A self-calibrating superconducting pair-breaking detector
AA self-calibrating superconducting pair-breaking detector
E. T. Mannila, ∗ V. F. Maisi, and J. P. Pekola QTF Centre of Excellence, Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland Physics Department and NanoLund, Lund University, Box 118, 22100 Lund, Sweden (Dated: February 5, 2021)We propose and experimentally demonstrate a self-calibrating detector of Cooper pair depairingin a superconductor based on a mesoscopic superconducting island coupled to normal metal leads.On average, exactly one electron passes through the device per broken Cooper pair, independentof the absorber volume, device or material parameters. The device operation is explained by asimple analytical model and verified with numerical simulations in quantitative agreement withexperiment. In a proof-of-concept experiment, we use such a detector to measure the high-frequencyphonons generated by another, electrically decoupled superconducting island, with a measurablesignal resulting from less than 10 fW of dissipated power.
Introduction.
A key prediction of the Bardeen-Cooper-Schrieffer theory of superconductivity is the ex-istence of an energy gap ∆ for single-particle excitations.The resulting exponentially suppressed density of ther-mal excitations makes superconductors very sensitive toradiation at frequencies higher than 2∆ /h with h thePlanck constant, which, although detrimental for super-conducting circuits used in quantum computing [1–3], en-ables applications as detectors. Pair-breaking supercon-ducting detectors, such as superconducting tunnel junc-tion [4], kinetic inductance [5, 6] and quantum capac-itance [7] detectors have found use in physics and as-tronomy, enabling single-photon detection at optical [8]and terahertz [7] frequencies. They can also be used asphonon-mediated detectors [9, 10], and tunnel junctiondetectors have been used for phonon spectroscopy [11–14].In all of these devices, inferring the number of bro-ken Cooper pairs from the measured response requirescalibration or modeling. In contrast, in this Letter wepresent a mesoscopic pair-breaking detector whose re-sponse is given simply by a current I = e Γ pb , (1)where e is the elementary charge and Γ pb is the rate atwhich Cooper pairs are broken. We describe the deviceoperation with a simple analytical model, which agreeswith the predictions of a full numerical model in quantita-tive agreement with experiment. In our proof-of-conceptexperiment, we measure the response of the detector topair-breaking phonons emitted by another superconduct-ing island, while ruling out that the response could be dueto non-pair-breaking mechanisms by comparison with anormal-metallic reference source. We extract the frac-tion of phonons transmitted from emitter to detectorand find that dissipated power as low as 10 fW in theemitter is enough to create a measurable signal in thedetector. Although the number of Cooper pairs gener-ated per absorbed phonon or photon depends on the fre-quency [15], due to its well-defined absorption volumeand self-calibrating operation we foresee our device as Γ pb (kHz) I = e Γ pb N =-1 N =2 N =0 N =1 Γ pb (a) (b) V g I e m i tt e r IV b ,emitter V b (c) (d) N QP Γ tunn I ( f A ) V b N QP Γ tunn detector2 µmphononemitter referencesource I ref V g ,emitter FIG. 1. (a) Sketch of device. Cooper pairs are broken whenradiation is absorbed in a superconducting island, and the re-sulting quasiparticles decay by tunneling to the normal metalleads (solid arrows) with a rate N QP Γ tunn for N QP excitationson the island. (b) Energies of states with N = − , , , n g = 0 .
5. Solid arrows in-dicate transitions which remove a quasiparticle through theleft (solid black arrows) or right (dashed black) tunnel junc-tion, while red arrows indicate transitions which add a quasi-particle into the island. The quasiparticle population is setby transitions between charge states N = 0 ,
1, while netcurrent flows via cycles involving the excited charge states N = − , +2. (c) Numerical verification of operation. Theideal response I = e Γ pb (solid line) is reproduced by our simu-lations (dashed line) within 1% up to Γ pb = 500 kHz. Symbolsshow the experimental response, where Γ pb is linearly propor-tional to the measured I emitter . (d) Proof-of-concept exper-iment and sketch of measurement setup, with pair-breakingdetector (top), superconducting phonon emitter (bottom left)and normal-metallic reference source (bottom right). particularly useful for studying propagation of athermalphonons. This is important in detectors [16–18] as well asapplications in quantum information based on supercon-ducting circuits, where phonons may cause quasiparticlepoisoning over large distances [19, 20]. a r X i v : . [ c ond - m a t . s up r- c on ] F e b Operating principle.
