Interplay Between Kinetic Inductance, Nonlinearity, and Quasiparticle Dynamics in Granular Aluminum Microwave Kinetic Inductance Detectors

PHYSICAL REVIEW APPLIED 11, 054087 (2019)

Francesco Valenti,1,2 Fabio Henriques,1 Gianluigi Catelani,3 Nataliya Maleeva,1 Lukas Grünhaupt,1 Uwe von Lüpke,1 Sebastian T. Skacel,4 Patrick Winkel,1 Alexander Bilmes,1 Alexey V. Ustinov,1,5 Johannes Goupy,6 Martino Calvo,6 Alain Benoît,6 Florence Levy-Bertrand,6 Alessandro Monfardini,6 and Ioan M. Pop1,4,* 1 Physikalisches Institut, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany 2 Institut für Prozessdatenverarbeitung und Elektronik, Karlsruhe Institute of Technology, 76344 Eggenstein-Leopoldshafen, Germany 3 JARA Institute for Quantum Information (PGI-11), Forschungszentrum Jülich, 52425 Jülich, Germany 4 Institute of Nanotechnology, Karlsruhe Institute of Technology, 76344 Eggenstein-Leopoldshafen, Germany 5 Russian Quantum Center, National University of Science and Technology MISIS, 119049 Moscow, Russia 6 Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France

(Received 9 November 2018; revised manuscript received 8 March 2019; published 31 May 2019)

Microwave kinetic inductance detectors (MKIDs) are thin-film, cryogenic, superconducting resonators. Incident Cooper pair-breaking radiation increases their kinetic inductance, thereby measurably lower- ing their resonant frequency. For a given resonant frequency, the highest MKID responsivity is obtained by maximizing the kinetic inductance fraction α. However, in circuits with α close to unity, the low supercurrent density reduces the maximum number of readout photons before bifurcation due to self-Kerr nonlinearity, therefore setting a bound for the maximum α before the noise-equivalent power (NEP) starts to increase. By fabricating granular aluminum MKIDs with different resistivities, we effectively sweep their kinetic inductance√ from tens to several hundreds of pH per square. We find a NEP minimum in the range of 30 aW/ Hz at α ≈ 0.9, which results from a trade-off between the onset of nonlinearity and a nonmonotonic dependence of the noise spectral density versus resistivity.

DOI: 10.1103/PhysRevApplied.11.054087

I. INTRODUCTION The first MKIDs consisted of thin-film-aluminum distributed-element resonators. Their numerous incar- Since their first implementation 15 years ago [1], nations now include lumped-element- (LE) resonator microwave kinetic inductance detectors (MKIDs) have geometries [21], solutions such as spiral resonators [22] played an important role in ground-based radioastron- or various kinds of antenna coupling [23–25], and a wealth omy [2–7], particle detection [8–12], and are promising of different film materials such as TiN [26–28], NbN [29], candidates for spaceborne millimeter wave observations PtSi [30], and W Si [31], including hybrid realizations [13–16]. This remarkable development was facilitated by x y [32] and multilayered films [33–36]. the technological simplicity of MKIDs, which consist of Here, we propose granular aluminum (grAl), a compos- thin-film, cryogenic, superconducting resonators in the ite material made of pure Al grains with median size of microwave domain. The MKID signal is the shift of its the order of a = 3 ± 1 nm in an aluminum oxide matrix resonant frequency due to an increase in the kinetic induc- [37,38], as an alternative material for MKIDs. As illus- tance of the film, which is itself proportional to the number trated in Fig. 1, we employ a coplanar-waveguide (CPW) of Cooper pairs (CP) broken by the incoming radiation. geometry, in which the ground plane and the feedline of These compact, low-loss and multiplexable detectors can the CPW are made of pure aluminum, 50 nm thick, and also provide a convenient tool to probe material properties, the lumped-element MKID resonators are entirely made of such as the change of dielectric constant due to a superfluid grAl. transition [17], the density of states of granular aluminum Granular aluminum is an appealing material because it and indium oxide [18], or to image phonons [19,20]. already demonstrated high microwave internal quality factors, from 105 at low photon number up to 106 in the strong drive regime [39–41], and ease of fabrication, which *[email protected] simply consists in aluminum deposition in a controlled

kinetic inductance fraction, nonlinearity and quasiparticle dynamics, and its eﬀect on the noise-equivalent power (NEP)—the main operational ﬁgure of merit of MKIDs. We discuss experimental methods and results in Sec. III, showing that we can exploit the tunability of the grAl nonlinearity and superconducting gap to achieve low NEP values. In Sec. IV we conclude by proposing guidelines to further reduce the NEP in future designs.

