Electrothermal Model of Kinetic Inductance Detectors
EElectrothermal Model of Kinetic InductanceDetectors
C.N. Thomas, S. Withington and D.J. Goldie
Cavendish Laboratory, JJ Thomson Avenue, Cambridge, CB3 0HE, UKE-mail: [email protected]
Abstract.
An electrothermal model of Kinetic Inductance Detectors (KIDs) isdescribed. The non-equilibrium state of the resonator’s quasiparticle system ischaracterized by an effective temperature, which because of readout-power heatingis higher than that of the bath. By balancing the flow of energy into the quasiparticlesystem, it is possible to calculate the steady-state large-signal, small-signal and noisebehaviour. Resonance-curve distortion and hysteretic switching appear naturallywithin the framework. It is shown that an electrothermal feedback process exists,which affects all aspects of behaviour. It is also shown that generation-recombinationnoise can be interpreted in terms of the thermal fluctuation noise in the effectivethermal conductance that links the quasiparticle and phonon systems of the resonator.Because the scheme is based on electrothermal considerations, multiple elements can beadded to simulate the behaviour of complex devices, such as resonators on membranes,again taking into account readout power heating. a r X i v : . [ a s t r o - ph . I M ] A p r lectrothermal Model of Kinetic Inductance Detectors PACS numbers: 85.25.Pb, 85.25.Oj, 07.57.Kp
1. Introduction
Kinetic Inductance Detectors (KIDs) [1, 2, 3, 4] comprise a superconducting thin-filmmicrowave resonator connected to an optical or submillimetre-wave absorber of somekind. The resonator operates at a temperature T well below the superconductingtransition temperature T C : T /T C ≈ .
1. In this regime, the resonant frequency isstrongly influenced by the surface kinetic inductance of the film, which changes whenan energetic photon or phonon is absorbed. The design is intrinsically frequency-multiplexable, as multiple devices tuned to different resonant frequencies can becoupled to the same readout transmission line [5]. The devices can then be readoutsimultaneously by using a software-defined radio receiver to probe the changes intransmission gain and phase shift along the line in response to illumination. KIDs arebeing developed for applications across the whole of the electromagnetic spectrum, withparticular emphasis on realizing large-format imaging arrays and chip spectrometers [6,7, 8].There is increasing evidence that the energy dissipated in the resonator by themicrowave readout signal places fundamental limits on device performance. AlthoughKIDs are read out at frequencies less than 10 GHz, well-below the direct pair breakingfrequency of most low- T C superconductors (e.g. 160 GHz for Al at T /T C = 0 . lectrothermal Model of Kinetic Inductance Detectors
2. Thermal model
Goldie [9] has investigated the non-equilibrium distributions of quasiparticles andphonons in a superconductor in the presence of sub-gap microwave readout power.The simulations were performed within the rate-equation framework of Chang andScalapino [17]. Goldie’s simulations result in a number of important observationsthat motivate an electrothermal model that can be used for device-level calculations:(i) The driven quasiparticle distribution is well-approximated globally by a thermaldistribution having an effective temperature T qp , which is higher than the effectivephonon temperature T ph . The non-equilibrium values of quantities such as surfaceimpedance can be calculated to good accuracy using equilibrium expressions and theeffective quasiparticle temperature. (ii) The energy flow between the quasiparticle andphonon systems is well described by a simple expression involving T qp and T ph , in ananalogous manner to low-temperature electron-phonon coupling in normal metals [9].(iii) The original modelling has been extended to the case where above-gap signalphotons and sub-gap readout photons are absorbed simultaneously, and to a range ofsuperconducting materials [18]. This work shows that although there are subtle state-blocking effects, it is sufficient for most purposes to describe the effects of the readoutand signal photon fluxes as heating processes.On the basis of these observations, it is possible to create an electrothermal modelthat takes into account quasiparticle heating, and that can be used to explore thebehaviour of membrane-supported devices. Figure 2 shows a generic model of a KID.It distinguishes the quasiparticle system of the resonator, the phonons in the resonator,the phonon system of the substrate, which may be a suspended membrane, and thephonon system of the bath. q S is the rate at which energy is supplied by a pair-breakingoptical signal, q R is the readout power dissipated in the resonator, and q H representsany thermal power applied to the substrate directly: say from a calibration resistoror microstrip RF load on a membrane. The arrows indicate assumed directions ofenergy flow. Models of this kind can be used to represent the behaviour of all devicesencountered in practice: for example, the membrane-supported optical KID shown inFigure 2. In fact when testing devices of the kind shown in Figure 2, we have found thathysteretic switching occurs at readout powers of order 100 times smaller than identicalsubstrate-mounted devices, indicating that strong electrothermal feedback processes are lectrothermal Model of Kinetic Inductance Detectors SuperconductorQuasiparticlesSuperconductorPhononsSubstratePhononsBathq R q H q I q P q L q S T qp T ph T su T b p qp,N p ph,N p su,N Figure 1.