Our device, a superconductingisland with charging energy E C smaller than the super-conducting gap ∆, is sketched in Fig. 1(a). We oper-ate at low temperatures k B T (cid:28) E C , ∆ such that theprobability of thermally excited quasiparticles is negligi-ble. Incident radiation (pink) breaks Cooper pairs in amesoscopic superconducting aluminum island (blue) ata rate Γ pb . The resulting quasiparticle excitations (lightblue circles) relax by tunneling to the normal metal leadsthrough the left or right tunnel junctions with a rate N QP Γ tunn for N QP excitations on the island. Since relax-ation can happen equally likely through either junction,this process carries no current, but sets the time-averagedquasiparticle population to h N QP i = Γ pb / Γ tunn . (2)As quasiparticles can tunnel out as both electrons andholes, the tunneling events occur between the lowest-energy charge states with N = 0 , n g ≈ .
5. Since we consider the situation wherequasiparticles are present on the island, we do not needto consider a free energy cost for occupying states withodd N .In the presense of a small applied bias voltage V b < ∆ /e , quasiparticles can also tunnel out in the directionof the bias to the higher-energy charge states N = − N QP Γ tunn . Due to the high energy cost of occupy-ing these states, the island will then return to its previouscharge state near-instantaneously, when a new quasipar-ticle tunnels in through the left (right) tunnel junction.This occurs again in the direction of the bias, as indi-cated by the red arrows in Fig. 1(b), leading to onenet electron transported through the device. The cy-cles N = 0 → − → N = 1 → → N QP Γ tunn . Combined with Eq. (2), we obtain the resultof Eq. (1): I = e h N QP i Γ tunn = e Γ pb . Here, we haveassumed equal tunneling rates in both junctions for sim-plicity, but the result holds for unequal junctions as well(see supplement). Because the quasiparticle tunnelingrates are independent of energy [21], the current due topair-breaking forms diamond-shaped plateaus as a func-tion of V b and n g . Although the size and location of theseplateaus depends on E C , they exist for all E C < ∆ (seesupplement) and neither E C , V b nor n g need to be tunedprecisely for the self-calibrating operation.We validate this simple picture by performing numeri-cal simulations based on a rate equation tracking the oc-cupation probabilities of states with N excess electronsand N Q P quasiparticle excitations on the island [22]. Thesimulations incorporate single-electron and Andreev tun-neling at finite temperature of the normal metal leads, as well as quasiparticle recombination through the electron-phonon coupling (see Ref. [22] and supplement for de-tails). We find that with our device parameters, the effectof finite temperature and Andreev tunneling is negligi-ble around n g = 0 . | V b | < µ V. The non-zeroelectron-phonon recombination rate, scaling as Γ R N with the prefactor Γ R depending on the device parame-ters, reduces the quasiparticle population from the valueof Eq. (2) and the current response. The condition thatrecombination be negligible compared to relaxation bytunneling, Γ R h N QP i (cid:28) Γ tunn h N QP i , can be expressedas Γ pb (cid:28) ζ (5) k B Σ V ∆ e R T , (3)where Σ is the electron-phonon coupling constant, V isthe absorber volume, ζ is the Riemann zeta function, and k B is the Boltzmann constant (see supplement). For ourdevice parameters, the right-hand side evaluates to 70MHz, corresponding to femtowatts of absorbed power.The simulated current is within 1% of the ideal value upto 500 kHz, well above the values of Γ pb reached in theexperiment, as shown in Fig. 1(c). Proof-of-concept experiment.
A scanning electron mi-crograph of our proof-of-concept device is shown in Fig.1(d). The pair-breaking detector is an aluminum islandwith V = 0 . × . × . µ m , ∆ = 200 µ eV and E C = 92 µ eV, tunnel coupled to normal metallic copperleads. We fabricate a superconducting phonon emitterand normal metallic reference source on the silicon sub-strate simultaneously with the detector. Measurementswere performed in a plastic dilution refrigerator at a basetemperature of 40 mK. In measurements with the alu-minum in the normal state, we extract device param-eters, including the electron-phonon coupling constantΣ Al ≈ × W K − m − , and verify that heat conduc-tion through the substrate by thermal phonons is negli-gible as long as the dissipated power is below 1 pW. (SeeSupplement for details on fabrication and normal-statecharacterization.)Figure 2 presents the operation of the self-calibratingdetector in the superconducting state. When the cur-rent through both the phonon emitter and the referencesource is zero, the current at n g = 0 . V b is zerowithin the measurement accuracy of roughly 1 fA [Fig.2(a-c)] as expected. At around n g = 1, the current isfinite due to Andreev reflection [23, 24]. When we in-crease the current through the phonon emitter to 30 pA[Figs. 2(d-f)] or 120 pA [Figs. 