Electrothermal Feedback in Kinetic Inductance Detectors

T Guruswamy,∗ C N Thomas, S Withington, and D J Goldie Quantum Sensors Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge, CB3 0HE, UK (Dated: August 3, 2021) In Kinetic Inductance Detectors (KIDs) and other similar applications of superconducting microres- onators, both the large and small-signal behaviour of the device may be affected by electrothermal feedback. Microwave power applied to read out the device is absorbed by and heats the superconductor quasiparticles, changing the superconductor conductivity and hence the readout power absorbed in a positive or negative feedback loop. In this work, we explore numerically the implications of an extensible theoretical model of a generic superconducting microresonator device for a typical KID, incorporating recent work on the power flow between superconductor quasiparticles and phonons. This model calculates the large-signal (changes in operating point) and small-signal behaviour of a device, allowing us to determine the effect of electrothermal feedback on device responsivity and noise characteristics under various operating conditions. We also investigate how thermally isolating the device from the bath, for example by designing the device on a membrane only connected to the bulk substrate by thin legs, affects device performance. We find that at a typical device operating point, positive electrothermal feedback reduces the effective thermal conductance from the superconductor quasiparticles to the bath, and so increases responsivity to signal (pair-breaking) power, increases noise from temperature fluctuations, and decreases the Noise Equivalent Power (NEP). Similarly, increasing the thermal isolation of the device while keeping the quasiparticle temperature constant decreases the NEP, but also decreases the device response bandwidth.

PACS numbers: 74.78.-w, 29.40.-n, 07.57.Kp, 85.25.Pb, 85.25.Oj

I. INTRODUCTION onator and allows calculation of experimentally relevant measurements. In this work, we explore numerically the Kinetic Inductance Detectors (KIDs) are ultrasensi- implications of the theoretical model for a typical device tive photon detectors based on superconducting microres- configuration: an Al thin film resonator with resonant fre- onators.1 Their low noise and ease of multiplexing leads quency 5 GHz at a bath temperature of 0.1 Tc = 0.118 K, to a wide variety of applications, particularly in astro- where Tc is the superconducting critical temperature of physics2 and particle physics.3 The principles of opera- Al. tion are very similar to those used in superconducting In particular, as the model of Ref. 14 and the im- qubits4–6 and superconducting quantum interference de- plementation described here include both readout power vice (SQUID) multiplexers.7 During the normal operation heating and superconductor quasiparticle-phonon cool- of a KID, absorbed signal power breaks Cooper pairs in ing, we can calculate the effect of electrothermal feedback the superconductor, changing the quasiparticle effective on device performance. Here we define electrothermal temperature. This causes a change in the superconductor feedback as the phenomenon where readout power heats impedance and so a shift in the amplitude and phase of a the superconductor quasiparticles of a superconducting microwave readout tone transmitted through the device. microresonator, changing the superconductor impedance, Most existing models of KIDs8 do not include a detailed and hence the readout power absorbed. In principle, elec- analysis of how power absorbed by the superconductor trothermal feedback could affect both the magnitude and from the readout tone affects the quasiparticle tempera- bandwidth of the device response to signal power, and ture, or the rate at which the quasiparticles cool via the noise, and could be positive or negative depending on the superconductor phonons. Recent work9,10 has shown that details of the readout frequency, power, and resonator both in theory and experiment, quasiparticle heating due characteristics. In this work we quantify the effect of to readout power is significant under typical device oper- electrothermal feedback on device responsivity and noise ating conditions, and leads to nonlinear device behaviour. for a typical device configuration as in de Visser et al. 9 Furthermore, detailed calculations of quasiparticle-photon We also investigate the effect on device performance of and quasiparticle-phonon interactions in superconduc- intentionally varying the thermal isolation of the device arXiv:1701.09145v1 [astro-ph.IM] 31 Jan 2017 tors11,12 have shown the quasiparticle-phonon power flow from the bath, which may be expected to change the in a superconductor has a significantly different functional device dynamic range, responsivity, noise characteristics, form to the normal state electron-phonon power flow ex- or even to control electrothermal feedback and onset of pressions typically used, and also changes with readout nonlinearities due to readout power heating. In practice, power, frequency, and bandwidth.13 Incorporating these the thermal isolation could be controlled by depositing the findings in a higher level device model, Thomas et al. 14 superconducting resonator on a membrane only connected described a extensible framework which combines quasi- to the bulk substrate by thin legs. This principle is particle heating with the electrical behaviour of the res- already standard for other types of photon detectors such 2 as Transition Edge Sensors,15,16 and is used in recent attributed entirely to quasiparticle losses within the su- KID designs.17–19 As the context of this work is using perconductor, calculated from the real part of the surface superconducting resonators as detectors, we investigate impedance Zs. However the inductance L is the sum of the effects of this thermal isolation on responsivity to the magnetic inductance Lgeo due to the geometry of the signal power and noise. film, as well as that due to the imaginary part of the su- The model represents the device with an equivalent perconductor surface impedance, Ls. As the quasiparticle series electrical circuit and a thermal model of several temperature changes from Tb to Tqp, we modify the values connected heat capacities with a number of external power of R and L as functions of readout frequency ω = 2πνr as sources, dependent both on the temperatures of each element, and a set of externally controlled parameters. This < Zs(ω, Tqp) R(ω, Tqp) = R(ω, Tb) , (1) framework allows the steady-state operating point (set of < Zs(ω, Tb) temperatures and readout output) and the small-signal ! behaviour about that operating point to be determined. = Zs(ω, Tqp) L(ω, Tqp) = L(ω, Tb) + Ls(ω, Tb) − 1 . This small-signal analysis allows us to define and calculate = Zs(ω, Tb) the responsivity to signal (pair-breaking photons) power, (2) required for operating these devices as detectors. We also follow Ref. 14 in introducing noise sources based For the purposes of this paper, both R and Ls are related on temperature fluctuations, and calculate the resulting to the surface impedance by the same geometric factor,23 Noise Equivalent Power (NEP) as device operating point, and so for consistency we require electrothermal feedback, and thermal isolation are varied. Other models investigating electrothermal feedback based 1 = Zs(ω, Tb) 20,21 Ls(ω, Tb) = R(ω, Tb) . (3) on bolometer theory exist, but we consider this work ω < Zs(ω, Tb) more general as we are able to simultaneously consider the large and small-signal behaviour of the device, and Defining the kinetic inductance fraction α = include more complex thermal configurations beyond a Ls(ω, Tqp)/L(ω, Tqp), (3) is equivalent to the known single thermal link between the superconductor and bath. expression for internal quality factor8 We are therefore able to consider bolometric operating modes, a subject of ongoing work.18,22 In section II we 1 = Zs(ω, Tqp) = QI . (4) describe the model in further detail and its numerical α < Zs(ω, Tqp) implementation for a typical KID design. Section III discusses the dependence of the experimentally relevant Considered as an inequality (α ≤ 1), (4) gives the lower measurement outputs on key parameters, in particular limit of QI possible in a model with only quasiparticle focussing on the key questions of how electrothermal feed- losses. back and varying the thermal isolation of the device affect The surface impedance Zs of a superconducting film is device responsivity and noise characteristics. calculated in the local, thick film limit as s iωµ0 II. METHODS Zs(ω, Tqp) = (5) σ(ω, Tqp)