II. THEORY FIG. 1. Optical images of the aluminum sample holder and The NEP is defined as the radiant power needed to a grAl lumped-element resonator, together with its equivalent have equal signal and noise amplitudes in a 1-Hz output circuit. A sample consists of an ensemble of 22 resonators bandwidth, and can be expressed as [33] coupled to a central coplanar microwave waveguide, which is used to perform transmission spectroscopy. The enlargement S shows a single resonator, where we highlight the interdigitated NEP = ,(2) capacitor in magenta (which gives both C and Cc) and the mean- R dered inductor (which gives L) in green. The sapphire substrate is shown in dark green. All resonators are fabricated from a where S is the noise spectral density (Sec. II A)andR is 20-nm-thick grAl film using e-beam liftoff lithography, while the the responsivity (Sec. II C). In the following subsections, central conductor of the coplanar waveguide (orange) and the we discuss in detail the influence of grAl parameters on ground plane (light green) are made of 50-nm-thick aluminum each of these two quantities. patterned by optical liftoff. The meandered inductor for samples A–D is shaped as a third-degree Hilbert curve (shown), while for samples E and F it is shaped as a second-degree Hilbert curve A. Noise spectral density (cf. text and Appendix√ A). In order to distribute the resonant fre- Fluctuations of the MKID resonant frequency in the quencies f = (2π LC)−1 of the 22 resonators, we sweep the 0 absence of incoming radiation constitute noise. The noise capacitances by changing the length of the interdigitated fingers. spectral density (NSD) is computed by recording the fluctuations of the resonant frequency over time, taking its oxygen atmosphere [37,38]. Furthermore, by varying the Fourier transform and dividing by the square root of√ the oxygen pressure during deposition, one can tune material output bandwidth, hence the NSD is quoted in Hz/ Hz 4 parameters such as the resistivity (from 1 to 10 μ cm), units (cf. Appendix B). These fluctuations can be either kinetic inductance, and superconducting gap. The kinetic dominated by the added noise of the measurement setup, or inductance of a square of thin film is determined by the by noise sources intrinsic to the resonator, such as dielec- ratio of the normal-state sheet resistance per square Rn and tric two-level systems [51], microscopic charge fluctuators the critical temperature Tc [42,43]: [52], or fluctuations in the number of quasiparticles (QPs) [53–55]. As discussed in Sec. III C, for the grAl MKIDs 0.18 Rn Lkin, = ,(1)in this work the latter mechanism appears to be dominant, kB Tc and reducing the QP density can be a fruitful approach to suppress the NSD. and, in the case of grAl, it can reach values as high as a few nH/ [40,41]. Equation (1) is valid under the assumption of the grAl BCS constant being reasonably close B. Phonon trapping and quasiparticle fluctuations to 1.76, which holds for resistivities up to the m cm The grAl MKIDs are surrounded by the comparatively regime [44–46]. The grAl microstructure, consisting of much larger aluminum ground plane (cf. Fig. 1), which superconducting grains separated by thin insulating shells, has a lower superconducting gap and can act as a phonon can be modeled as a network of Josephson junctions trap [56–58], possibly reducing the number of QPs gen- (JJs), which simplifies to a one-dimensional (1D) JJ chain erated by nonthermal phonons from the substrate, as for resonators in the thin-ribbon limit (length width schematically illustrated in Fig 2. In the following, we thickness) [47]. We can use this model to quantitatively develop a theoretical framework to quantify the phonon- estimate the nonlinearity of grAl resonators and extract trapping effect of the ground plane, i.e., how the difference the self-Kerr coefficient of the fundamental mode K11 [47], between superconducting gaps of the circuit and ground similarly to the case of resonators made of mesoscopic JJ plane is related to a suppression of the QP generation- arrays [48–50]. recombination noise, and with that, the NEP. In order to This article is organized as follows: in Sec. II,we model the effect of phonon traps, we start from the expres- propose a model to quantify the interplay between sion of the NEP (in the ideal case of unit conversion

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that cannot break pairs in grAl (cf. Fig. 2). We model the dynamics of hot phonons and quasiparticles in a phenomenological way, with rate equations of the Rothwarf-Taylor type. For quasiparticles in grAl, the relevant processes are generation from pair breaking by hot phonons (rate bG) and recombination (rate rG). Similarly, for quasiparticles in Al, we have generation by pair breaking (bA) and recombination (rA), but also scattering to lower energies (rate sA). For the phonons, we assume some generation mechanism with rate gP, in addition to generation and recombination in both grAl and Al. The rate equations are then

˙ =− 2 + FIG. 2. Phonon trapping and quasiparticle number reduction: NG 2rGNG 2bGNP,(4) schematic depiction of the dynamics described in Eqs. (4)–(6). N˙ =−2r N 2 + 2b N − s N ,(5) We show the superconducting energy band diagram in the exci- A A A A P A A tation picture for both grAl (in blue) and Al (in red). Excita- ˙ = − + 2 − + 2 NP gP bANP rANA bGNP rGNG.(6) tions and relaxations are represented by solid black and dashed gray arrows, respectively. Wiggly arrows represent phonons and We consider now the steady-state solution. The first the labels represent their corresponding frequency. Substrate phonons with characteristic frequency larger than the grAl spec- equation simply gives tral gap fG, generated at rate gP, can break grAl CPs at rate bG. The resulting grAl QPs recombine at rate rG, emitting phonons bGNP that can travel through the substrate and reach the Al ground NG = (7) plane, which covers most of the chip (cf. Fig. 1), breaking Al rG CPs at rate bA. The excited QPs can scatter to a lower energy via electron-phonon interaction at rate sA and recombine to form an and the last two terms in the last equation cancel out. Then Al CP at rate rA. In both cases the emitted phonons have char- we are left with the system acteristic frequencies lower than the spectral gap of grAl, thus being unable to break grAl CPs. =− 2 + − 0 2rANA 2bANP sANA,(8) = − + 2 efficiency) dominated by QP generation-recombination 0 gP bANP rANA.(9) noise [54,55], We can solve the first equation for N in terms of N , A P substitute into the second equation, and find NG NEP = 2G ,(3) τG 2 = gP + 4rAgP NP 2 . (10) bA bAsA where G, NG,andτG are the grAl superconducting gap, QP number, and QP lifetime, respectively. In the following √ For “weak” scattering, s 2 r g , we then have we use the indexes G and A to refer to thin-film grAl and A A P aluminum. We assume that all phonons in grAl, Al, and 2 the substrate quickly reach the steady state after a high- ≈ 4rAgP NP 2 . (11) energy generation event [19,20], allowing us to describe bAsA their density as position independent. Moreover, since the temperature T is small (1.76kBT G, A, G − A), Note that Eq. (11) diverges as the scattering rate decreases. we neglect any effect of thermal phonons and we focus This unphysical result is due to the fact that we neglect on phonons with energy above the highest grAl gap (EP > other mechanisms that can decrease the hot phonon num- max G ), so they can create quasiparticles in grAl by breaking ber, for example, escape from the substrate into the sample Cooper pairs at a rate bG independent of G. We indicate holder, or phonon scattering that cools them below the grAl with NP the number of these “hot” phonons. In addition gap. In a relaxation-time approach, such a contribution to quasiparticles in grAl with number NG, we also have would add a term −ePNP to the right-hand side of Eq. (6) in quasiparticles in Al. order to account for escaped phonons. Then we can show / 2 / We are interested in the quasiparticles generated by that Eq. (11) remains valid so long as 4eP bA sA 4rAgP. hot phonons. They can recombine by emitting a hot Having NP from Eq. (11), we can calculate NG using Eq. phonon again, or scatter by emitting lower-energy phonons (7). As for the quasiparticle lifetime, linearizing Eq. (4)

