fullyin-laboratory searching techniques. The general concept of a LSW experiment is shown in Fig. 1(b).A laser beam is sent through a long magnet, allowing for the coherent photon-axion conversion due to thePrimakoff effect. The wall acts as photon barrier, thus blocking the laser beam, while allowing the axion topass through (since its interaction with massive matter is negligibly small). A second magnet placed afterthe wall causes the photon-axion back conversion. Since both conversions are very rare (depending on g ), very intense sources are necessary. The highest luminosity photon sources currently available are thegyrotrons, that operate typically below the THz region, with a maximum power of 1 MW at about 100GHz. In this spectral region, single-photon detection is extremely difficult. In fact, LSW experiments at themicrowave have been proposed [12] but not realized yet.To implement microwave LSW experiments, the key ingredient is thus the development of newultrasensitive single-photon detectors operating at unprecedented low frequencies f ≤
100 GHz [13,14].Nowadays, state of the art detectors for astrophysics are mainly based on transition edge sensors (TESs)[15,16] and kinetic inductance detectors (KIDs) [17–19]. A strong reduction of the thermal exchangesin the sensing elements is fundamental to push single-photon detection to lower bounds. To this end,miniaturization and Josephson effect [20] have been exploited [21–24].Recently, two microwave single-photon detectors, nanoscale transition edge sensors (nano-TES) [25]and the Josephson escape sensor (JES) [26], have been designed by employing a one dimensional fullysuperconducting Josephson junction (1DJ) as radiation absorber. The nano-TES and the JES point towardsunprecedented frequency resolutions of about 2 GHZ thus enabling the possibility to implement LSWexperiments. In addition, the sensitivity of these sensors can be in situ tuned by simple current biasing.This paper reviews these sensors by covering their theoretical and experimental properties. Inparticular, Section 2 presents the theoretical description of the 1DJs, while Sec. 3 shows their experimentalelectronic and thermal transport properties. Section 4 introduces the operation principles of the nano-TEsand JES detectors. Section 5 presents the detection performance of the nano-TES and the JES. Section 6presents the experimental methods used for the sensors characterization. Finally, Sec. 7 resumes the resultsand opens to new applications for the nano-TES and the JES detectors.
2. Theoretical modelling of a one-dimensional fully superconducting Josephson junction
A Josephson junction (JJ) is a structure where the possibility of a superconductor to carry adissipationless current is strongly suppressed. Typically, the discontinuity of the supercurrent flowis realized by interrupting the superconductor by means of a weak link . A weak link can consist of athin insulating barrier forming a superconductor/insulator/superconductor SIS-JJ, a short section ofnormal metal creating superconductor/normal metal/superconductor SNS-JJ, a physical constriction inthe superconductor producing an SsS-JJ (known as Dayem bridge), or a short section of lower energygap superconductor realizing a SS’S-JJ. Here, we focus on the one-dimensional version of a SS’S-JJ wherethe two superconducting lateral electrodes ( S ) are separated by a one-dimensional wire ( A ) made of adifferent superconductor. In addition, both the thickness ( t ) and width ( w ) of A are smaller than theLondon penetration depth ( λ L , A ) and the Cooper pairs coherence length ( ξ A ). The one-dimensionalityensures constant superconducting wave function and homogeneous supercurrent density along the wirecross section, and uniform penetration of A by an out-of-plane magnetic field. In the following, we willname such a structure as 1DJ for simplicity. The general structure of a 1DJ is shown in Fig. 2(a). of 15 (a) (b) axion mass (meV)1 a x i on c oup li ng ( G e V - ) -16 -14 -12 -10 photon frequency (GHz)1 10 10 -1 -2 laser magnets wall axionphotons cavities detectorphoton Figure 1.