A. Electrical circuit where σ(ω, Tqp) = σ1(ω, Tqp) − iσ2(ω, Tqp) is the complex conductivity calculated using the Mattis-Bardeen We represent the electrical behaviour of the device equations.24 The specific form of the surface impedance using a series equivalent circuit as illustrated in Thomas expression is not crucial to the model; the thin film regime 1 8 et al. 14, figure 3. A superconducting film (resistance R (Zs ∝ σ− ) could equally be considered. and inductance L) is coupled by a capacitance C to a For readout frequencies νr close to the resonance frequency ν , the de-embedded values of the transmission driving generator and readout load of impedance Z0. 0 One of the goals of this work is to be as independent of (S21) and reflection (S11) S-parameters, as measured at the device plane, are14 device geometry as possible. We therefore parametrise√ the circuit in terms of a resonance frequency ν0 = 1/2π LC, 1 Q ν − ν − internal quality factor QI = 2πν0L/R, and coupling qual- T r 0 S11(νr,Tqp) = − 1 + 2iQT , (6) ity factor QC = 4πν0L/Z0. Instead of choosing a device QC ν0 geometry and calculating R, L, and C, we specify the S21(νr,Tqp) = 1 + S11(νr,Tqp) , (7) generator and load impedance Z0, resonator frequency ν0 and quality factors QI and QC at resonance and when the where νr is the readout frequency, total quality factor superconductor quasiparticle temperature Tqp is equal to QT = QI QC /(QI + QC ), and we emphasise that all of ν0, the bath temperature Tb. This uniquely specifies R(Tb), QI , and QC are dependent on quasiparticle temperature L(Tb), and C. Tqp. We are able to calculate the scattering parameters In this electrical equivalent circuit, the resistance R is as functions of detuning of the readout frequency ∆ν = 3