054087-3 FRANCESCO VALENTI et al. PHYS. REV. APPLIED 11, 054087 (2019) around the steady state, one can see that 1/τG = 4rGNG. twice the fraction of broken CPs. For practical reasons, the Therefore, from Eq. (3) we get choice of values for the operational frequencies f0 is limited by the availability and cost of readout electronics, and it is typically in the range of a few GHz. bGrA G NEP ≈ 4G bGNP = 8gP . (12) To estimate P , we assume that every collected pho- b s abs A A ton with hf > 2 breaks a CP. The amount of impedance matching between the resonator plane and the medium The quasiparticle scattering rate s depends on the A through which the photons propagate (for example, vac- quasiparticle energy above the gap . At low tempera- A uum or dielectric substrate) defines the detector absorp- ture, we approximate the scattering rate due to electron- tance A (cf. Appendix A). The power absorbed by the phonon interaction by its zero-temperature expression, resonators is then P = δP A, where δP is the change which according to Ref. [59] can be written in the form: abs in in in radiant power under illumination through the optical 1 + − ω setup. Under operational conditions, for any MKID design, s = dωω2 A one aims to minimize the impedance mismatch for the A τ 3 0A A 0 ( + − ω)2 − 2 A A A incoming radiation, in order to obtain a value for as close as possible to unity. 2 × 1 − A , (13) (A + )(A + − ω) D. Voltage responsivity R where the prefactor 1/τ accounts for the strength of the Following Eq. (15), the responsivity scales linearly 0A with the kinetic inductance fraction α. However, in the electron-phonon interaction. For A we then find sA ∝ 7/2 limit α → 1 the performance of MKIDS is limited by (/A) (1 + /8A)/(1 + /A), while for A we 3 the early onset of nonlinear phenomena, i.e., the res- have sA ∝ (/A) . Here ≈ G − A is of order A, so the first expression applies. Since we are interested in onators bifurcate at low readout voltages. In order to maximize the microwave signal-to-noise ratio, one wants the dependence of NEP on G, dropping prefactors we arrive at to operate MKIDs at the highest possible readout power before bifurcation [47,63,64] and before degrading the 7/2 detector by breaking CPs [65–67]. Typically, in high- A 1 NEP ∝ 2 . (14) kinetic-inductance detectors, bifurcation sets the maxi- G − + 7 G A G A mum number of circulating photons, which in our case 5 − 6 Equation (14) indicates an anticorrelation between the is in the range of 10 10 (cf. Fig. 3), and can be as low as 102, as reported in Ref. [41]. The NEP defined NEP and the superconducting gap G. We recall that Eq. (14) applies only for the case of a system in which in Eq. (2) is thus implicitly dependent on the maximum the ground plane has a relevant phonon-trapping effect, microwave readout voltage, which in turn scales with the square root of the maximum number of circulating pho- i.e., G − A 1.76kBT, otherwise it would imply an unphysical divergence of the NEP when resonators and tons before bifurcation nmax. This dependence can be made ground plane are fabricated with the same material. In explicit by defining a voltage responsivity Sec. III C we present experimental results, which are qual- √ R ≡ α n . (16) itatively consistent with this phonon-trapping model, and V max we show that the increase of the grAl gap at the top of The voltage responsivity is trivially zero if α is zero, but the so-called superconducting dome [37,38,44–46,60–62] it also vanishes in the limit Lkin →∞(α → 1), when the is responsible for a significant improvement in detector resonator nonlinearity also increases, suppressing nmax.In performance. the following, we quantify this nonmonotonic dependence. The kinetic inductance fraction α = Lkin/Ltot can be C. Responsivity estimated by knowing the geometry, resistivity, and critical The responsivity of a MKID can be expressed as [33] temperature of a resonator, as we discuss in Appendix C. We show that for a grAl lumped-element resonator with |δf | αf δx kinetic inductance dominating over the geometric induc- R = 0 = 0 QP , (15) Pabs 4 Pabs tance we can write (cf. Appendix D) √ 2 where δf0 is the shift in resonant frequency due to pair 4 C nmax = √ √ , (17) breaking, P is the radiant power absorbed by the detec- 2 abs 3 3Qc(πea) Lkin tor, α is the kinetic inductance fraction with respect to the total inductance, f0 is the unperturbed resonant frequency, where is the meandered inductor length and C is the inter- and δxQP is the shift in the quasiparticle density, defined as digitated capacitance (cf. Fig. 1), Qc is the coupling quality