Electrothermal model of a generic KID. T qp , T ph , T su and T b are theeffective quasiparticle, phonon, substrate and bath temperatures respectively. q R and q S denote the dissipated read-out and signal powers respectively. q H represents anypower dissipated in the substrate directly, say by a calibration heater. q I , q P and q L are internal power flows. The terms of the form p ..., N are effective noise sources. Resonator bodyActive section (absorber)Micromachined suspended island Substrate
Figure 2.
SiN membrane-supported optical-photon counting KID fabricated in thePhysics Department, University of Cambridge. present. In the case of KIDs on bulk substrates, it is only necessary to use the elementsin the dashed box.Assuming the resonator can be modelled as an impedance shunting a losslessreadout line, the dissipated power is given by q R = 2 (cid:60) [ S ( ν r ) (1 − S ∗ ( ν r ))] P R , (1)where S is the forward S -parameter of the resonator defined using reference planes onthe readout line, ν r is the readout frequency, P R is the incident readout power, and (cid:60) lectrothermal Model of Kinetic Inductance Detectors S is discussed in Section 3.1. An importantquestion is into which block in Figure 2 does q R flow? Ohmic dissipation transfersenergy to the quasiparticle system, whereas dielectric or Two-Level-System (TLS)loss [19] transfers energy to substrate phonons. In the case of a transmission-line, thesemechanisms can be described by a distributed series resistance and a shunt conductancerespectively, giving attenuation coefficients α and α . It is then straightforward toshow that the total power dissipated divides according to the ratios α / ( α + α ) and α / ( α + α ), allowing q R to be partitioned between the quasiparticle and phonon blocksin Figure 2.The power dissipated by a pair-breaking optical signal is q S = ηP S , (2)where P S is the incident optical power and η is the optical efficiency. Other models ofthin-film superconducting devices normally include a quasiparticle generation efficiencyin the optical term [20, 18]. This accounts for the fraction of energy lost on shorttime scales when high-energy (potentially pair-breaking) phonons escape from the film.However, this behaviour is included explicitly in our model via the heat flow betweenthe superconductor and substrate phonon systems.The rate of energy flow between the quasiparticle and phonon systems, q I , can beapproximated by q I ( T qp , T ph ) = V Σ s η g ( P abs ) [ T qp exp ( − g ( T qp ) /k b T qp ) − T ph exp ( − g ( T ph ) /k b T ph )] , (3)where V is the active volume of the resonator, T ph is the phonon temperature, Σ s is a material-dependent constant and η g ( P abs ) is the fraction of q I resulting fromrecombination of quasiparticles into Cooper pairs [9]. In what follows, all temperaturesare effective temperatures unless otherwise stated. Strictly, (3) is valid for T qp /T C < . T b rather than the superconductor phonontemperature T ph . This was a convenience introduced in the original papers because itwas found that T ph ≈ T b . Secondly, in [9] and [13], (3) includes an additional factor of[1 + τ l /τ pb ] − that accounts for the loss of pair-breaking phonons from the film into thesubstrate; τ l and τ pb are, respectively, the characteristic timescales over which loss andpair-breaking processes occur. In this paper we represent phonon-loss by an explicitheat flow term between the phonon systems of the superconductor and substrate, sothe factor is unnecessary. This partitioning allows the inclusion of the effects of thesuperconductor phonon heat capacity in the small signal model, which is not possiblein the original formulation.The heat flow between the phonon system of the superconductor and the phonon lectrothermal Model of Kinetic Inductance Detectors q P , is given by q P ( T ph , T su ) = A Σ[ T n ph − T n su ] , (4)where T su is the temperature of the substrate phonons, A is the area of the interface,and Σ is a material-dependent parameter. It is common to take n = 4 on the basis ofthe acoustic- and diffuse-mismatch models of phonon scattering at a boundary [21].When a suspended membrane is used, the heat flow from the substrate to the bath, q L , has a similar form to (4), but n can take on a range of values depending on thecross-section and length of the legs used to support the membrane. Calculations of thiskind are discussed in [22] and [23], and are well known in bolometer theory.While we have assigned the superconductor phonons a single effective temperature,models also exist in which the phonons are divided into two populations, with differenteffective temperatures, based on their ability to break Cooper pairs [24]. This andother more detailed models can be treated within the framework of this paper bymodification of the underlying thermal model. For example, the two-temperature modelsjust described can be incorporated by splitting the superconductor phonons in Figure2 into two heat capacities representing the sub-populations, then partitioning the heatflows accordingly.