νr − ν0 and quasiparticle temperature Tqp, initialising where V is the volume of the superconductor, the calculations with only the device parameters Z0, ν0, Σs is a material-dependent constant, in this case 10 3 1 12 and quality factors QI and QC at Tb, as well as relevant 3.23 × 10 W m− K− for Al, and ∆(T ) is the super- material properties, but with no dependence on a specific conducting gap energy at temperature T . This expres- device geometry. In test calculations of S21 as a function sion for qrec has been explored both in understanding of readout frequency, as the quasiparticle temperature is nonequilibrium behaviour in resonators11,12 and in super- increased, the resonance peak shifts to a lower frequency conducting tunnel junctions.30 The scattering portion of and becomes shallower and broader in agreement with the power flow could also be described by an approximate detailed microstrip-based models. As the readout power expression for a quasiparticle distribution in quasiequilib- 30 absorbed by the resonator qR is rium, but we find after considering the nonequilibrium quasiparticle distributions resulting from heating by a 2 2 qR = PR 1 − |S21| − |S11| (8) single frequency microwave tone, it is better accounted for in this case by scaling qrec by a factor 1/ηr, where ηr is the power and microwave frequency dependent frac- for an incident readout power PR, we are now able to proceed to a full electrothermal model. tion of the total power flow carried by recombination, derivable from the full solutions of the nonequilibrium kinetic equations as outlined in Refs. 11 and 12. Here, B. Thermal configuration readout frequency does not vary significantly, so only the power dependence of ηr is retained. For Al and a 5 GHz microwave readout, η < 0.6 and has a logarithmic depen- We model the thermal behaviour of a generic KID with r dence on power12 above power densities of 2.83 mW m 3. three heat capacities: the superconductor quasiparticles, − The phonon trapping factor 1 + τ /τ introduced in the superconductor phonons, and the substrate phonons, l pb Ref. 11 is eliminated in favour of an explicit heat flow as shown in the block diagram inset of figure 2. Each term between the superconductor phonons and substrate has its own effective temperature: T , T , and T qp ph su phonons. respectively. The heat capacity of the superconductor These expressions for q and q are plotted against quasiparticles is given by BCS theory;25 while the heat I R quasiparticle temperature T in figure 1 for an example capacity of the superconductor phonons are assumed to qp resonator. Considering only T , possible steady state follow a Debye model. The substrate material of Si N , qp x y operating points where there is no net power flowing an amorphous dielectric, is known to be dominated by into the quasiparticles are at the intersections of q and two-level systems and has a heat capacity linear in tem- I q . We note that for negative detuning of the readout perature26 such that C = c V T , where V is the R su su su su frequency (∆ν = ν − ν < 0), in this example there total volume of the substrate, and c = 26.44 J m 3 K 2 r 0 − − are three possible operating points – two stable and one is an empirically derived constant for Si N .27 Three pos- x y unstable – indicated by the arrows. On a frequency sweep, sible external sources of power are included. The device this shows as hysteresis with respect to sweep direction of readout, at frequencies below the superconducting gap the quasiparticle temperature10 and hence the scattering frequency, heating the quasiparticles (q ); a signal, of R parameter |S |. We also contrast the superconducting photons at frequencies above the gap frequency, also heat- 21 two-temperature exponential model for q (10) to a normal ing the quasiparticles (q ); and any power heating the I S state power law electron-phonon expression for metals substrate upon which the device is patterned (qH ). The readout power absorbed q has been described normal 5 5 R qI = V Σn(Tqp − Tph) , (11) in section II A. Power flow between the superconduc- 9 3 5 tor quasiparticles and phonons qI is described by a two- with Σn = 1.68 × 10 W m− K− for Al from the material temperature exponential approximation to the total power properties used. At low temperatures, the normal state flow.11,12 In general, superconductor quasiparticles and power flow is several orders of magnitude greater than phonons interact both by inelastic scattering and by quasi- the superconducting value from (10). If implemented particle recombination into Cooper pairs. By appropri- in calculations, this would reduce or hide the effects of ately integrating the Chang & Scalapino kinetic equa- readout power heating. tions28,29 describing the rate of change of the quasiparticle For the power flow between the superconductor phonons energy distribution due to these interactions, we derived and the substrate phonons we use the Kapitza expression 4 4 the net power flow due to recombination qrec, and the qP = ΣsuAres(Tph −Tsu), where Ares is the area of the su- total power flow qI , when Tqp Tc and Tph < Tqp in perconducting resonator in contact with the substrate and 2 4 clean superconductors as Σsu is a material-dependent constant, 850 W m− K− for 9 an Al/SixNy interface. The power flow from the sub- qrec = V Σs Tqp exp −2∆(Tqp)/kBTqp strate to bath is modelled as effectively limited by narrow (9) legs. The thermal power carried by SixNy legs is found 31 n n −Tph exp −2∆(Tph)/kBTph empirically to be qL = gsuKb(Tsu − Tb ), where gsu is a geometric factor equal to the ratio of the width to length q = q /η , (10) I rec r of the legs, Kb is a material-dependent constant, and the 9 10−

10 10− 4 11 10− then includes all contributions to the dynamical behaviour 9 of the device, including electrothermal feedback. Typical 10− values for the heat capacities C and thermal conductances 12 G are provided in the block diagram inset of ﬁgure 2. The 10− 10 10− substrate-bath conductance G is varied by changing the (W) L geometry factor g over a few orders of magnitude. q su 11 13 10− The dynamical behaviour of the device about the chosen 10− operating point is described by

12 10− ∂∆T(t) 1 (W) qR, ∆ν < 0 = −C0− · G0 · ∆T(t) , (12) 14 q ∂t 10− 13 10− qR, ∆ν = 0 which can be analysed in the frequency domain as an