Laboratory axion search. ( a ) Photon-axion coupling ( g ) versus axion mass, where axions originsare 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 detectorspresented in this review. ( a ) 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 isgenerated by the conversion of a photon through Primakoff effect and passes through the wall (grey). Theaxions converts back in a photon of same energy in the second magn etic field area and is revealed by asingle-photon detector (red). k B T/E
J,0 R / R N I/I
C,0 δ U / E J , I C /I C,0 k B T/E
J,0 d R / d T ( a . u . ) I/I
C,0
I/I
C,0 w tII R N J δUφ/ π 420 U ( φ ) / E J , -12 I/I
C,0 Figure 2. Structure and transport properties of a 1DJ. ( a ) Top: Scheme of the structure a 1DJ. Twosuperconducting electrodes ( S , blue) are separated by a weak link composed of a superconducting wire( A , bronze). The width ( w ) and the thickness ( t ) of A are indicated. The two superconducting energy gapsfollow ∆ A (cid:28) ∆ S . The current ( I ) flowing along the 1DJ is shown. Bottom: 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 1DJcalculated for different values of I . The energy barrier for the escape of the phase particle from the WP ( δ U )decreases by rising the bias current, thus the probability of the transition of the 1DJ to the normal-stateincreases with I . The phase particle in the WP is indicated. ( c ) Energy barrier ( δ U ) normalized with respectto the zero-temperature Josephson energy ( E J ,0 ) calculated by varying the critical current (top) and the biascurrent (bottom), respectively. ( d ) temperature dependence of the normalized resistance ( R / R N ) calculatedfor selected values of I . R N is the normal-state resistance of the 1DJ. ( e ) Temperature derivative of theresistance (d R /d T ) calculated for the same values of I in panel (d). of 15 The behavior of the 1DJ can be described by means of the overdamped resistively shunted junction(RSJ) model [27], where the JJ is shunted by its normal-state resistance [see Fig. 2(a)]. Here, the bias current( I ) dependence on the stochastic phase difference [ ϕ ( t ) ] over the junction reads2 e ¯ h ˙ ϕ ( t ) R N + I C sin ϕ ( t ) = I + δ I th ( t ) , (1)where e is the electron charge, ¯ h is the reduced Planck constant, R N is the wire normal-state resistance,while I C is its critical current. The normal-state resistance of the 1DJ acts as shunt resistor providing athermal noise contribution to the flowing current given by (cid:104) δ I th ( t ) δ I th ( t (cid:48) ) (cid:105) = K B TR N δ ( t − t (cid:48) ) , where k B is theBoltzmann constant and T is the temperature. The transition to the normal-state of a JJ or a superconductingnano-wire is usually attributed 2 π quasiparticle phase-slips [27,28], because a full phase rotation entails topass through I C =
0. Within the RSJ model, the phase slip is represented as the motion of a phase particlein a tilted washboard potential (WP) under the presence of friction forces. The WP can be written U ( ϕ ) = − ¯ hI e ϕ − δ U cos ϕ , (2)where δ U ( I , E J ) is the escape energy for the phase particle. We note that, the only parameter dependenton the JJ geometry is δ U ( I , E J ) . For a 1DJ, it takes the form [29] δ U ( I , E J ) ∼ E J ( − I / I C ) = Φ I C π ( − I / I C ) . (3)Equations 2 and 3 show that both bias current and Josephson energy ( E J = Φ I C /2 π with Φ (cid:39) × − Wb the flux quantum) define the WP. In particular, δ U is suppressed by lowering the Josephsonenergy and rising the bias current. The latter also produces the tilting of the WP, as shown in Fig. 2(b).It is interesting to quantitatively compare the effects of I and E J on the WP. To this end, we substitutedthe Josephson energy with the critical current in Eq. 3. Without current bias ( I = δ U ∼ I − C .The comparison between the two methods to suppress the energy barrier is shown in Fig. 2(c). Thus, thecurrent bias is the most efficient method to control the supercurrent flowing in a 1DJ.The normal-state resistance of a 1DJ is very low. Therefore, it can be described by means of theoverdamped junction limit of the RSJ model. In this approximation, the temperature dependence of thevoltage drop build across a 1DJ can be written [30] V ( I , E J , T ) = R N I − I C ,0 Im I − iz (cid:16) E J k B T (cid:17) I − iz (cid:16) E J k B T (cid:17) , (4)where I C ,0 is the junction zero-temperature critical current, I µ ( x ) is the modified Bessel function withimaginary argument µ , and the imaginary argument takes the form z = E J k B T II C . Therefore, V stronglydepends on I C (thus E J ) and I . The current derivative of the voltage drop calculated at different values oftemperature provides the R ( T ) characteristics R ( I , E J , T ) = d V ( I , E J , T ) d I . (5)By solving Eq. 5 for different values of I , we can evaluate the impact of the bias current onthe resistance versus temperature characteristics of ta 1DJ. In particular, Fig. 2(d) highlights that the of 15 (a) (b) temperature T C T e R / R N k B T / E J , I/I
C,0 T C T e 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 criticaltemperature ( T C ) are indicated. ( b ) Critical temperature ( T C , blue) and escape temperature ( T e , orange) as afunction of the bias current ( I ) calculated by means of the RSJ model of a 1DJ. temperature of the superconducting-to-resistive state transition of the JJ decreases by rising I . Furthermore,high values of bias current have a second important effect on the R ( T ) : the temperature width of thetransition narrows. The temperature derivative of R ( T ) confirms the positive impact the current bias onthe transition width, as shown by Fig. 2(e). This behavior is related to the decrease of δ U and to the currentinduced tilting of the WP (providing a preferred direction of the phase-slips).Two different temperatures related to the superconductor-to-normal-state transition can be defined, asshown in Fig. 3(a). On the one hand, the effective critical temperature ( T C ) is the temperature correspondingto half of the normal-state resistance [ R ( T C ) = R N /2]. On the other hand, the escape temperature ( T e ) isthe maximum value of temperature providing a zero resistance of the 1DJ [ R ( T e ) = T C and T e , as shown in Fig. 3(b). In particular, the effective critical temperaturedecreases much faster than the escape temperature by rising I , thus providing T C ∼ T e for I → I C .This behavior highlights once more the narrowing of the superconducting-to-normal-state transition froincreasing bias current.