2(g-i)], the current levelat the plateau around n g = 0 . | V b | ≈ µ V in-creases, as predicted by our simple model. The currentincreases also around n g = 0, which is due to Andreevcurrent flowing once the odd charge states are populateddue to quasiparticles [25, 26].In contrast, when we increase the current through thereference source up to 7.5 nA, while the phonon emitter -1 0 10204060 - I ( f A ) -1 0 1 2-200-1000100200 V b ( µ V ) -2 -1 0 1 2 -1 0 10204060 - I ( f A ) -1 0 1 2-200-1000100200 V b ( µ V ) -2 -1 0 1 2 -1 0 10204060 - I ( f A ) V b ( µ V ) -2 -1 0 1 2 -1 0 10204060 - I ( f A ) -2 -1 0 1 2-200-1000100200 V b ( µ V ) -2 -1 0 1 2 experiment simulation experiment+simulation(k) T N = 50 mK n g n g n g (g) I emitter = 120 pA n g n g n g (d) I emitter = 30 pA n g n g n g (c) n g n g n g I (A) -15 -14 -13 -12 -11 I (A) -15 -14 -13 -12 -11 I (A) -15 -14 -13 -12 -11 I (A) -15 -14 -13 -12 -11 (b) Γ pb = 50 Hz(e) Γ pb = 12 kHz(h) Γ pb = 56 kHz (i) I emitter = 120 pA, Γ pb = 56 kHz(f) I emitter = 30 pA, Γ pb = 12 kHz(c) I emitter = 0, Γ pb = 50 Hz(l) I ref = 7.5 nA, T N = 50 mK(j) I ref = 7.5 nA(a) I emitter = 0 FIG. 2. Measured (left column and symbols in right column) and simulated (middle column and solid lines in right column)subgap current through our pair-breaking detector. On the first row, no current is applied to the emitter or reference source,while the current increases substantially when the current through the phonon emitter is increased to 30 pA (second row) or120 pA (third row). This is reproduced quantitatively in the simulations by changing only the rate of pair-breaking radiationΓ pb . In contrast, increasing the current through the reference emitter up to 20 nA (bottom row) creates nearly no change inthe detector response, proving that the response is indeed due to pair-breaking phonons. White dashed lines in panels (d,e,g,h)indicate the regions where the self-calibrating response is expected. is kept grounded [Fig. 2(j-l)], the measured current stayszero within the 2 e -periodic Coulomb diamonds. Themain difference between the phonon emitter and the ref-erence source is that the emitter island is superconduct-ing aluminum, which will emit phonons whose energydistribution is peaked above 2∆ ≈ k B × pb for dif-fering I emitter , and in Fig. 3, we see that a linear relationΓ pb = AI emitter /e with A = 7 . × − is a good fit atall but the lowest emitter currents. Thus our simulationaccounts for all the relevant processes, which verifies thatthe self-calibrating mode can be used in experiments.Next, we turn to measure the phonon transmissionfrom the phonon emitter to the detector. Modeling of therecombination phonon emission (see supplement) allowsus to estimate the emission rate of phonons with energy ≥
2∆ as Γ = ηI emitter /e with the proportionality con-stant η = 0 .
68. We find that the measured detector cur-rent is also linear in I emitter , as shown in Fig. 3(b). Hencewe extract the proportion of phonons emitted that are ab- I ( f A )
0 100 200 300 400 5000204060 I ( f A ) pb (kHz) n g = 1 n g = 0 n g = 0.5 phonon emitter referencesource (a)(b) (c) -1 0 1 n g I ( f A ) pb ( k H z ) -12 -10 -8 I emitter / I ref (A)
62 41161114249412 I emitter (pA) I emitter (pA) FIG. 3. (a) Detector current versus gate offset n g at V b =110 µ V with varying I emitter in the experiment (symbols) anddifferent Cooper pair breaking rates Γ pb = AI emitter /e in thesimulations (solid lines). (b) I extracted at the self-calibratingoperating point ( n g = 0 .
5, circles) and for comparison at n g = 1 and n g = 0, where the current is due to Andreevreflection. (c) Measured I at n g = 0 . sorbed in the detector as x = Γ pb / Γ ≈ . × − . Thisfraction is difficult to estimate by other means due tothe long mean free paths of the phonons at low tempera-tures, but the order of magnitude is in line with measure-ments in Refs. [13, 14]. We also quantify the difference inCooper pair breaking caused by the phonon emitter andreference source in Fig. 3(c). The reference source causesno detectable signal until currents of almost 20 nA areapplied, even when we obtain a measurable signal fromas little as 16 pA passing through the superconductingphonon emitter, which corresponds to less than 10 fW ofdissipated power.As a detector of phonons with energy 2∆, our devicehas a noise equivalent power of 3 × − W/ √ Hz, basedon assuming that 25% of the incident phonons are ab-sorbed and the measured noise level (see supplement).This is over two orders of magnitude better than in themicroscale phonon detectors of Ref. [13], while our de-vice also provides the advantage of a self-calibrating op-erating mode and well-defined absorption volume. Thedevice performance could be enhanced at other operatingpoints once the response has been calibrated in the self-calibrating mode. In our device, the responsivity at low phonon flux increases by roughly a factor of 2 at n g = 0(Fig. 3(b)), and could be increased further at higher V b . Conclusions.