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(a) (b) (c)

× ×

FIG. 3. Microwave characterization of MKIDs via transmission measurements. (a) Amplitude (normalized by the sample-holder response) and phase of the transmission coefficient S21 for a resonator in sample A, at a readout power in the single-photon regime (n¯ ≈ 1). Raw data is shown as green circles and the solid black line is the fit to the complex scattering parameters from Ref. [68]; the fitted values are given in the top panel. (b) Internal quality factors as a function of the average number of readout photons circulating in the resonator, shown up to the bifurcation threshold n¯ = nmax in log-log scale for all samples. The average photon number is ¯ = /ω2 calculated from the estimated on-chip readout power as n 2QcPcold 0 (cf. Appendix D). The single-photon regime corresponds to Pcold ≈−150 dBm. Notice that the maximum photon number decreases for higher resistivity films, due to the higher nonlinearity [cf. Eq. (17)]. (c) Change in resonant frequency as a function of temperature, averaged for all resonators in√ each sample [colored circles, using the same palette as in (b)]. We fit the measured values with the BCS equation δf0(T)/f0 =−3.32α Tc/T exp (1.76Tc/T), adapted from Ref. [69] (colored lines), where we use f0 = f0(T ≈ 25 mK). The fitting parameters are the kinetic inductance fraction α and the critical temperature Tc, shown in the legend. factor, e is the electron charge, and a is the size of an alu- hosting 22 resonators each. We label the chips from A to minum grain. By plugging α and Eq. (17) into Eq. (16) one F according to the grAl sheet resistance, of 20, 40, 80, finds 110, 450, 800 /, respectively. The reader may refer to Appendix E for more details on the material properties of 3/4 Lkin the fabricated samples. The 50-nm-thick surrounding Al RV ∼ , (18) Lkin + Lgeom CPW ground plane is patterned in a second optical liftoff lithography step. with Lkin ∼ Rn/Tc [cf. Eq. (1)], where Rn is the normal- Two different measurement setups are required in order state sheet resistance per square, and Tc is the critical to characterize (a) the intrinsic properties of the MKID temperature. From Eq. (18) one can see that RV tends resonators, at frequencies in the GHz range, and (b) the to zero for both limit cases Lkin = 0 (no detection) and operational MKID response to millimeter wave radiation. Lkin →∞(fully nonlinear system). The voltage responsiv- To measure their microwave properties, we anchor the ity increases sharply with Lkin until it reaches a maximum MKIDs to the base plate of a so-called “dark” dilution at α = 3/4, after which it slowly decreases (cf. Fig. 4). cryostat, with a base temperature of about 25 mK. In Thus, RV quantifies the interplay between kinetic induc- this setup, we couple to room temperature electronics via tance and nonlinearity, and is a convenient metric to deter- heavily attenuated and filtered radio-frequency (rf) lines, mine whether a given kinetic inductance fraction α gives a including IR filters (cf. Ref. [78]), with the goal of reduc- sensitive enough detector, while not being severely limited ing stray radiation from the higher temperature stages of by nonlinearity. the cryostat. On the other hand, in order to measure the NEP, we III. EXPERIMENTAL RESULTS need to shine millimeter wave radiation onto the resonators and operate them at high readout powers. We thus use A. Measurement setups a much less shielded dilution refrigerator, with an opti- The MKIDs discussed in this work are lumped-element cal opening and a base temperature of approximately 150 resonators, composed of a meandered inductor shaped as mK, which we refer to as the “optical” cryostat. The sam- a Hilbert curve of third or second degree (H3 and H2), ple chip is mounted in an aluminum box with an optical and an interdigitated capacitor (cf. Appendix A). The 20- window on one side (cf. Fig. 1 and Appendix F). For nm-thick grAl film is patterned on a sapphire substrate via measurements performed in the optical cryostat, the opti- e-beam liftoff lithography. We fabricate 2 × 2cm2 chips cal window is facing a cryogenic millimeter wave optical

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(a)

(b)