3. Experimental demonstration of a 1DJ
This section is devoted to the experimental demonstration of bias current tuning of the R versus T characteristics of a 1DJ. In particular, Sec. 3.1 aims to proof that the structure under study isone-dimensional, while Sec. 3.2 will show the tuning of the superconducting-to-normal phase transitionby varying I . A typical 1DJ is realized in the form of a 1.5 µ m-long ( l ), 100 nm-wide ( w ) and 25 nm-thick ( t ) Al/Cubilayer nanowire-like active region sandwiched between the two Al electrodes. The detailed fabricationprocedure is described in Sec. 6. To ensure that the JJ is one-dimensional ( ξ A > t , w and λ L , A > t , w ), a fullspectral characterization of A is necessary. To this end, the test device is equipped with two additionalAl tunnel probes, as shown by the false-color scanning electron micrograph (SEM) in Fig. 4(a). The IV tunnel characteristics of A are performed by applying a voltage ( V ) and measuring the current ( I ) flowingbetween one lateral electrode and a tunnel probe. The experimental set-up is described in detail in Sec. 6.The energy gap of a superconductor is temperature independent up to T ∼ T C thus implying ∆ ( T ) = ∆ , with ∆ its zero-temperature value [27]. Since aluminum thin films typically show a T C ≥ of 15 (b) (c)500 nm(a) V A -0.25 0 0.25 0.5 -30 -
30 20 mK
250 mK I ( n A ) -0.5 V (mV) I ( p A )
150 250
20 mK250 mK V (µV) I
23 µV
Figure 4. Measurement of the density of states in a 1DJ. ( a ) False-color scanning electron micrographof 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. To this enda 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 toextract ∆ A ,0 (cid:39) µ eV and ∆ P ,0 (cid:39) µ eV as the crossing between the black dotted lines and I = bilayer). As a consequence, this experimental set-up can be employed to study the superconductingproperties of A . In particular, the zero-temperature energy gap ( ∆ A ) will be helpful to demonstrate theone-dimensionality of the nanowire.To obtain ∆ A , the IV characteristics were measured at base temperature ( T =
20 mK) and well abovethe expected critical temperature of A but below 0.4 T C , Al ( T =
250 mK), as shown in Fig. 4(b). At the basetemperature, both A and P are in the superconducting state. Therefore, the voltage bias needs to reach V = ± ( ∆ A ,0 + ∆ P ,0 ) / e (with ∆ P the zero-temperature gap the Al probe) to switch to the normal-state[32]. On the contrary, at T bath =
250 mK the nanowire is in the normal-state thus the transition occurs at V = ± ∆ P ,0 / e . The resulting zero-temperature energy gap of the Al probe is ∆ P (cid:39) µ eV [see the blowin Fig. 4(c)], therefore indicating a critical temperature T C , P = ∆ P ,0 / ( k B ) (cid:39) ∆ A ,0 (cid:39) µ eV thus indicatinga critical temperature T C , A (cid:39)
150 mK.A 1DJ requires that the intrinsic superconducting properties of the nanowire are uniform and dominateover the proximity effect induced by the lateral banks. The latter could induce an energy gap in anon-superconducting Al/Cu bilayer given by E g (cid:39) hD A / l (cid:39) µ eV [33], where D A is the diffusionconstant of the active region. The latter ca be calculated as D A = ( t Al D Al + t Cu D Cu ) / ( t Al + t Cu ) (cid:39) × − m /s, where t Al = D Al = × − m s − are the thickness and the diffusionconstant of the Al thin film, respectively, while t Cu =
15 nm and D Cu = × − m s − are the thicknessand the diffusion constant of the Cu layer, respectively. Since E g ∼ ∆ A ,0 , the superconductingproperties of A are dominated by the Al/Cu bilayer.If the Al/Cu bilayer lies in the Cooper limit [34,35], it can be considered a uniform superconductor.The Cooper limit has two requirements: negligible contact resistance between the two layers and thicknessof each layer lower than its coherence length. Since its large surface area, the Al/Cu interface resistanceis negligibly small with comparison to the nanowire normal-state resistance, thus fulfilling the firstrequirement. In addition, the superconducting Al film fulfils ξ Al = √ ¯ hD Al / ∆ Al (cid:39)
80 nm (cid:29) t Al = ∆ Al (cid:39) µ eV is its measured superconducting energy gap. At the same time, the Cu layer obeysto ξ Cu = (cid:112) ¯ hD Cu / ( π k B T ) (cid:39)