qqRR, ∆, ν∆ <ν0 > 0 eigenvalue problem. The modes and dynamical time con- 14 1 15 10− q , ∆ν = 0 stants are found by diagonalising G − · C . As G is 10− qRI (exponential) 0 0 0 qR, ∆ν > 0 not diagonal, the effective time constants may be quite 15 10− qqII(exponential)(power law) different from the time constants expected from C/G of any individual elements. There are three dynamical 16 qI (power law) 10− 16 modes: in the regime where the limiting conductance is 0.1610− 0.18 0.2 0.22 0.24 0.26 0.28 0.16 0.18 0.2 0.22 0.24 0.26 0.28 the quasiparticle-phonon cooling qI , the modes are a fast cooling of the superconductor phonons into the substrate TqpTqp(K)(K) phonons; a moderately slow cooling of the entire device Figure 1. Example of readout power absorbed by supercon- into the bath; and a slow cooling of the superconductor quasiparticles into the superconductor phonons. In ductor quasiparticles, qR, below resonant frequency (∆ν = νr ν0 < 0), at resonant frequency (∆ν = 0), and above general in this regime we find Tqp > Tph ≈ Tsu ≈ Tb. resonant− frequency (∆ν > 0), all as a function of quasiparticle Figure 2 shows how the three time constants associated temperature Tqp. The power flow out of the superconductor with these change as the device’s thermal isolation from quasiparticles and into the superconductor phonons, qI (10), the bath is increased, by changing the geometric factor is included as a dashed line. For comparison the normal state gsu associated with the substrate-bath conductance GL. power law expression (11) is also included. Where q and R As GL is reduced, the dynamical modes change so that qI intersect, as indicated by the arrows, represents a possi- the overall device cooling is slower than the supercon- ble steady-state operating point for the quasiparticles. Al ductor quasiparticle to phonon cooling, and now we find resonator, with PR = 20 pW. Tph > Tb and Tsu > Tb. In the figure, the time constants are labelled based on the heat capacities and conductances involved in the modes they are associated with; for exam- exponent n depends on the detailed physics of the heat ple, τ1, the smallest time constant, is the eigenvalue for flow. Experiments in our group for 200 nm thick SixNy the fastest dynamical mode involving power flowing from n 15 find, typically, Kb = 60 pW K− and n = 2. the superconductor phonons into the substrate phonons, Including both the signal power qS = ηSPS absorbed as for typical parameters this involves a large conductance by the quasiparticles, and heater power qH = ηH PH GP compared to the quasiparticle-phonon conductance 9 absorbed by the substrate phonons, allows consideration GI . of both a direct photon detection mode of operation, and a bolometric mode in which the KID resonator is thermally coupled to some other absorbing system. In this work, we C. Numerical implementation assume the absorbed power is simply the incident power (PS and PH ) multiplied by an efficiency (ηS and ηH ) but The software implementation of the model contains in general more complex models for absorbed power could two parts: finding the operating point given a set of be incorporated. parameters (“large-signal”), and linearising the behaviour Following Ref. 14, we combine the elements into a tem- of the device in response to small perturbations about perature vector T, a heat capacity matrix C0(T, v) and a that operating point (“small-signal”). power flow vector q(T, v) which describes the net power Low temperature (T Tc, the superconducting criti- flow into each element of the model. The vector v includes cal temperature) analytical approximations for supercon- all other parameters which do not depend on temperature, ductor complex conductivity25 and Debye phonon heat such as readout frequency νr, readout applied power PR, capacity were used. Superconductor quasiparticle heat ca- signal frequency νs, signal power PS, and power into the pacity and superconducting energy gap were calculated by substrate PH . Finding the steady-state operating point of numerical integration, assuming the quasiparticle energy the device involves finding the temperature vector T0 such distribution could be approximated by Fermi functions at 25 that q(T0, v0) = 0, for a given parameter vector v0. The the quasiparticle effective temperature T . Solving for qp effective thermal conductance matrix G0 = − ∂q/∂T the operating point q(T, v) = 0 for T = T T T v0 qp ph su 5

1 device consists of a resonator of dimensions 4 mm×3 µm× 10− 200 nm, patterned on a SixNy membrane of the same 2 τ2 ΣC/GL 10− ∼ dimensions, connected eﬀectively by a single thin SixNy leg of dimensions 2 µm × 100 µm × 200 nm to the thermal τ3 Cqp/GI 3 ∼ 10− bath, which is always at Tb = 0.1 Tc = 0.118 K. The superconducting Al was set to have a zero-temperature 4 10− superconductor energy gap of ∆0 = 180 µeV, critical temperature Tc = Cqp aJ/K quasiparticles (Tqp) ∼ 9 1.18 K, and residual resistivity ρn(0 K) = 5.8 × 10− Ω m, 5 2 10− GI pW/K (s) all typical of ﬁlms deposited by our group. ∼ τ superconductor At a bath temperature T = 0.1 T , and a resonance 6 C aJ/K b c 10− phonons (T ) ph 8 ph ∼ frequency of ν0 = 5 GHz, we require QI ≥ 10 from (4).

7 GP nW/K In all the simulations reported here we choose QI (Tb) = 10− ∼ 8 5 substrate 6 × 10 and QC (Tb) = 3 × 10 , with a generator/load C fJ/K phonons (T ) su 8 su ∼ impedance Z0 = 50 Ω. These are the values of QI and 10− GL pW/K nW/K QC at zero incident power; at a typical operating point, ∼ − Cph 9 bath (T ) τ1 when Tqp > Tb, QI is much reduced. 10− b ∼ GP We assume the incident signal power PS is dissipated 10 in the quasiparticles with 100% efficiency (ηS = 1), inde- 10− 2 1 0 10− 10− 10 pendent of signal frequency, as with the incident power gsu PH on the substrate (ηH = 1). We also ignore any cable loss or change to the signal from the mixer or filtering Figure 2. Variation of the dynamical time constants (from (RI ,RQ,R0 = 1 from Ref. 14). the eigenvalues of (12) in the frequency domain) as the device thermal isolation from the bath is decreased by changing the geometry factor gsu. With PR = 25 fW, and zero readout III. RESULTS frequency detuning (νr = ν0(Tqp)). Inset: Block diagram of the thermal model of a generic Kinetic Inductance Detector with thermal isolation. Three heat capacities, each with their A. Operating point own effective temperatures Ti, and relevant power flows qi are considered. Typical heat capacities C and thermal con- By sequentially solving for the operating point, we cal- ductances G for the device design and operating conditions culate how the operating point changes as we sweep the described in the text are labelled. readout frequency, both in the forward and reverse directions. The inset of figure 3 plots the scattering parameter |S | for forward and reverse frequency sweeps at three given v = P ν P ν P was done by multidimen- 21 R r S s H different incident readout powers. At the highest read- sional Newton-Raphson iteration with a quadratic line out power plotted, the forward (solid line) and reverse search,32 and with all temperatures bounded between T b (dashed line) do not follow the same path – readout power and T . Analytical expressions for derivatives were used, c heating induced hysteresis.10 The exact readout power at except for ∂S /∂T and ∂S /∂T which were approx- 21 21 which hysteresis appears is dependent on the resonator imated numerically using first-order central differences. volume, as well as the efficiency with which the resonator In some cases, we want the operating point to follow the couples to the readout line, and the bath temperature. resonant frequency even as it changed with quasiparticle At optimal coupling Q = Q , and for low bath temper- temperature or other parameters. However, the steady I C atures, we expect larger effects due to heating, and so state operating point is needed to calculate the resonant visible nonlinearities at lower powers. frequency, and changing the readout frequency changes In practice, an IQ readout is usually used for reading the operating point. Therefore these calculations, de- out the device, which allows measurement of both the am- scribed as at fixed detuning rather than at a fixed readout plitude and phase of the readout tone as the in-phase and frequency, were done by iteratively solving for the oper- quadrature outputs. The main plot of figure 3 shows the ating point and appropriate readout frequency, until the same frequency sweep as the inset, now plotted in the IQ relative readout frequency change in one iteration was plane. For a perfectly symmetric resonance, a frequency less than 10 9. − sweep traces out a circle. As the readout power increases, increasing the quasiparticle temperature, the circle radius decreases; at the highest powers, the circle distorts and D. Modelled device parameters shows hysteresis between the readout frequency up (solid line) and down (dashed line) sweeps. In this work, we consider a device configuration in This phenomena is similar in effect, but different in which an Al superconducting resonator is patterned on source, to hysteresis due to the intrinsic nonlinearity of the 8,33 a SixNy membrane, which in turn is suspended by thin kinetic inductance, which is not included in this work, SixNy legs from the surrounding bulk substrate. The as the emphasis here is on electrothermal feedback due 6