FIG. 4. Kinetic inductance fraction, maximum number of readout photons, and voltage responsivity as a function of the sheet resistance per square Rn scaled to the critical temperature Tc. (a) We plot the kinetic inductance fraction α in green and the maximum number of readout photons before bifurcation nmax in red. Dashed and dotted lines are analytical predictions for H3 and H2 geometries 5 used in this work, for both α (cf. Appendix C)andnmax [estimated with Eq. (17) at a fixed Qc = 10 ]. The color shaded regions overlapping with the nmax analytical lines represent the range of maximum photon number values corresponding to the different interdigitated capacitances C of the various resonators in each sample [cf. Eq. (17)]. Full markers show values measured in this work: squares and circles relate to H3 and H2 geometries, respectively. For comparison, empty markers show α values reported in the literature for various materials (cf. Refs. [11,27,31,69–77] and Appendix C). The measured nmax values are approximately two times lower than predicted for all samples, which might be due to a systematic underestimation of the on-sample readout power. The reported values are averaged over all functional resonators for each sample. (b) Voltage responsivity RV, defined as the product between the kinetic inductance fraction α and the square root of the maximum number of readout photons before bifurcation nmax [cf. Eq. (16)]. We report values for H3 and H2 geometries with dashed and dotted lines (analytical predictions) and square and circle markers (measured data), respectively. The color shaded regions overlapping with the lines represent the range of different interdigitated capacitances C. The maximum voltage responsivity is obtained at α = 3/4 [cf. (a)], as indicated by the vertical black lines across the two panels (left line for H3 geometry, right line for H2). setup used to focus the blackbody radiation onto the sam- resonant frequency of one of the MKIDs in sample A, ple, as described in Ref. [33]. For measurements performed for a readout power in the single-photon regime. We in the dark cryostat, the optical window is covered with employ the circle fit routine detailed in Ref. [68] to extract aluminum tape, and the sample holder is placed in a series resonator parameters of interest, namely the resonant fre- of successive cryogenic infrared and magnetic shields, quency f0, the internal quality factor Qi, and the coupling similar to Ref. [78]. quality factor Qc. An example of fitted on-resonance frequency response is shown in Fig. 3(a), from which we B. Microwave characterization extract an internal quality factor on the order of 105 in the Figure 3 shows the results of measurements performed single-photon regime. in the dark cryostat. In Fig. 3(a) we show a typical result In order to compute the average number of readout pho- for the transmission coefficient S21 in the vicinity of the tons in the resonator, n¯, we estimate the on-chip power

054087-6 INTERPLAY BETWEEN KINETIC... PHYS. REV. APPLIED 11, 054087 (2019) by summing the total attenuation on the input line of the C. Measurement of the NEP cryostat (see Appendix G). It is important to note that, The opening of the optical cryostat is coupled to a matte, due to the uncertainty in the attenuation figure of the rf high-density polyethylene disk, which is cooled down by components over a broad frequency range, this method is a pulse-tube cryocooler and used as a blackbody source. A only accurate within an order of magnitude. In Fig. 3(b) we layer of Eccosorb [82] absorber sponge can be manually ¯ present measurements of Qi as a function of n, for read- interposed between the source and the cryostat, acting as a ¯ ≈ out powers ranging from n 1 up to the critical number room-temperature blackbody. This procedure allows us to of readout photons nmax [reported in Fig. 4(a) along with switch between a 100-K and a 300-K source. By account- its analytical calculation], at which the resonator bifur- ing for this change in temperature, and the optical coupling cates. The internal quality factor increases monotonically between the source and the cryostat, we obtain the shift with the average number of photons in the resonators. This in radiant power on the sample δPin [33], in the range of type of dependence was previously observed for grAl res- 0.3 pW for the H2 resonator design, and 0.1 pW for the onators in Ref. [41], and it is also routinely measured H3 design. For each of our samples we estimate the power in thin-film aluminum resonators [51]. The increase of coupled into the MKID, Pabs = δPinA, by calculating the the internal quality factor with power can be interpreted film absorptance A, using analytical formulæ corroborated as the combined effect (cf. Fig. 2(c) from Ref. [41]) of by finite-element simulations (cf. Appendix A). the saturation of dielectric losses [51,79], together with Following Eq. (15), we obtain the MKID responsiv- photon-assisted unpinning of nonequilibrium QPs, which ity by measuring the resonant frequency shift, δf0, when are then allowed to either diffuse away from regions of high the illumination source changes from cryogenic black- current density or to recombine into CPs [80,81]. body to room temperature. The results for all samples By sweeping the sample temperature up to about are plotted in purple in Fig. 5(a). The brown curve in 600 mK we observe a downward shift in the resonant Fig. 5(a) shows the measured shift in QP density for frequency, as shown in Fig. 3(c), which we fit using a absorbed power, δxQP/Pabs, obtained by using the second BCS model [69] to obtain the kinetic inductance frac- and third terms of Eq. (15). The QP signal is remark- tion α and the critical temperature Tc. In the inset of ably constant for all resistivities, with an average value −4 −1 Fig. 3(c), we report the resulting fit parameters averaged δxQP/Pabs ≈(4.8 ± 1.6) × 10 pW , indicating that over all functional resonators in each chip. The obtained the measured fluctuations in R are simply due to different α is in good agreement with the analytical prediction fundamental mode frequencies, f0, of MKIDs in different [cf. Fig. 4(a)], and Tc shows a dome-shaped dependence samples. with resistivity [cf. Figs. 3(c) and 5(d)] in agreement with We calculate the NSD by recording δf0(t) in the absence Refs.[37,38,44–46,60–62]. of incoming radiation, when the opening of the optical Using the model of Eq. (17) for the maximum num- cryostat is covered. Since perfect optical sealing can not ber of circulating photons before bifurcation, in Fig. 4(a) be achieved in the optical cryostat, we repeat the mea- we plot the calculated nmax and α versus the ratio of surement in the dark cryostat to obtain the intrinsic noise the normal-state sheet resistance and critical temperature, figure of the MKIDs. The comparison between the two Rn/Tc, which determines the kinetic inductance [see Eq. measured NSDs is shown in Fig. 5(b). Note that the depen- (1)]. For comparison, in Fig. 4(a) we also overlay the mea- dence of the NSD versus resistivity shows a minimum for sured values of α and nmax for samples A–F, together with a measurements performed in both cryostats, and the val- literature survey of reported α values for superconducting ues measured in the dark are a factor of 2 lower. This resonators fabricated with various other materials. nonmonotonic dependence versus normal-state resistivity By replacing the expressions of α and nmax in Eq. (16), of the film is correlated with the superconducting gap we can compare analytical predictions of the voltage value reported in Fig. 5(d), which suggests quasiparticle responsivity RV with measured values, as shown in generation-recombination as the dominant source of noise Fig. 4(b). Considering that Tc is bounded within the 2 − [54,55,83,84]. This would also explain the lower NSD 2.3 K interval for all samples, the voltage responsivity measured in the dark cryostat, where the superior shield- is almost exclusively a function of resistivity. Although ing and filtering results in a lower density of nonthermal RV shows a maximum for grAl sheet resistances in the QPs. range of 10 − 20 , where we would expect the detec- We compute the NEP as the ratio of the NSD measured tor performance to be optimal, we find it remarkable that in the optical cryostat and the responsivity R, as per Eq. RV does not rapidly degrade at high resistivities, several (2), and we plot the results in Fig. 5(c). The dependence of orders of magnitude greater than the ones currently used in the NEP versus grAl film resistivity shows a minimum at R MKID technology. This slow decrease of V with increas- ρ ≈ 200 μ cm, corresponding to a sheet resistance Rn ≈ ing kinetic inductance opens the way for the study, and 100 , which is an order of magnitude larger than the typi- possible use, of MKIDs with very large kinetic inductance cal values used in the MKID community [16,32,33,84–86]. (α ≈ 1). We plot the values of the grAl superconducting gap in