255 nm (cid:29) t Cu =
15 nm, where k B is the Boltzmann constant and T = A can be considered as formed from asingle superconducting material. of 15 (c) I/I C Sample T e ( m K ) I/I C Sample δ T C ( m K ) (b)500 nm R L IV ac V out (a) (b) T (mK) 12060300 R ( Ω )
80 90 180 I (nA)15 Sample Figure 5. Current control of the resistance versus temperature characteristics in a 1DJ. ( a ) False-coloredscanning electron micrograph of a 1DJ. The nanowire is made of an Al/Cu bilayer (yellow) separating twothick Al electrodes (blue). The 1DJ is AC current-biased (amplitude I ), while the voltage drop across thewire ( V out ) is measured through a lock-in amplifier. The load resistor ( R L (cid:29) R N ) guarantees constant biascurrent 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 twodifferent samples. ( b ) Width of the phase transition ( δ T C ) versus I for two different 1DJs. We can now discuss the one-dimensionality of A . In particular, the superconducting coherence lengthin A is given by ξ A = (cid:113) l ¯ h / [( t Al N Al + t Cu N Cu ) R N e ∆ A ,0 ] (cid:39)
220 nm, where R N = Ω is the nanowirenormal-state resistance, N Al = × J − m − and N Cu = × J − m − are the density of statesat the Fermi level of Al and Cu, respectively. Since the Cooper pairs coherence length in A is much largerthan its thickness ( ξ A (cid:29) t = t Al + t Cu = ξ A (cid:29) w =
100 nm. In addition, the London penetration depth for the magnetic field of A takes the form λ L , A = (cid:112) ( ¯ hwt A R N ) / ( πµ l ∆ A ,0 ) (cid:39)
970 nm, where µ is the magnetic permeability ofvacuum. Therefore, the nanowire is 1D with respect to the London penetration depth, since λ L , A (cid:29) t , w .Concluding, the Al/Cu bilayer embedded between two Al electrodes forms a 1DJ. Therefore, thisstructure can be used to investigate the impact of I on the R ( T ) characteristics. To investigate the impact of the bias current on the transport properties of a 1DJ, the resistance R vs temperature characteristics were obtained by conventional four-wire low-frequency lock-in techniqueby varying the excitation current amplitude from 15 nA to 370 nA . The current was generated byapplying a voltage ( V ac ) to a load resistor ( R L ) of impedance larger than the device resistance ( R L = Ω (cid:29) R N (cid:39) Ω ), as shown in Fig. 5(a). For the details regarding the device fabrication and experimentalset-up see Sec. 6.The magnetic field generated at the wire surface by the maximum employed bias current is B I , max = µ I max / ( π t ) (cid:39) µ T, where I max =
370 nA and µ is the vacuum magnetic permeability. This value is of 15 (a) (b) phonons, T bath T bath T A T bath P in P e-ph P A-B P A-B I Bias RR S LI Figure 6. General thermal and electrical model of the nano-TES and JES. ( a ) Schematic representationof a typical biasing circuit for the nano-TES and JES. The parallel connection of the sensor (of variableresistance R ) and the shunt resistor ( R S ) is biased by the current I Bias . The role of R S is to limit the Jouleoverheating of A when transitioning to the normal-state. The variations of the current ( I ) flowing throughthe sensing element are measured thank to the inductance L . For instance, an inductively-coupled SQUIDamplifier could serve as read-out element. ( b ) Thermal model of the nano-sensors where the main thermalexchange channels are shown. P in is the power coming from the incoming radiation, P e − ph is the heatexchanged 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 . orders of magnitude lower than the critical magnetic field of A that was measured to be about 21 mT [26].So, the self-generated magnetic field does not affect the properties of the 1DJ.The resistance versus temperature characteristics shift towards low temperatures by rising the currentfrom ∼
3% and ∼
65% of I C ,0 . In addition, the R ( T ) characteristics preserve the same shape up to thelargest bias currents. The use of an AC bias allowed to resolve the R vs T characteristics near the criticaltemperature. In fact, values of DC bias higher than the retrapping current [36] ( I R , that is the switchingcurrent from the resistive to the dissipationless state) would cause the sudden transition of the deviceresistance to R N . Instead, the AC bias has always a part of the period lower than I R thus enabling theprecise measurement of the entire R ( T ) traces.The electronic temperature of the nanowire ( T A ) at the middle of the phase transition under currentinjection does not coincide with T bath , since Joule dissipation (for R (cid:54) =