3. Microwave response
In our scheme, the state of a resonator is parameterized by the effective quasiparticletemperature, T qp , and therefore it is necessary to consider how perturbative changes in T qp about some quiescent value lead to small changes in the amplitude and phase of thetransmitted microwave signal. In what follows, we shall use ˜ g ( ν ) to denote the one-sidedspectral representation of a time-domain signal g ( t ): g ( t ) = (cid:90) ∞ (cid:60) (cid:2) ˜ g ( ν ) e πiνt (cid:3) dν. (5) ν will be used for frequencies near the readout frequency ν r , and f will be used for signalfrequencies parametrically down-converted into the range 0 ≤ f (cid:28) ν r . To calculate microwave response, a KID can be modelled in an architecture-independentmanner as a series RLC resonator shunting a readout line: Figure 3. V ( t ) represents theapplied readout voltage, Z the characteristic impedance of the readout line and V ( t )the transmitted voltage. R , L and C are the effective lumped resistance, inductanceand capacitance of the KID which, for generality, are all assumed to depend on T qp .Analysing this circuit, we find 2 V ( t ) = V ( t ) − Z I ( t ) andˆ F ( T qp ) I ( t ) = ∂ t V ( t ) , (6)where the differential operator ˆ F is defined asˆ F ( T qp ) = 2 L ( T qp ) ∂ t + [2 R ( T qp ) + Z ] ∂ t + 2 C ( T qp ) − . (7) lectrothermal Model of Kinetic Inductance Detectors V(t)Z Z R(T qp )L(T qp )C(T qp ) I(t)V (t) Output Figure 3.
Equivalent circuit of a KID resonator.
Strictly (6) and (7) assume that the rate of change of T qp is smaller than the rates ofchange of V and I , so that L and R can be extracted from the time derivatives. However,this requirement is always met under normal operating conditions. For constant T qp ,the solution of (6) in the spectral domain is˜ V ( ν ) = 12 S ( ν, T qp ) ˜ V ( ν ) (8)˜ I ( ν ) = − Z − S ( ν, T qp ) ˜ V ( ν ) (9) S ( ν, T qp ) = − Q T Q C (cid:20) iQ T ν − ν ν (cid:21) − (10) S ( ν, T qp ) = 1 + S ( ν, T qp ) , (11)where ν = 1 / π √ LC is the resonant frequency, Q C = 4 πν L/Z is the coupling qualityfactor, Q I = 2 πν L/R is the internal quality factor and Q T = Q I Q C / ( Q I + Q C ) isthe total quality factor. S and S are the de-embedded values of the reflection andtransmission S -parameters, as would be measured at the device plane.Operationally, a homodyne readout scheme is usually used, in which a microwavetone is split between two signal paths, one through the KID and one bypassing it. Thebypass and through signals are then used to drive the LO and RF ports respectivelyof an I/Q mixer. The in-phase ( I ) and quadrature ( Q ) outputs of the mixer can beexpressed as I ( t ) = (cid:90) t −∞ R I ( t − t (cid:48) ) V in ( t (cid:48) ) cos 2 πν r t (cid:48) dt (cid:48) (12) Q ( t ) = − (cid:90) t −∞ R Q ( t − t (cid:48) ) V in ( t (cid:48) ) sin 2 πν r t (cid:48) dt (cid:48) , (13)where ν r is the readout frequency, V in ( t ) is the input voltage to the mixer, and R I ( t ) lectrothermal Model of Kinetic Inductance Detectors R Q ( t ) are response functions that characterize