sorbed powers; if we instead use the normal-state power 15 0.8 law expression (11), Tqp changes by the same amount as

| 0.6 Tph and Tsu, as in a linear system where G is constant.

21 This eﬀect may be useful for isolating the responsivity S | 0.4 of the device from small changes in bath temperature, a 10 0.2 potential source of noise.

) 60 40 20 0 20 40

2 − − − − 5 ∆ν (kHz) (K) 2 10− 5 b gsu = 2 10− (10 T × 4 g = 6 10−

− su Q 3 × 10− 2 qp gsu = 2 10− 0 T × 40 20 0 20 40 − −

0.2 fW 3 5 (K) 10− 20 fW b

− T 5 10−

80 fW −

ph 7 0 5 10 15 20 T 10− 2 40 20 0 20 40 I (10− ) − −

(K) 10−

Figure 3. IQ mixer outputs during up (solid) and down b

T 5 (dashed) sweeps of readout frequency, at selected incident 10− − readout powers PR. At high readout powers, hysteretic be-

su 7 haviour is observed – the up and down frequency sweeps follow T 10− diﬀerent trajectories in the IQ plane. Inset: The forward 40 20 0 20 40 − − scattering parameter S21 measured at the device plane for ∆ν (kHz) the same frequency sweeps.| |

Figure 4. Steady-state temperatures of superconductor quasi- to readout power heating. In particular, readout power particles (Tqp), superconductor phonons (Tph), and substrate heating causes a dissipative nonlinearity which causes phonons (Tsu) as a function of readout frequency detuning (∆ν = νr ν0), at three different values for the thermal conduc- both a resonant frequency shift and a significant change − 10 tance of the device to the bath (by changing gsu). Maximum in the depth of the resonance, whereas a purely reactive heating is at zero detuning. Increasing the thermal isola- nonlinearity such as the kinetic inductance nonlinearity tion of the device increases Tph and Tsu but not Tqp. With 33 causes primarily a shift in resonance frequency. Micro- PR = 0.2 fW. scopically, the heating effect corresponds to the heating of quasiparticles by the microwave field, while the intrinsic kinetic inductance nonlinearity corresponds to the changes in the Cooper pair states due to the microwave B. Small-signal analysis field.34 Which phenomenon is more significant depends on the detailed geometry of the device, as the heating Once the operating point of the device is fixed, we can effect is a function of the absorbed power density, while consider the response to small perturbations. Linearising the kinetic inductance nonlinearity is a function of the the IQ-response about the operating point for small time- current density. dependent changes in the parameter vector ∆v(f), we Typically, the limiting thermal conductance is the quasi- find particle to phonon conductance GI . The steady-state op- ∆P (f) erating point therefore usually has Tqp > Tph ≈ Tsu ≈ Tb. R ∆I(f) = FI (f) xˆ · K0(f) · ∆v(f) + HI (f) We plot the operating point temperatures in figure 4 as 2PR a function of readout frequency detuning; the largest (13) readout power absorbed, and consequently heating, is at ∆PR(f) zero detuning. If we increase the thermal isolation of the ∆Q(f) = FQ(f) xˆ · K0(f) · ∆v(f) + HQ(f) , 2PR device from the bath by decreasing gsu and hence GL, (14) the operating point shifts so both Tph > Tb and Tsu > Tb, but still Tph ≈ Tsu; interestingly, however, Tqp does not in the frequency domain. FI,Q(f) describes the I and Q change by the same magnitude – when there is signifi- response to changes in quasiparticle temperature; K0(f) cant quasiparticle heating, Tqp is isolated from changes maps changes in the parameter vector v to changes in to the other temperatures. This is due to the functional the temperature vector T; and HI,Q(f) describes the form of GI being highly nonlinear even for moderate ab- change in I and Q directly due to changes in readout 7

power, independent of quasiparticle temperature. A full PR = 0.2 fW derivation and discussion is available in Ref. 14. 0.4 GR Full expressions for the readout contributions to K0 in Gqp Thomas et al. 14 account for the resonator dynamics – in 0.2 particular, its electrical response time QT /ν0. This in- 0 troduces imaginary components in the frequency domain, (fW/K) G 0.2 and physically corresponds to the possibility of energy − oscillating between the electrical and thermal systems. 0.4 − All other contributions to K0 are assumed quasistatic, and 40 20 0 20 40 so are entirely real. − − PR = 20 fW