0) causes the quasiparticlesoverheating in A yielding T A > T bath [32]. Therefore, from the R vs T curves we can only specify thecurrent-dependent escape temperature [ T e ( I ) ]. The values of T e are shown in Fig. 5(c) as a function of I / I C for two different samples. The escape temperature is monotonically reduced by rising the bias currentwith a minimum value ∼
20 mK for I =
370 nA, that is ∼
15% of the intrinsic critical temperature of theactive region, T iC ∼
130 mK.The superconducting-to-normal-state transition ( δ T C ) narrows in temperature by increasing thecurrent injection, as depicted in Fig. 5(d). In particular, δ T C is suppressed by a factor of 4 at the largest biascurrent. We stress that this behavior is in full agreement with the theoretical behavior of a 1DJ shown inSec. 2. Therefore, in the following we will focus on the detection properties of a 1DJ.
4. Operation principle of the nano-TES and JES
The 1DJ was employed to design two single-photon detectors operating in the GHz band: thenanoscale transition edge sensor (nano-TES) [25] and the Josephson escape sensor (JES) [26]. These sensorstake advantage of the strong resistance variation of the superconducting nanowire while transitioningto the normal-state, such as in a conventional TES [15]. Differently from the TES, the sensitivity of thenano-TES and the JES can be in situ controlled, since the resistance versus temperature characteristics ofa 1DJ can be tuned by varying the bias current. As a consequence, the 1DJ serves as the active region of of 15 these sensors. The main difference between the nao-TES and the JES is the operating temperature. Indeed,the nano-TES operates at T C , i.e., at the middle of the superconductor-to-normal-state transition [see Fig.3(a)], while the JES operates at T e , i.e., deeply in the superconducting state. Notably, these temperaturescan be very different at large bias currents [see Fig. 3(b)].For both sensors, the absorption of radiation provokes an increase of the electronic temperature in thesuperconducting nanowire ( T A ) thus driving its transition to the normal-state. The latter would generateJoule heating in the active region when biased with a constant current with consequent thermal instability.To solve this issue, the the nano-TES and the JES could be biased with the circuitry shown in Fig 6(a).The shunt resistor ( R S ) limits the current ( I ) flowing through the sensor ( R ) when the A undergoes thesuperconducting-to-normal-state transition. This is called negative electrothermal feedback (NEFT) [15]For the nano-TES, the sensor is biased at T C ( R = R N /2), therefore the condition for the shunting resistorreads R S = IR N / [ ( I Bias − I )] , where I Bias is the current provided by the generator. For the JES, the deviceis operated at T e ( I ) ,i.e., at R =
0, and the role of R S is to limit the current flow through the sensing elementbelow I R . This happens for R S ≤ R N I R / I bias and brings A quickly back to the superconducting state afterradiation absorption. Therefore, the sensor always operates in the superconducting state. For both thenano-TES and the JES, the variations of I , due radiation absorption, can be measured via a conventionalSQUID amplifier coupled to the inductance L [25].The ability of a superconducting sensor to resolve a single-photon depends on their ability to convertthe power of the incoming radiation into a change of electronic temperature in the active region. Thelatter is related to the predominant thermal exchange mechanisms occurring in A . Figure 6(b) shows thethermal model describing the active region of both the nano-TES and the JES, where P in is the powerassociated to the external radiation, P e - ph is the heat exchange with lattice phonons, and P A - B represents theenergy out-diffusion for the active region to the lateral leads. When