In conclusion, we have proposed and ex-perimentally implemented a mesoscopic superconductingdetector of high-frequency radiation leading to Cooperpair depairing. We have used the proof-of-concept de-vice to detect the nonequilibrium phonons emitted by an-other superconducting detector, while using a referencesource to rule out mechanisms other than pair breaking.Due to the well-defined microscale absorption volume andself-calibrating operation, our detector could be partic-ularly useful for studying athermal phonon propagation,relevant both in the context of low-temperature detec-tors [16–18] and in mitigating phonon-mediated quasi-particle poisoning of superconducting quantum circuits[19, 20, 29]. As a first step in this direction, we ex-tract the fraction of phonons transmitted from emitterto detector over a distance of 8 µ m. As low a power as10 fW dissipated in the phonon emitter caused a mea-surable increase in the Cooper pair creation rate on theabsorber island. Hence phonon-mediated poisoning is aplausible explanation of the charge detector backaction ofRefs. [30, 31], and our results highlight the importanceof avoiding dissipation in superconducting quantum de-vices. Finally, we expect that monitoring the individualquasiparticle relaxation events with a fast charge detec-tor as in [32] would enable detecting every single phononor photon absorbed.We acknowledge useful discussions with O. Mailletand J. T. Peltonen. This work was performed as partof the Academy of Finland Centre of Excellence pro-gram (project 312057). We acknowledge the provi-sion of facilities and technical support by Aalto Uni-versity at OtaNano - Micronova Nanofabrication Centreand OtaNano - Low Temperature Laboratory. 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Pekola QTF Centre of Excellence, Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland Physics Department and NanoLund, Lund University, Box 118, 22100 Lund, Sweden (Dated: February 5, 2021)This supplementary information includes details on sample fabrication and normal-state charac-terization, and modeling of phonon emission, and calculations for the limits on the operation of theself-calibrating mode.
DEVICE FABRICATION AND NORMAL-STATE CHARACTERIZATION
The three single-electron transistors in our experiment, shown in Fig. 1(d) of the main text, are fabricated onthe same silicon chip approximately 8 µ m from each other. The substrate is a 525 µ m thick silicon (100) boron-doped substrate (resistivity 0.001-0.005 Ωcm) that stays conducting at low temperatures, coated with 200 nm ofthermal silicon oxide. Due to the insulating oxide, the distance between the SETs, and the conducting substrate,there is no direct resistive or capacitive coupling between the islands of the SETs, although there is some capacitivecrosstalk between the bulky gate electrodes. The devices are fabricated in one deposition step with standard electron-beam lithography and two-angle evaporation. A thin aluminum oxide tunnel barrier separate between the 80 nmsuperconducting aluminum and 100 nm normal metal copper films, which form the island and leads, respectively, ofthe detector and phonon emitter. In the reference source, the island is copper and leads aluminum, but the designis otherwise as similar as possible to that of the phonon emitter. The phonon emitter island has a relatively largevolume V to obtain a large phonon emission rate with a given temperature.The silicon chip is attached with vacuum grease to a brass sample holder thermally anchored to the mixing chamberof a home-made plastic dilution refrigerator with a base temperature of 40 mK. The sample holder is closed with asingle threaded cap, which together with the ground plane formed by the conducting substrate suppresses photon-assisted tunneling due to stray microwaves [1, 2] adequately for this experiment, although the sample stage is possiblynot fully microwave-tight. The measurement wiring lines are filtered with approximately 2 m of Thermocoax [3]between the 1 K and mixing chamber stages. Direct current electrical measurements were performed with room-temperature voltage sources and current amplifiers. In the subgap measurements shown in Figs. 2 and 3 in the maintext, we have subtracted a leakage current linear in the gate voltage corresponding to a resistance of 10 Ω, mostlikely originating from the room-temperature breakout box.We characterize the devices with the aluminum driven into the normal state by a magnetic field of 70 mT appliedperpendicular to the substrate. For each device, we simultaneously fit current-voltage characteristics measured atbath temperatures T bath between 41 mK and 441 mK (Fig. S1(a-c)) with a standard rate equation model for anormal-metallic single-electron transistor. In the simulations, we allow the island temperature T is to deviate from T bath and solve T is self-consistently from the heat balance equation [4]˙ Q tunn = ˙ Q e-ph , (1)where ˙ Q tunn is the heat current due to the tunneling electrons. The heat flux to the phonon