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FIG. 6. Noise-equivalent power as a function of the superconducting gap of granular aluminum. Samples A–D appear to follow the anticorrelation predicted by Eq. (14)—higher granular aluminum gap results in a lower NEP. The outlier nature of samples E and F (grayed out) is addressed in the main text.

suggesting that phonon trapping in the Al ground plane plays an important role. The fact that the high-resistivity samples E and F deviate from the trend could point to an additional complexity in the quasiparticle dynamics that is currently unaccounted for, e.g., a drastically reduced dif- fusion length due to disorder. Furthermore, the resistivity of samples E and F is at the edge of the threshold beyond which the grAl BCS coefficient increases to about 2.1–2.2 [44–46]. While the change in the BCS factor is a small FIG. 5. Properties of grAl MKIDs as a function of film resistiv- effect in itself, the underlying physics—yet to be fully δ α ity. (a) We use the measured values of f0, f0, , and the estimated understood—likely has manifold implications on quasi- power absorbed by the resonators Pabs, to compute the respon- particle dynamics and the corresponding noise, that go sivity and the shift in quasiparticle density [cf. Eq. (15)], which beyond our current phonon-trapping model and should be we plot in purple and brown, respectively. (b) NSD measured for the same samples in what we denote “optical” (in orange) addressed in future work. and “dark” (in turquoise) cryostats (cf. Sec. III), evaluated at 10 Hz. The NSD shows a minimum at ρ ≈ 200 μ cm for measurements taken in both cryostats (see the main text and Appendix B IV. CONCLUSIONS for a detailed discussion). (c) Noise-equivalent power, calculated We use granular grAl as an alternative thin-film mate- as the ratio of measured responsivity and NSD in the optical cryo- rial to fabricate MKIDs with resistivities ranging from 40 stat. The resistivity dependence of the NEP is dominated by that to 1600 μ cm, corresponding to kinetic inductances up of the NSD. (d) Measured grAl superconducting gap G, calculated from the critical temperature reported in Fig. 3(c) under the to orders of magnitude higher than those found in cur- BCS assumption = 1.76kBTc. Note that values of NEP and G rent MKID technology. To minimize the NEP, we find an are anticorrelated (see text for a detailed discussion). interplay between kinetic inductance fraction α and the nonlinearity limiting the maximum number of readout photons before bifurcation to nmax, resulting in an optimal Fig. 5(d) in order to highlight its anticorrelation with the α = 3/4. This value does not depend on resonator geome- NEP, as expected from the model discussed in Sec. II B try, which can be optimized for maximum absorptance. In [cf. Eq. (14)], under the assumption of dominating QP order to quantify the outcome of this interplay, we√ intro- generation-recombination noise. duce the concept of voltage responsivity RV = α nmax. In Fig. 6 we plot the measured NEP versus G.For For α>3/4 we expect an increase of the NEP due the samples A–D, we observe an anticorrelation between the slow decrease of RV; however, experimentally, we find NEP and the height of the superconducting grAl gap, the NEP to be minimum at α ≈ 0.9. This is due to the