the critical temperature of the lateralelectrodes ( T C , B ) is much higher than operating temperature ( T C for the nano-TES and T e for the JES), theybehave as energy filters, the so-called Andreev mirrors [37], thus ensuring perfect thermal insulation of A ( P A - B → P e - ph is the predominant thermal relaxation channel in the active region.See Ref. [25] for the application limits of this assumption.For the nano-TES, the active region operated almost in the normal-state (at R N /2). Therefore, theelectron-phonon coupling of a normal-metal diffusive thin film ca be employed [15,32] P e - ph , n = Σ A V A (cid:16) T A − T bath (cid:17) , (6)where V A is the volume of A , while Σ A is its electron-phonon coupling constant. The resulting thermalconductance for the active region of a nano-TES ( G th , nano - TES ) can be calculated through the temperaturederivative of the electron-phonon energy relaxation [15] G th , nano - TES = d P e - ph , n d T A = Σ A V A T A . (7)Differently, the JES operates deeply in the superconducting state at T e with A . Therefore, at verylow temperatures the electron-phonon heat exchange is exponentially suppressed with respect to thenormal-state [38] P e - ph , s ∝ P e - ph , n exp [ − ∆ A / ( k B T A )] , (8)where ∆ A is the superconducting energy gap in A . The thermal conductance of the active region of a JES(operating in the superconducting state) takes the form [39] G th , JES ≈ Σ A V A T e ς ( ) (cid:20) f (cid:18) ∆ (cid:19) cosh ( ˜ h ) e − ˜ ∆ + π ˜ ∆ f ( ∆ ) e − ∆ (cid:21) , (9)where the first term refers to the electron-phonon scattering, while the second term stems from therecombination processes. In Eq. 9, ς ( ) is the Riemann zeta function, ˜ ∆ = ∆ A / k B T is the normalizedenergy gap of A , ˜ h = h / k B T represents exchange field (0 in this case), f ( x ) = ∑ n = C n x n with C ≈ C ≈ C ≈ C ≈ f ( x ) = ∑ n = B n x n with B = B = B =
5. Single-photon detection performance of the nano-TES and the JES.
Microwave LSW experiments for axions search require single-photon detectors of frequency resolutionon the order of a few GHz. In the next sections, we will show all the theoretical relations describing thesensing properties of a nano-TES and a JES single-photon detector and the performance deduced from theexperimental data.
In order to determine the performances of a sensor in single-photon detection, the frequencyresolution is the most used figure of merit, since it defines the lowest energy that the detector can reveal.For a nano-TES, it can be written [15] δν TES = h (cid:115) α (cid:114) n k B T C C e , nano − TES , (10)where ¯ h is the Planck constant, α = d R d T TR is the electrothermal parameter accounting for sharpness of thephase transition from the superconducting to the normal-state [15], n = C eTES is the electron heat capacitance. It is interesting to note the stronglydependence on α value which determines the NETF mechanism [15].Since the nano-TES operates at the critical temperature, the electron heat capacitance of the activeregion is written [32] C e , nano - TES = γ A V A T C , (11)where γ A is the Sommerfeld coefficient of A .The time response of the detector defines the speed of the read-out electronics necessary to correctlyreveal the incoming single-photon. By considering the circuitry implementing the NETF [see Fig. 6(b)], thepulse recovery time takes the form [15] τ e f f = τ nano - TES + α n , (12)where τ nano - TES is the intrinsic recovery time of A . The latter can be calculated by solving the timedependent energy balance equation that takes into account all the exchange mechanisms after radiationabsorption [32]. The re-thermalization of the quasiparticles to the equilibrium depends exponentially on time with a time constant ( τ nano - TES ) given by the ratio between the thermal capacitance and the thermalconductance of A τ nano - TES = C e , nano - TES G th , nano - TES . (13)Since α (cid:29) n , the pulse recovery time is much shorter than the intrinsic time constant of A ( τ e f f << τ nano - TES ). Therefore, the overheating into the active region is decreased by the NETF, thus compensatingfor the initial temperature variation and avoiding the dissipation through the substrate.