bath is given by˙ Q e-ph = Σ V ( T − T ) , (2)where Σ is the electron-phonon coupling constant and V the volume. From these fits, we determine the total tunnelresistance R T and charging energy E C = e / C Σ , where C Σ is the total capacitance of the island, given in Table I.The fits shown in Fig. S1 are obtained with Σ Al = 2 × WK − m − , consistent with results in the literature [5–7].However, our fit would be consistent with values Σ Al between 1 × to 10 × WK − m − and also power-lawexponents in Eq. 2 differing from 5. We determine ∆ from similar measurements in the superconducting state at zeromagnetic field. For the copper island of the reference source, we use the literature value Σ Cu = 2 × WK − m − [4, 8].In the numerical simulations used in the superconducting state shown in Figs. 2 and 3 in the main text, we useinstead a rate equation tracking the probabilities of having N excess electrons and N QP quasiparticles on the island,with the rates given in Eq. [9]. In addition to the parameters determined from large-scale IV fits, we fit the effective a r X i v : . [ c ond - m a t . s up r- c on ] F e b (a) T bath = 41 mK I ( n A ) (c) T bath = 296 mK (b) T bath = 160 mK P heater (W)10 -15 -12 -9 V b (µV)-200 0 200-1-0.500.51 -200 0 200 V b (µV)(d) P heater = 11 pW T sub = 81 mK -1-0.500.51 I ( n A ) T s ub ( m K ) FIG. S1. (a-c) Measured and simulated I - V b characteristics of the detector, measured in the normal state with a magneticfield B ⊥ = 70 mT applied parallel to the sample at different bath temperatures T bath . Gray dots show the measured currentwhen the gate voltage is swept over several periods of the gate offset n g , while black lines show simulations with T is solvedself-consistently at n g =0 (minimum current) and n g =0.5 (maximum current) with parameters given in Table I. Dashed linesshow simulations where the overheating of the islands is neglected and T is = T bath . Red circles show the maximum voltageat a current of 10 pA, used as a thermometer. (d) Detector I-V characteristics measured at base temperature but with 11pW heating power applied to the reference source, superimposed with simulations assuming a bath temperature of 81 mK.(e) Extracted substrate temperature when heater power P heater is applied to the phonon emitter (circles) or reference source(triangles) in the normal state at T bath = 40 mK.TABLE I. Device parameters. The tunnel resistance R T , charging energy E C and superconducting gap ∆ were determined byfitting large-scale IV curves measured in the normal ( R T , E C ) or superconducting state (∆) as described in the text. Junctionasymmetry is defined as the ratio between the areas and hence resistances of the two junctions in each device. The island andjunction areas were estimated from micrographs of the devices.Device Island material Σ (10 WK − m − ) V ( µ m ) ∆ ( µ eV) R T (kΩ) E C ( µ eV) Junction asymmetryDetector Al 2 0.9 × × × × × × conduction channel area A ch ≈
15 nm setting the magnitude of the Andreev current from the slope of the currentat n g = 1, consistent with the values of Ref. [10].We characterize the heat conduction of the substrate due to thermal phonons with the devices in the normal state.The current through a normal-metallic SET is sensitive to the temperature of the metals, and effectively we canuse one of the SETs as a Coulomb blockade thermometer [11], although in the unusual regime E C > k B T . As aprobe of the substrate temperature T sub , we use the maximum voltage drop over the detector at fixed bias currentof 10 pA, indicated by red circles in Figs. S1(a-c). The voltage drop is measured as a function of T bath to obtain acalibration. We then repeat the measurement at T bath ≈
40 mK but using either the emitter or reference source as aheater dissipating a measured power P heater = IV . We have verified by comparing full IV curves measured at finite P heater with simulations to verify that in the normal state, the only effect of the heaters is to increase the substratetemperature T substrate seen by the detector (Fig. S1(d)). We observe no change in T sub below P heater ≈ MODELING OF PHONON EMISSION
To obtain the emission rate of phonons from a superconductor, one could solve the kinetic equations for thenonequilibrium quasiparticle and phonon distributions within the superconductor with the voltage bias serving as -500 0 500 V b, emitter (µV) T S ( m K ) I emitter (pA) ( G H z ) simulationlinear fit -15 -12 -9 I emitter (A) T S ( m K ) (a)(b) (c)(d) -500 0 500 V b, emitter (µV) -505 I e m i tt e r ( n A ) n g, emitter = 0 n g, emitter = 0.5measured pb ( k H z ) (e)(f) n g, emitter I ( f A ) n g, emitter I e m i tt e r ( A ) n g, emitter = 0 n g, emitter = 0.5 n g, emitter = 0 n g, emitter = 0.5 FIG. S2. (a) Current-voltage characteristics of the phonon emitter in the superconducting state. Measurements (gray) areswept over several periods of n g, emitter , while the simulations show the minimum and maximum current at each V b, emitter ,obtained at n g, emitter = 