We identify the contribution to G0 from GR = GR − ∂qR ∂Tqp as the electrothermal feedback. As shown in 0.5 Gqp ﬁgure 5, GR has the opposite sign to the other contributions to the quasiparticle temperature-dependent elements

of the thermal conductance matrix {G0} = Gqp, and (pW/K) 1,1 0 so has the effect of reducing the net effective thermal con- G ductance from the quasiparticles to the superconductor phonons. In principle, it may be possible to find a device 40 20 0 20 40 operating point where the contribution of electrothermal − − feedback is made into negative feedback, and so enhances ∆ν (kHz) the net effective thermal conductance, by varying the readout frequency, device quality factors and readout powers. Figure 5. (a): Effective thermal conductance between super- However, stable device operation is only possible for a conductor quasiparticles and bath as a function of readout limited range of parameters, and we find for a typical frequency detuning ∆ν = νr ν0. GR = ∂qR/∂Tqp (solid) − − configuration the contribution of electrothermal feedback is the contribution of electrothermal feedback to the effective is consistently positive. At high readout power, when the thermal conductance, and Gqp = ∂qR/∂Tqp + ∂qI /∂Tqp (dashed) is the total component of− thermal conductance de- operating point (in particular Tqp) is asymmetric about the resonance frequency due to readout power heating, pendent on quasiparticle temperature. (b): At high readout powers, Tqp asymmetry about the resonant frequency also GR is also asymmetric about the resonance frequency. leads to an asymmetric GR.

K0(f) maps small changes in the parameter vector v to changes in the temperature vector T. Its bandwidth C. Responsivity is determined by the time constants associated with the dynamical cooling modes of the device. We can com- pare the magnitude of the response to changes in signal To quantify the responsivity of the device to signal power to changes in substrate heating power to under- power PS, we can investigate the expressions ∆I/∆PS stand the operation of the device as a phonon-mediated and ∆Q/∆PS, where ∆I and ∆Q are the changes in the detector, where a separate structure absorbs the primary I and Q outputs for a change in signal power ∆PS. We signal and is connected to the device substrate only ther- can consider the resulting trajectory in the IQ plane, mally, and so substrate phonons must carry power to combining these two expressions to obtain a direction of the KID for detection.18,22 Figure 6 shows the photon maximum responsivity. A single frequency modulation in signal power response in (a) is several orders of magni- signal power creates an ellipse centred on the operating tude greater than the substrate heating power response point. The major and minor axes of this ellipse are in (b). This would again appear to be due to the nonlin- q earity of the quasiparticle-phonon conductance effectively ∆A(f) = <∆I(f) 2 + <∆Q(f) 2 (15) isolating the quasiparticle temperature from the rest of q the device – if using the normal-state power law expres- ∆B(f) = =∆I(f) 2 + =∆Q(f) 2 . (16) sion (11) the responsivities are equal, but when using the appropriate superconducting expression (10), orders By choosing the maximum of ∆A/∆PS and ∆B/∆PS we of magnitude more substrate heating power is required have a single measure of the responsivity of the device than pair-breaking photon power in order to cause the when using an optimal readout technique. same change in quasiparticle temperature. FI,Q is the One of the primary aims of this work is to extract response of the IQ output to changes in quasiparticle from the model the dependence of the responsivity on temperature Tqp. The rolloff of the electrical response electrothermal feedback and the thermal isolation of the outside the bandwidth of the detector (ν0/QT ) is seen in device. If we construct an effective overall thermal con- figure 6(c) and (d). This rolloff of the electrical response ductance between the quasiparticles and the bath Gtotal, is also dependent on readout frequency detuning. which must always be positive at a stable operating point, 8

6 15 with feedback 100 (c) < 10 without feedback

= /K) 5 50 3

5 −

0 (10

0 I

F 4 3) (signal) (K/fW) , (a) 5 (1 50 0 − − 3 1 5 3 1 5 (1/pW) K 10− 10 10 10− 10 10

S 3 P

1 40 ∆

(b) (d) A/

∆ 2

0.5 20 /K) 3 −

0 0 (10 1 Q F 5) (heater) (K/pW) , (1

0 0.5 20 − 3 1 5 3 1 5 − 0 K 10− 10 10 10− 10 10 40 20 0 20 40 − − f (Hz) f (Hz) ∆ν (kHz)