Since the current injection does not change the energy gap of the active region ( ∆ A ∼ const), only theeffective critical temperature of A changes with I , while the intrinsic values of critical temperature ( T iC ) isunaffected. As a consequence, being at T e ( I ) , the JES operates deeply in the superconducting state, thusensuring high sensitivity (the thermalization is exponentially suppressed by the energy gap, see Eqs. 8and 9).The frequency resolution of a JES ( δν JES ) can be calculated from [24] δν JES = h (cid:113) k B T e C e , JES . (14)The electron heat capacitance needs to be calculated at the current-dependent escape temperature [ T e ( I ) ],thus in the superconducting state, and takes the form C e , JES = γ A V A T e Θ Damp = C e , nano - TES Θ Damp , (15)where the electronic heat capacitance is given by C s = γ A T iC (cid:18) ∆ A k B T e (cid:19) − e − ∆ A / k B T e . (16)Furthermore, Θ Damp is the low temperature exponential suppression with respect to the normal metalvalue, an it takes the form [40] Θ Damp = C s γ A T e . (17)Since the JES does not operate at the middle of the superconducting-to-normal-state transition, theJES speed does not depend on the electrothermal parameter. Indeed, it is given by the relaxation half-time( τ ), which reads [24] τ = τ JES ln 2, (18)where τ JES is the JES intrinsic thermal time constant. The latter is calculated by substituting the JESparameters in Eq. 13, thus considering C e , JES and G th , JES in deep superconducting operation.
In this section, we show the sensing performance deduced for two different 1DJs (samples 1 and 2 ofFig. 5) operated both as nano-TES and JES.Table 1 resumes the figures of merit calculated in case of nano-TES operation. The intrinsic relaxationtime of the active region is limited by G th , nano − TES to a few microseconds for both devices ( τ (cid:39) µ sand τ (cid:39) µ s). On the one hand, the electron heat capacitance is C e , nano − TES = × − J/K and
Sample T c τ τ e f f δν ν / δν (mK) ( µ s ) ( µ s ) (GHz) 100 GHz 300 GHz1 128 6 0.01 100 1 32 139 5 0.2 540 0.18 0.55 Table 1. Main figures of merit deduced for the nano-TES.
The time constant τ , the pulse recovery time τ e f f , the frequency resolution δν , and the resolving power ν / δν (at 100 and 300 GHz) are reported for twofabricated nano-TESs. the thermal conductance takes value G th , nano − TES = × − W/K for sample 1. On the other hand, C e , nano − TES = × − J/K and G th , nano − TES = × − W/K for sample 2. The thermal responseof the nano-TES full detector is strongly damped from the eletrothermal parameter. Consequently, τ e f f isfrom one to two orders of magnitude smaller than the thermal response time ( τ e f f << τ ). In particular, thedetector response time is τ e f f ,1 = µ s and τ e f f ,2 = µ s for sample 1 and sample 2, respectively. Thefrequency resolution depends on the electrothermal parameter ( α − ), too. Therefore, the two nano-TESsshow different values of δν . In particular, δν (cid:39)
100 GHz ( δ E (cid:39) δν (cid:39) δ E (cid:39) ν / δν ) reaches valueslarger than 1 for ν ≥
100 GHz for sample 1.The performance in the JES operation are expected to strongly depend on he bias current. Indeed, Fig.7(a) emphasizes the variations over 3 orders of magnitude of δν JES on I . The best frequency resolutionis ∼ ν / δν JES ). Figure 7(b) shows ν / δν JES calculated as a function of the frequency of the incident photons. In particular, ν / δν JES can reach ∼
80 at 100 GHz and ∼
240 at 300 GHz for 370 nA.The dependence of the JES time constant ( τ ) on I is shown in Fig. 7(c). In particular, τ monotonically increases by rising I , and varies between ∼ µ s at low current amplitude and ∼
100 ms at370 nA. Notably, these values are orders of magnitude larger than that of nano-TESs. As a consequence,the read-out of single photons with the JES allows to employ slower and thus cheaper electronics.Concluding, both the nano-TES and the JES show frequency resolutions enabling the search of axionsthrough LSW experiments in the microwave frequency band. In particular, the JES allow to performexperiments in a wide range of energies down to about 8 µ eV. Furthermore, the slow response time of theJES would simplify the read-out circuitry involved in the experiment.