0 (solid lines) and n g, emitter = 0 . T S on the superconducting emitter island versus voltage. (c) The emitter temperature as a function of I emitter .The temperature for all values of n g, emitter collapse on top of each other, and the emitter starts to overheat already at currentsbelow 1 pA. (d) Rate of recombination phonon emission Γ according to Eq. 3, calculated from the simulated T S shownin panels (b) and (c) (circles), and a linear fit between Γ Delta and I emitter (solid line). (e) Detector current I and thecorresponding Cooper pair breaking rate Γ pb , measured at fixed emitter voltage bias V b, emitter = 395 µ V but varying emittergate offset n g, emitter . (f) Measured I emitter corresponding to the data shown in panel (e). a quasiparticle injection term [14, 15]. However, here we use a simpler model assuming a thermal distribution ofquasiparticles at an elevated temperature T S , coupled to a thermal phonon distribution at T bath . Although thedistribution of the quasiparticles may not follow a thermal form, we believe a thermal model adequate here for thefollowing reasons: First, the bias voltage across the two tunnel junctions of our emitter satisfies V b, emitter < . /e ,so that the relaxation phonons emitted by the injected quasiparticles will not have enough energy to break furtherCooper pairs. Second, the current in a normal metal-insulator-superconductor structure biased above the gap is ratherinsensitive to the number or distribution of quasiparticles. Third, the recombination rate itself does not dependstrongly on the distribution of quasiparticles but rather on their total number. Thus we use T S as a convenientquantity for parametrizing the number of quasiparticles. The effects of phonon trapping [16] can be included in thevalue of the electron-phonon coupling constant Σ, as discussed below.We obtain T S by solving the heat balance equation, Eq. (1), but with both the tunneling rates and electron-phonon coupling modified by superconductivity. The rates and heat deposited in single-electron tunneling are givenfor instance in [4]. For the electron-phonon heat-flow we use Eq. (3) from Ref. [9]. We solve self-consistently theemitter current I emitter , temperature T S and superconducting gap ∆( T S ) of the island as a function of n g, emitter and V b, emitter , as shown in Fig. S2. In the regime I emitter < . T S ) is suppressedfrom the zero-temperature value by less than 2%. We also include a small parasitic heat flow P = 10 − W into theisland to improve numerical convergence, which does not affect the results when T S >
200 mK, the range relevantfor phonon emission. After solving T S , we then calculate the emission rate of phonons with energy ≤
2∆ from therecombination rate [9] Γ = ˙ Q R
2∆ = π V Σ6 ζ (5) k B (cid:20) k B T S ∆ + 74 ( k B T S ) ∆ (cid:21) e − kBTS , (3)where ζ is the Riemann zeta function.Both the current I emitter and island temperature T S depend on the gate offset n g, emitter = V g, emitter C g, emitter /e whencalculated at fixed V b, emitter , as shown in Fig. S2(a-b). However, for a fixed I emitter , the simulated T S does not dependon n g, emitter , but instead all the curves collapse on each other [Fig. S2(c)]. The island is also overheated very rapidlydue to the exponential suppression of the electron-phonon coupling: a current of 1 pA is enough to heat the emitterup to 270 mK. The 2∆ phonon emission rate in Fig. S2(d) calculated with Eq. 3 is close to linear in I emitter . Wefit a linear dependence between the 2∆ phonon emission rate and emitter current [solid lines in Fig. S2(d)], yieldingΓ = ηI emitter /e with η = 0 .
68. The simulations shown in Figs. 2 and 3 of the main text correspond to settingΓ pb = AI emitter /e , where the proportionality constant A = 7 . × − . This allows us to deduce Γ pb = x Γ , where x = A/η = 1 . × − is the fraction of 2∆ phonons emitted that are absorbed in the detector.At the smallest currents ( I emitter <
50 pA), the measured detector current shown in Fig. 3(a) of the main text issomewhat smaller than the simulation assuming the linear relation between I emitter and Γ . This could be explainedby a relatively large part of the current at the smallest I emitter being carried by Andreev reflection, which does notheat the superconductor and thus causes no phonon emission. Also, the linear fit in Fig. S2(d) overestimates thephonon emission somewhat compared to the simulated Γ .Our simulations in Fig. S2(a) do not reproduce the rapid rise of I emitter at n g, emitter = 0 and V b, emitter slightlyabove 2∆ /e . We attribute this to inelastic cotunneling [17–19], which is not included in our model, but argue thatthis does not substantially change the rate of 2∆ phonon emission or the functional dependence on current. In aninelastic cotunneling event, two electrons simultaneously tunnel through the whole device, leaving two excitations onthe island. However, in two successive single-electron tunneling events, two excitations on the island are involved aswell. Because V b, emitter is close to 2∆ /e , the energies of all the excitations will be close to ∆, and we expect thatinelastic cotunneling will lead to a similar dependence of T S and hence Γ on I emitter as single-electron tunneling.We can also estimate the nonequilibrium phonon emission from the superconducting leads of the reference source byconsidering quasiparticle diffusion in the spirit of Ref. [20]. We assume a simplified geometry with superconductingleads with a rectangular cross section with area 80 nm ×