Figure 6. (a) and (b): Real (solid) and imaginary (dashed) Figure 7. Responsivity of IQ output, in the direction of maxi- parts of elements of the parameter response matrix K0, which mum responsivity A, to zero-frequency changes in signal power v maps changes in the parameter vector to changes in the PS , with (solid) and without (dashed) the effects of electrother- temperature vector T, as a function of the modulation fre- mal feedback, as a function of readout frequency detuning v quency of ∆ (f). The response is bandwidth limited by the ∆ν = νr ν0. At this moderately high incident readout power, electrothermal dynamical response time of the overall device. heating leads− to an operating point asymmetric about reso- K0 is the superconductor quasiparticle temperature Tqp { }1,3 nance, and leads to the peak responsivity being slightly below response to changes in absorbed signal power PS , and K0 resonance. Including the contribution of electrothermal feed- { }1,5 is the Tqp response to changes in absorbed substrate heater back decreases the effective thermal conductance, increasing power PH . (c) and (d): Real (solid) and imaginary (dashed) the responsivity. With PR = 25 fW. parts of IQ output response FI,Q which map changes in superconductor quasiparticle temperature Tqp to changes in I and Q mixer outputs, as a function of the modulation fre- When the substrate-bath conductivity GL is decreased by quency of ∆v(f). This response is limited by the electrical a factor of 103, so it is comparable to the superconductor bandwidth of the resonator, ν /Q . With zero detuning of 0 T quasiparticle-phonon conductance G , the device response readout frequency, and P = 20 aW. I R bandwidth shows two separate time constants; in this case one is associated with cooling of the quasiparticles into the superconductor and substrate phonons, and the other we would expect the responsivity essentially depends on 1 is the cooling of the entire device into the bath. the inverse of this effective thermal conductance, ∝ Gtotal− , as in standard bolometer theory.18 Therefore the effect of feedback – under typical conditions, a negative con- D. Noise and NEP tribution reducing the positive Gtotal – is to increase responsivity, and this is confirmed by the detailed model as shown in figure 7. For a symmetric resonance, the max- Noise is introduced into the model in a way similar to imum responsivity is obtained at zero detuning, when the equilibrium analysis – we consider a Langevin source such that the internal energy of each of the thermal elements readout frequency νr is equal to the resonant frequency fluctuates according to its temperature ν0(Tqp). When the resonance is asymmetric due to readout power heating, the optimal readout frequency may be below the resonant frequency, also shown in figure 7. h∆U(t)∆U(t)†i = kBC0 · T0 · T0 , (17) Readout schemes exploiting nonlinear, asymmetric resonances may be possible for devices operating within a where {T0}mn = {T0}mδmn is a diagonal matrix con- carefully controlled parameter space.33 structed from T0. This leads to a noise power vector ∆P added to the power flow vector q, with spectral power Since the responsivity ∝ G 1 , decreasing the effective total− density14 thermal conductance of the device by increasing the thermal isolation from the bath also increases the magnitude of the responsivity. Figure 8 shows this as well as the h∆P(f)∆P(f)†i = 2kB G0 · T0 · T0 + T0 · T0 · G0† corresponding decrease in the device response bandwidth. (18) 9

0.7 thermal responsivity14 and is increased overall by the 5 A, gsu = 2 10− electrothermal feedback. The noise in the IQ output is × A, g = 0.02 plotted in figure 9, with and without the effects of elec- 0.6 su 5 B, gsu = 2 10− trothermal feedback. Assuming a uniform spectrum for × B, gsu = 0.02 the temperature fluctuations as for Langevin sources, the 0.5 bandwidth of the noise in the IQ output is determined by the same dynamical time constants as the responsivity.

(1/fW) 0.4

S 16 P with feedback ∆ 0.3 14 without feedback A, B/ ∆ 0.2 12 /Hz) 12 − 10

0.1 (10 i † } 8 0 3 2 1 0 1 2 3 4 I,Q

10− 10− 10− 10 10 10 10 10 {

∆ 6 f (Hz) } I,Q Figure 8. Responsivity of IQ output in principal directions A { 4 ∆ h (solid) and B (dashed) to a small change in signal power PS , at selected values of thermal isolation of the device from the 2 bath, as a function of the modulation frequency f of the signal power change. The change in the response bandwidth is due 0 to the changes in the dynamical response time of the device 40 20 0 20 40 as the eﬀective thermal conductance changes. Al resonator, − − ∆ν (kHz) with zero detuning of readout frequency, and PR = 0.2 fW.