054087-8 INTERPLAY BETWEEN KINETIC... PHYS. REV. APPLIED 11, 054087 (2019) pronounced minimum of the NSD at α ≈ 0.9, which coin- cides with the region of maximum grAl superconducting gap as a function of resistivity. We explain the anticorrelation between NSD and grAl gap using a phonon-trapping model in the surrounding Al ground plane. The measured NEP values for grAl MKIDs, scaled for maximum absorptance to allow for a fair comparison between√ different film resistivities, are in the range of 30 aW/ Hz, and are FIG. 7. First three iterations of a Hilbert fractal, i.e., Hilbert comparable to state of the art [16,32,57,65,76,83–89]. curve of the first three degrees. In green we higlight the recur- Guided by these results, future research should focus rence of (n − 1)th structures in the nth Hilbert fractal iteration, on increasing the grAl superconducting gap (e.g., using a connected by black lines. cold deposition method as found in Ref. [90]), employing a thicker and lower gapped (e.g., titanium) ground plane, and engineering the meandered inductor geometry in order The meandered inductor is shaped as a Hilbert curve [93], to maximize the optical impedance matching between the shown in Fig. 7. resonator film and the photon-collecting medium, as sug- The nth-order fractal can be ideally decomposed into n − gested in Ref. [91]. We believe that the flexibility, low 4 1 zeroth-order structures, which are stripes of width + ≈ losses, and capability of reaching high resistivities with- w and length s w s, oriented horizontally and ver- out being severely limited by the onset of nonlinearities tically with approximately equal distribution. The filling = / suggest grAl as a viable candidate for future ultrasensitive factor is defined as F w s. This geometry renders the MKID applications. detector sensitive to two polarizations at once. Increas- ing the degree n increases the inductance, and decreases ACKNOWLEDGMENTS the impedance. The two geometries we used in this work are second and third degree Hilbert curves (H2 and H3), We are grateful to A. Karpov, L. Swenson, P. de Visser, since n = 2andn = 3 offer a good compromise between and J. Baselmans for insightful discussions, and to L. resonant frequencies that are within the bandwidth of the Radtke and A. Lukashenko for technical support. Facil- readout electronics and impedances that are easily matched ities use is supported by the KIT Nanostructure Service to the substrate. We estimate the resonator sheet resistance Laboratory (NSL). Funding is provided by the Alexander per square needed to assure impedance matching as von Humboldt Foundation in the framework of a Sofja Kovalevskaja award endowed by the German Federal Min- s = Z0 ⇐⇒ ≈ × istry of Education and Research, and by the Initiative and Rn,match √ Rn,match F 100 , w sapphire Networking Fund of the Helmholtz Association, within (A1) the Helmholtz Future Project Scalable solid state quantum computing. This work is partially supported by the Min- where Z ≈ 377 is the impedance of vacuum and istry of Education and Science of the Russian Federation 0 sapphire is the relative dielectric permittivity of the sapphire sub- in the framework of the Program to Increase Competitive- strate. We compare this formula to finite-element simu- ness of the NUST MISIS, Contracts No. K2-2016-063 and lations, showing an accuracy within 5%, and use it to No. K2-2017-081. estimate the detector absorptance as APPENDIX A: GEOMETRY CHOICE AND |Rn − Rn,match| IMPEDANCE MATCHING A(Rn) = , (A2) Rn + Rn,match We give a brief overview of the choice of resonator geometry, following in the footsteps of the much more which we then use to calculate Pabs = δPinA. The fab- detailed treatment found in Ref. [92]. The detectors used ricated MKIDs have = 2.5 mm, w = 2 μm (H3) and in this work are back-illuminated lumped-element res- = 1.2 mm, w = 12 μm (H2). Notice that H2 resonators onators, composed by a meandered inductor and interdig- have about three times larger surface. itated capacitor, and patterned on sapphire. They offer a practical advantage over distributed resonators since the APPENDIX B: NOISE SPECTRAL DENSITY lumped-element capacitance can be swept by changing the length of the capacitor fingers, with no effect on the In MKIDs, the detector signal is the resonant frequency inductance. This is needed to obtain a fine comb of res- shift δf0 caused by millimeter wave illumination. As a onant frequency dips, limited only by the loaded quality consequence, any fluctuation of the resonant frequency factor of the resonators, which is an important consider- observed in the absence of incoming millimeter wave pho- ation towards densely packed kinetic inductance arrays. tons constitutes noise. The noise is calculated by observing

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TABLE I. Geometric parameters with which Eq. (C2) is tested APPENDIX C: KINETIC INDUCTANCE against the exact solution found in Ref. [94]. FRACTION Length (μm) Width (μm) Thickness (nm) The kinetic inductance fraction α is defined as α = /( + ) Span 200–2000 2–20 10–50 Lkin Lkin Lgeom . The kinetic inductance of a supercon- Step 100 2 5 ducting film can be expressed as [40,42,43] = Eq.= (1) 0.18 Rn the resonant frequency fluctuations over time δf (t) under Lkin Lkin, ,(C1) 0 w kB w Tc zero illumination condition and computing the NSD as

where is the length, w is the width, Rn is the normal-state 2 |F{δf0(t)}| |δf0(f )| sheet resistance per square, and T is the critical temper- S(f ) = = √ ,(B1) c B B ature. In the case of the loop-free meandered inductors employed in our resonators, the geometric inductance is where B is the output bandwidth deﬁned as B = only given by self inductance. While the self inductance 1/2texposure. We recall that the change in resonant frequency of a rectangular bar can be computed with an exact closed is caused by a change in kinetic inductance, formula (as reported in Ref. [94]), it is lengthy and cum- bersome. A much more compact formula can be found α δ Lkin in Ref. [95], which in the thin-ribbon limit ( w t) δf0 =−f0 .(B2) 2 Lkin reads The kinetic inductance scales with the inverse of the 2 L ≈ 2 × 10−7 ln .(C2) Cooper pair density nS, allowing one to write geom w