200 nm, coupled to a perfect quasiparticle trap at a distanceof 1 µ m. In the experiment, the lead widens much faster and overlaps with the imperfect quasiparticle trap formed bythe normal shadow already at half a micron distances, so this leads to an upper limit for the quasiparticle temperatureat the tunnel junction. We use ρ = 90 Ωnm as the normal-state resistivity of the aluminum film [21]. These valueslead to 330 mK temperature at the junction with I ref = 10 nA. The volume 80 nm ×
200 nm × µ m at such atemperature would emit recombination phonons at a rate of 500 kHz, two orders of magnitude lower than the emissionrate from the phonon emitter at I emitter = 15 pA. However, recombination phonons emitted by the leads could bea plausible explanation for the current-dependent quasiparticle poisoning due to the backaction of a charge detectorwith much longer isolated superconducting leads and a normal island in Ref. [22], as well as explaining the backactionfrom the fully superconducting device in Ref. [23]. ESTIMATION OF PHONON TRAPPING EFFECTS AND PHONON ABSORPTION PROBABILITY
In this section, we estimate the effects of phonon trapping (phonon reabsorption before it escapes to the substrate)in the phonon emitter, as well as the fraction of incident phonons that are absorbed in the detector. The estimateshave relatively large uncertainties, but we note that due to the self-calibrating operation our device could be usefulin measuring the phonon transmission and absorption probabilities if it were subjected to a known phonon flux.In the phonon emitter, as the timescales for phonon absorption (Cooper pair breaking) are several orders of mag-nitude faster than the recombination times [24], the effect of phonon trapping is to simply increase the recombinationtimes by some material- and substrate-dependent factor F , which we can view as decreasing the electron-phononcoupling constant Σ by the same factor. We can estimate the phonon trapping factor as F = 4 d/η Λ [25], where d = 80 nm is the film thickness, η ≈ . F is even closer to unity. As our value for Σ measured in the normal state is somewhat on the low sidecompared to the literature values, we do not further correct for phonon trapping.For the phonon absorption probability in the detector, we may consider that Ref. [29] found good agreementbetween measurements and simulations using the acoustic mismatch model with a transmission coefficient T Si → Al between 0.3 and 0.55 for 60 nm Al films, which assumed that the phonon is absorbed with an unity probability onceit entered the film. In contrast, Ref. [30] assumed a transmission probability from Si to Al larger than 0.9 but phononabsorption probabilities in 140-220 nm thick films on the order of 0.25. In contrast to these works, in our case the Sisubstrate is covered by a 200 nm SiO film, and to our knowledge there are no experimental data available for sucha system. As the acoustic impedances are better matched between Al and SiO than Al and Si [25], we believe 25%to be a reasonably cautious estimate for the fraction of incident phonons that are eventually absorbed in the Al film.This parameter is only used in the calculation of the noise-equivalent power. ADDITIONAL DERIVATIONS
In this section, we show that the self-calibrating response is obtained even if the tunneling rates through the twojunctions are unequal, derive the condition of Eq. (3) given in the main text, as well as the range of V b and n g forthe self-calibrating mode. Unequal tunnel resistances
Here, we consider a system where the tunneling rates for quasiparticles leaving the island differ, which in practicemay be due to different junction areas and resistances. Instead of the single rate Γ tunn , the quasiparticles may exitthe island through the left and right junctions with rates Γ L and Γ R , respectively, but we still assume that the returnfrom the charge states N = − L or Γ R . Now, we can write arate equation for the probability P N QP to have N QP quasiparticles present: ddt P N QP = − Γ pb P N QP + Γ pb P N QP − − (Γ L + Γ R ) N QP P N QP − (Γ L + Γ R )( N QP + 1) P N QP +1 (4)The probabilities P N QP for negative N QP are set to zero, and in the steady state the time derivative is zero. Multiplyingby N QP and summing over all N QP we obtain the steady state quasiparticle population h N QP i = 2Γ pb / (Γ L + Γ R ) . (5)To calculate the total current, we need the rate at which the first step in the cycle N = 0 → − → N = +1 → +2 → +1) occurs. The first step is a quasiparticle tunneling out through the left (right) junction. From a state with N QP quasiparticles, this occurs at a rate N QP Γ L ( N QP Γ R ). The total current reads then I/e = Γ L X N QP even P N QP N QP + Γ R X N QP odd P N QP N QP . (6)We now sum Eq. (4) over the even values of N QP :0 = − Γ pb X N QP even P N QP + Γ pb X N QP even P N QP − − (Γ L + Γ R ) X N QP even N QP P N QP + (Γ L + Γ R ) X N QP odd N QP P N QP , (7)which leads to P N QP even N QP P N QP = P N QP odd N QP P N QP . Now, as P N QP even N QP P N QP + P N QP odd N QP P N QP = P N QP N QP P N QP = h N QP i by definition, we obtain I/e = Γ L h N QP i / R h N QP i / pb . (8) Limitation of response due to recombination