Figure 9. Noise squared spectral density in IQ output due to temperature fluctuations, with (solid) and without (dashed) This expression can be compared to the expression for the effects of electrothermal feedback, as a function of readout noise power limited by phonon fluctuations in a thermal frequency detuning ∆ν = νr ν0. In this model only tem- conductance G, well known in bolometer theory,15 of perature fluctuation noise is included,− so noise peaks when 2 h∆P (f)∆P (f)∗i = 4kBT G. We note that in the IQ quasiparticle temperature Tqp is at a maximum. Electrother- output, this noise corresponds only to noise in one direc- mal feedback decreases the effective thermal conductance and tion in the IQ plane, as the only source is temperature so decreases the noise in the power flows, but also increases the responsivity such that the noise in the IQ output increases fluctuations in Tqp. This fluctuation in quasiparticle effective temperature in magnitude. With PR = 25 fW. is, microscopically, the generation-recombination noise of the superconductor quasiparticles.35 However, there Noise Equivalent Power (NEP) is a common figure of are three important points: this expression should be merit for detectors; it is defined in this case for signal evaluated at the effective temperature of the quasipar- power into a detector as ticles T , which may be significantly higher than the qp p bath temperature Tb; the overall noise is scaled by 1/ηr, h∆x∆x†i NEPx = , (19) as in addition to quasiparticle recombination, there is dx/dPS also quasiparticle-phonon scattering contributing to a fluctuation in the effective temperature; and finally, the where x is the output as experimentally measured, for electrothermal feedback further changes the effective G example the IQ output. Lower NEPs correspond to more and so changes the magnitude of the observed noise. The sensitive detectors; KIDs were originally proposed to op- 19 1/2 1 overall effect of the first two factors is to increase the noise erate at NEPs below 10− W Hz− . The NEP contri- estimates over a calculation assuming Tqp = Tb, while the bution from the intrinsic noise of the device is plotted in electrothermal feedback then reduces it slightly. Overall, figure 10, as a function of the modulation frequency of the we find for typical conditions the noise power is higher signal power. Since in this model the axis of thermal noise than an estimate assuming Tqp = Tb. We must also distin- in the IQ plane and the axes of maximum and minimum guish the noise in the power flows (18), which reduces due IQ responsivity to signal power do not need to align, to electrothermal feedback, compared to the consequent the NEP contribution is split up into two components noise in the IQ output, which includes the electrical and along A and B. The NEP contribution is flat within 10 the response bandwidth of the device, then increases out- shows the effects of readout power heating, which can be side that bandwidth. In practice, there would also be increased by increasing the thermal isolation of the device. other noise sources, for example from the readout system This readout power heating and electrothermal feedback or photon noise, which would also limit the achievable lead to nonlinear resonances, asymmetric about the reso- NEP. At low frequencies in particular, there can be a sig- nant frequency; at high readout powers, we observe hys- nificant 1/f component to the noise,36 constraining the teresis and multiple stable operating points. Two regimes performance of very narrow bandwidth devices with high of operation are possible; one in which the superconductor thermal isolation or Q-factors. Overall we find the con- quasiparticle-phonon thermal conductance is the limiting tribution of intrinsic thermal noise to NEP, as measured factor, and so the superconductor phonons and substrate in the IQ output, increases with the effective total ther- phonons are approximately at the bath temperature; and 1/2 one in which the substrate-bath thermal conductance mal conductance as NEP ∝ Gtotal. This means that the electrothermal feedback reducing Gtotal has a net benefit, is the limiting factor, and in steady-state operation the reducing the NEP, as shown in figure 10. For comparison, superconductor phonons and substrate phonons may be significantly above the bath temperature. We find that the NEP of a bolometer limited√ by phonon noise in a 16 2 thermal conductance G is 4kBT G in equlibrium. when quasiparticle heating is significant, the nonlinear functional form of the quasiparticle-phonon thermal con-

16 ductance appears to isolate the quasiparticle temperature 10− A with feedback from other parameters, such as substrate heating power A without feedback and bath temperature. This eﬀect will be explored and B with feedback quantiﬁed in further work. )

2 17

/ 10− B without feedback

1 Linearising the behaviour of the device about the − steady-state operating point, we consider the ellipses created in the IQ plane by sinusoidal modulations of 18 10− the temperature-independent parameters, and deﬁne the optimal responsivity of the device by ﬁnding the two principal axes. Following this readout strategy allows us to

19 show that the maximum responsivity is at the resonant 10− frequency when readout power heating is not significant, but slightly below when heating is significant. The device output responsivity to signal (pair-breaking) power is in- NEP contribution (W Hz 20 10− versely proportional to the effective overall conductance 1 from the quasiparticles to the bath Gtotal− , as is the noise squared in the IQ output due to temperature fluctuations. This leads to a Noise Equivalent Power ∝ G1/2 . There- 3 2 1 0 1 2 3 4 5 6 total 10− 10− 10− 10 10 10 10 10 10 10 fore increasing the thermal isolation of the device while f (Hz) keeping quasiparticle temperature constant (for example by reducing readout power), decreasing the effective ther- Figure 10. Noise-equivalent signal power (NEP) in IQ output, mal conductance from the superconductor quasiparticles contribution from intrinsic temperature fluctuations, along the to the bath decreases the NEP of the device. For typ- directions of A (solid) and B (dashed) from (16) in the IQ ical operating conditions we also find the contribution plane, with and without the effects of electrothermal feedback. of electrothermal feedback to be positive, reducing the NEP peaks at the limits of the response bandwidth of the overall thermal conductance, and so helping to reduce the device. Electrothermal feedback reduces the effective thermal NEP of the device. However, there are tradeoffs between conductance and so reduces the NEP. With zero detuning of reducing thermal conductance and the dynamic range readout frequency, and PR = 2 fW. and bandwidth, as well as between the increased noise from a higher quasiparticle temperature, and responsivity to bath temperature fluctuations. For highly thermally isolated devices, we also expect nonlinearities will become IV. CONCLUSIONS prominent at lower absorbed powers. These conclusions may inform device design of the next generation of KIDs 20 1/2 Both the large-signal operating point and small-signal attempting to reach NEPs of 10− W Hz− , required device behaviour of a Kinetic Inductance Detector may be by a range of far-infrared astronomy applications.15 affected by readout power heating of the superconductor The extensible nature of this model leaves many possi- quasiparticles – electrothermal feedback. Implementing a ble avenues for further work, including investigating how general, extensible model of a typical Kinetic Inductance background signal loading affects device performance; in- Detector allows us to investigate the effect on device be- cluding more noise sources, for example from the readout haviour. The steady-state operating point, for a given system or phase noise, and calculating their effect on readout frequency and applied readout power, clearly NEP and the optimal readout scheme; and including 11 other common nonlinear phenomena affecting KIDs, such considered here, the model is also applicable to devices as Two-Level Systems (TLS) which as significant at low intended to detect pulsed signal power, and so may be readout powers, and the intrinsic nonlinearity of kinetic useful in calculating expected pulse shapes for high-energy inductance due to the change in Cooper pair states in photon detection applications. superconductors carrying high currents. Although not

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