δLkin δnS δxQP − = =− ,(B3)We test this formula against the exact one for all combi- Lkin nS 2 nations of , w,andt listed in Table I. Errors are always below 10%. To estimate the kinetic inductance fraction α where xQP is the normalized grAl quasiparticle density. Under the assumption of quasiparticles dominating over of our resonators starting from their geometry, we combine other sources of noise, we can then link the fluctuations of Eqs. (C1) and (C2). the resonant frequency over time with fluctuations of the In Table II we summarize a brief literature survey of pre- quasiparticle density viously measured kinetic inductance fractions for MKIDs made of various thin-film materials. These values are used α in Fig. 4 in the main text. δf (t) =− f δx (t).(B4) 0 4 0 QP APPENDIX D: MAXIMUM NUMBER OF The resonant frequency fluctuates by up to roughly 1 READOUT PHOTONS IN GRAL 1D RESONATORS kHz, thus for α ≈ 1andf0 of the order of GHz, we −6 obtain δxQP ≈ 10 , comparable to the background values We give a brief recapitulation of the model proposed recorded in Ref. [41]. in Ref. [47]. Granular aluminum is a composite material

TABLE II. Summary of surveyed MKID parameters used in Fig. 4 in the main text, including both distributed- and lumped-element resonators.

Material LE Rn () Tc (K)α(%) Ref. Al Yes 0.66 1.2 6 [11] Al No 0.13, 0.66, 1.3 1.2 7, 45, 63 [69] Al No 0.13 1.2 28 [70] Al No 0.13 1.2 7 [72] Al Yes 4 1.46 40 [73] Al Yes 4 1.2 50 [71] NbTiN No 8.75 14 9 [74] NbTiN No 5.6 14 35 [75] TiN Yes 25 4.1 74 [76] TiN Yes 45 2 ≈ 100 [27] WSi2,W3Si5 No 45, 5.6 1.8, 4 92, 42 [77] WSi2,W3Si5 Both 7, 13, 44, 4.5 1.8, 4 43, 75, 96, 77 [31]

054087-10 INTERPLAY BETWEEN KINETIC... PHYS. REV. APPLIED 11, 054087 (2019) made of aluminum grains in a nonstoichiometric aluminum oxide matrix. The structure of grains, separated by thin insulating barriers, is modeled as a network of JJs [48–50,96]. A stripline grAl resonator, or a lumped- element grAl resonator with a thin enough meandered inductor, can be considered as a one-dimensional chain of eﬀective JJs, which allows the resonator self-Kerr [97] nonlinearity to be derived

ω2 0 K11 = Cπea , (D1) jcV where a is the characteristic size of an aluminum grain, V = wt is the volume of the resonator, jc is the critical current density, e is the electron charge, ω0 = 2πf0 is C the resonant frequency, and is a geometric parameter of FIG. 8. Technical drawing of the aluminum sample holder. The order unity, which in our case is C = 3/16. The maximum inset shows an optical image of a single H3 resonator. number of photons in the resonator at bifurcation is [64] κ APPENDIX F: SAMPLE-HOLDER DESIGN nmax = √ , (D2) 3K11 We show a technical drawing of the aluminum sample holder in Fig. 8. The inset enlarges the sapphire chip (A) κ = / where f0 Qtot is the instantaneous bandwidth of the to show one of the 22 resonators (B), where we highlight resonator. The total kinetic inductance of the JJ array is the interdigitated capacitor (C), meandered third-degree = × / Lkin LJ a, where LJ is the inductance of a single Hilbert curve inductor (D) and CPW feedline (E). Note effective JJ, allowing us to write that H2 resonators have a meander width of 12 μm. The chip also hosts test stripes (F) used for room-temperature jc = . (D3) dc measurements of the sheet resistance (more stripes are 2eaLkinwt present on the wafer prior to dicing). The feedline is wire bonded to the printed circuit boards (G) that couple to co- Under the assumption of strongly overcoupled resonators axial connectors (H). The sample holder is closed with a (Q ≈ Q ), and kinetic inductance dominating over the tot c √ solid aluminum lid (I), with an aperture on the backside geometric inductance (1/f ≈ 2π L C), we can use Eqs. 0 kin (J), allowing for millimeter wave illumination. (D1) to (D3) to write √ 4 2 C APPENDIX G: PHOTON NUMBER CALIBRATION nmax = √ √ . (D4) IN THE DARK CRYOSTAT 2 3 3Qc(πea) Lkin We give a brief description of the dark-cryostat exper- APPENDIX E: SUMMARY OF RESONATOR imental setup, and of the method used to estimate the PARAMETERS number of photons circulating in a resonator. As schematically shown in Fig. 9, the sample under test is thermally We present a summary of the material properties of the anchored to the dilution stage of a cryostat, and we per- fabricated samples in Table III. form rf transmission measurements. We use a vector network analyzer (VNA) to generate the input tone and to TABLE III. Summary of the material properties of the grAl analyze the output. We add room temperature and cryo- films presented in this work. genic attenuators to the input line, with a typical total − ρ α attenuation in the range of 90 dB. A number of fixed Sample Rn Lkin, Tc NEP√ (/)(μ )( /)()() / attenuators and a low pass filter are used at different tem- cm pH K % (aW Hz) perature stages of the cryostat to prevent rf heating and to A2040162.029086± 34 thermalize the input rf field. Once the rf signal is trans- B4080322.119538± 3 mitted through the sample, an isolator is used to prevent C 80 160 64 2.23 97 47 ± 10 ± back-propagating noise from the amplifiers. The signal is D 110 220 88 2.27 98 29 8 filtered and travels through superconducting cables (green) E 450 900 360 2.29 100 58 ± 10 F 800 1600 640 2.22 100 155 ± 48 before being amplified by a HEMT amplifier at 4 K, and a room-temperature amplifier.

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