In the general case with non-negligible recombination rates, the quasiparticle population on the island satisfies ddt h N QP i = − R h N QP i − tunn h N QP i + 2Γ pb . (9)The prefactors are Γ R = Σ∆ / [12 ζ (5) D ( E F ) k B V ] [9] and Γ tunn = ( e R T V D ( E F )) − [31]. Here, ζ is the Riemannzeta function, the density of states D ( E F ) = 2 . × J − m − [32], and R T is the resistance of the two junctionsof the device in series. The first and third factors of two are due to quasiparticles being created and recombining inpairs, while the second is due to the quasiparticles relaxing through the two junctions: as in the main text, Γ tunn isthe relaxation rate through a single junction. We can neglect the recombination term if h N QP i Γ R (cid:28) Γ tunn , which issatisfied if Γ pb (cid:28) Γ / Γ R = 12 ζ (5) k B Σ∆ e R T V . (10) δ E R ( ) = ++ δ E L ( + ) = -- δ E L ( + ) = Δ - δ E L ( + ) = Δ - δ E R (- ) = Δ + δ E R ( ) = Δ + -0.5 0 0.5 1 1.5-200-1000100200 V b ( µ V ) -2-1012 I / e Γ pb -0.5 0 0.5 1 1.5 (a) (b) n g n g δ E R ( ) = ++ δ E L ( + ) = -- δ E L ( + ) = Δ - δ E L ( + ) = Δ - δ E R (- ) = Δ + δ E R ( ) = Δ + FIG. S3. Simulated current through the detector with E C = 0 .
46∆ and thresholds for self-calibrating operation arising fromsingle-electron (solid lines) or Andreev (dashed lines) tunneling, calculated in (a) at T N = 5 mK (essentially zero temperature)and (b) at T N = 42 mK. Thresholds for self-calibrating equation
The thresholds for the selfcalibrating operation can be derived from considering the energy gains for adding (+) orremoving (-) an electron from the island with N electrons initially through the left (L) or right (R) junctions, whichinclude the contribution from the charging energy E C ( N − n g ) and the voltage source V b : δE ± L ( N ) = 2 E C ( ∓ ( N − n g ) −
12 ) ∓ eV b / δE ± R ( N ) = 2 E C ( ∓ ( N − n g ) −
12 ) ± eV b / . (12)Here we assume a bias voltage V b / > − V b / − ∆ < δE < ∆, transitions removing a quasiparticle can happenat the rate Γ tunn N QP where N QP is the number of quasiparticles. If δE > ∆, transitions adding a quasiparticle intothe island are energetically allowed, and occur roughly at an ”ohmic” rate ( δE − ∆) / ( eR T ), which is several ordersof magnitude higher for typical device parameters than the rate for removing a quasiparticle.The self-calibrating operation occurs when the following conditions are satisfied: • The transitions N = 0 → +1 and N = +1 → − ∆ <δE + L (0) , δE + R (0) , δE − R (1) , δE − R (+1) < ∆. This condition ensures that the quasiparticle population satisfies Eq.(2) of the main text. • The transition N = +1 → +2 ( N = 0 → −
1) can remove but not add a quasiparticle through the right (left)junction: − ∆ < δE + R (1) , δE − L (0) < ∆. This condition guarantees that the first step of the current-carryingcycles occur with a rate h N QP i Γ tunn ; • The transition N = 2 → − →
0) can add a quasiparticle to the island through the left (right) tunneljunction, but not through the right (left) junction: δE − L (2) , δE + R ( − > ∆, but δE − L (2) , δE + R ( − < ∆. Thisguarantees that the second step of the current-carrying cycles is fast compared to the first step and actuallytransports an electron through the entire device.These conditions lead to a set of constraints for n g and V b for given values of E C and ∆. If E C > ∆, theconstraints are not satisfied simultaneously. For 0 . < E C < ∆, the tightest constraints are by the conditions thatthe quasiparticles can tunnel out also against the bias between the states N = 0 , +1: δE + L (0) , δE − R (1) > − ∆, andthat quasiparticles can tunnel out to the excited states in the direction of the bias: δE − L (0) , δE + R (1) > − ∆. For E C < . δE − L (2) , δE + R ( − > ∆, and the constraint that tunneling events in the direction of the bias between N = 0 , +1cannot add a quasiparticle: δE − L (+1) , δE + R (0) < ∆.At small R T , additional constraints are due to the fact that Andreev tunneling should not be energetically allowed.The energy gains in Andreev tunneling δE ±± L/R are double the single electron gains in Eqs. (11), (12) because twoelectrons are transported. This leads to the additional conditions that two-electron tunneling in the direction of thebias should not be energetically allowed, δE −− L (+1) < δE ++ L (1) < E C = 0 .