6. Materials and Methods
All the devices presented in this review were fabricated by electron-beam lithography (EBL) and3-angles shadow evaporation through a suspended resist maske onto a silicon wafer covered with300-nm-thick SiO thermally grown on an intrinsic silicon wafer. To obtain the resist suspended mask, abilayer composed of a 950-nm-thick MMA(8.5)MMA layer and a PMMA (A4, 950k) film of thickness ofabout 300 nm was spin-coated on the substrate. The ratio between the electron irradiation doses to makethe resists soluble is DOSE
MMA : DOSE
PMMA (cid:39) − Torr by keeping the target substrateat room temperature. First, 13-nm-thick Al layer was evaporated at an angle of -40 ◦ . Second, the film wasthen oxidized by exposition to 200 mTorr of O for 5 minutes to obtain the tunnel probes of the devicedevoted to the spectral and the thermal measurements. Third, the Al/Cu bilayer ( t Al = -6 -3 -9 τ ( s ) I/I C (c)0.7(a) I/I C δ ν J E S ( H z ) (b) ν (GHz) 10 ν / δ ν J E S TESJES I C =575 nA10 -2 I (nA)= Figure 7. Performance calculated for the JES. ( a ) Frequency resolution δν JES as a function of I for sample1 (blue triangles) and 2 (yellow squares). The dashed lines indicate the frequency resolutions of thecorresponding nano-TESs. ( b ) Resolving power ( ν / δν JES ) as a function of the frequency ( ν ) of the incidentsingle-photons calculated for sample 1. The dashed lines indicate the resolving power of the correspondingnano-TES. ( c ) Time constant versus bias current for sample 1 (blue triangles) and 2 (yellow squares). Thedashed lines indicate the time constants of the corresponding nano-TESs. t Cu =
15 nm) forming the superconducting nanowire is evaporated at an angle of 0 ◦ . Fourth, a second40-nm-thick Al film was evaporated at an angle of + ◦ to obtain the lateral electrodes completing the 1DJ.The angle resolution of each evaporation was ∼ ◦ . The average film thickness can be controlled duringthe evaporation process with the precision of 0.1 nm at the evaporation rate of about 1.5 angstrom/s. The electronic and the spectral characterizations presented in this review were performed at cryogenictemperatures in a He- He dilution refrigerator equipped with RC low-pass filters (cut-off frequency ofabout 800 Hz). The lowest electronic temperature obtained was 20 mK.The bias current tuning of the transport properties of the 1DJ is realized by standard lock-in technique.The AC current bias is produced by applying a voltage V ac at a frequency 13.33 Hz to a load resistance R L =
100 k Ω ( R l >> R N ) in order to obtain a bias current independent from the resistance of the 1DJ. Thevoltage drop V the device is measured as a function of T bath via a voltage pre-amplifier connected to alock-in amplifier. The use of a pre-amplifier allows to decrease the noise in the measurement. The controlof the transport properties of the 1DJ is thus performed by varying V ac .The energy gap of the superconducting nanowire was determined by tunnel spectroscopy. To thisend, a voltage was applied between one tunnel probe and one lateral electrode by means of a low noiseDC source, while the flowing current was measured through a room temperature current pre-amplifier.
7. Conclusions
This paper reviewed two innovative hypersensitive superconducting radiation sensors: the nanoscaletransition edge sensor (nano-TES) and the Josephson escape sensor (JES). Both devices are based on aone-dimensional Josephson junction (1DJ) with the capability to in situ fine tune their performance bysimple current bias. The nano-TES and the JES have the potential to drive single-photon detection in thegigahertz band towards unexplored levels of sensitivity. In fact, the nano-TES shows a frequency resolutionof about 100 Ghz, while the JES is able to resolve single-photons down to 2 GHz. Therefore, these sensorsare the perfect candidate for the implementation of light shining through walls (LSW) experiments for thesearch of axions operating at micro- and milli-electroVolt energies. Furthermore, the nano-TES and the JEScould have countless applications in several fields of quantum technology where single-photon detectionis a fundamental task, such as quantum computation [41] and quantum cryptography [42,43].
Funding:
This research was funded by the European Union’s Horizon 2020 research and innovation programme underthe grant No. 777222 ATTRACT (Project T-CONVERSE) and under FET-Open grant agreement No. 800923-SUPERTED,and by the CSN V of INFN under the technology innovation grant SIMP.
Acknowledgments:
We acknowledge N. Ligato, G. Germanese, V. Buccheri, P. Virtanen, P. Spagnolo, C. Gatti, R.Paoletti, F.S. Bergeret, G. De Simoni, E. Strambini, A. Tartari, G. Signorelli and G. Lamanna for fruitful discussions.
Conflicts of Interest:
The authors declare no conflict of interest.
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Nat. Photonics2019