post-conversion filtering. V in ( t ) isrelated to V ( t ), the voltage across the KID, by V in ( t ) = (cid:90) t −∞ R ( t − t (cid:48) ) V ( t (cid:48) ) dt (cid:48) , (14)where the effects of amplification and cabling have been collected into a system responsefunction R ( t ). In the remainder of the paper we assume that the post-conversionbandwidth is always small enough that 0 ≤ f (cid:28) ν r , and then in the spectral domain(12) and (13) become˜ I ( f ) = + 18 ˜ R I ( f ) (cid:104) ˜ R ( ν r + f ) ˜ V ( ν r + f )+ ˜ R ∗ ( ν r − f ) ˜ V ∗ ( ν r − f ) (cid:105) ˜ Q ( f ) = − i R Q ( f ) (cid:104) ˜ R ( ν r + f ) ˜ V ( ν r + f ) − ˜ R ∗ ( ν r − f ) ˜ V ∗ ( ν r − f ) (cid:105) . (15)When the quasiparticle temperature is constant, T qp = T , (8) can be used in (15) toshow ˜ I (0) = 18 (cid:60) (cid:104) ˜ R I (0) ˜ R ( ν r ) S ( ν r , T ) V (cid:105) (16)˜ Q (0) = 18 (cid:61) (cid:104) ˜ R Q (0) ˜ R ( ν r ) S ( ν r , T ) V (cid:105) . (17)The outputs of the I/Q mixer therefore measure the real and imaginary parts of theproduct of the transmission characteristic of the measurement system and the desiredtransmission characteristic, S , of the KID. Now assume that T qp is varied according to T qp ( t ) = T + ∆ T qp ( t ), where T is thequiescent, or operating-point temperature of the quasiparticle system. Techniques forfinding T will be discussed later. The resonator current can be written I ( t ) + ∆ I ( t ),where I ( t ) is the current when T qp = T , and ∆ I ( t ) is the perturbation produced by∆ T qp ( t ). Because the temperature dependence is in the coefficients of (6), the operatorˆ F ( T qp ( t )) may be writtenˆ F ( T qp ( t )) ≈ ˆ F ( T ) + ∆ T qp ( t ) · [ ∂ T qp ˆ F ( T qp )] T . (18)(18) and the perturbed resonator current can then be substituted into (6). Cancellingsteady state terms and neglecting those that are second order in the perturbations, wefind that ∆ I ( t ) obeysˆ F ( T )∆ I ( t ) = − ∆ T qp ( t ) · [ ∂ T qp ˆ F ( T qp )] T I ( t ) , (19)where we again emphasise that ∆ T qp ( t ) varies slowly with respect to the microwavevoltages and currents. (19) can be recognised as a variant of (6), where the effective lectrothermal Model of Kinetic Inductance Detectors I ( t ). Remembering that I ( t ) = ˜ I exp [2 πiν r t ], theright-hand side of (19) can be expressed as − (cid:60) (cid:26) πiν r V [1 − S ( ν r , T )] − [ ∂ T qp S ] ν r ,T × (cid:20)(cid:90) ∞ ν r ∆ ˜ T qp ( ν − ν r ) e πiνt dν + (cid:90) ν r ∆ ˜ T ∗ qp ( ν r − ν ) e πiνt dν (cid:21)(cid:27) , (20)where we have replaced ∆ T qp ( t ) with its spectral representation and made use of the fact ∂ T [ ˆ F ( T )] T I ( t, T ) = − ˆ F ( T ) ∂ T [ I ( t, T )] T when I ( t, T ) is a solution of (6). In additionwe have dropped a term that involves evaluating ∆ ˜ T qp ( ν ) at frequencies ν > ν r , inkeeping with our assumption that T qp varies more slowly than the current. It is thenstraightforward to solve (19) in the spectral domain using (20), giving∆ ˜ I ( ν ) = − V Z ν r ν [1 − S ( ν r , T )] − [1 − S ( ν, T )] × [ ∂ T qp S ] ν r ,T (cid:40) ∆ ˜ T ∗ qp ( ν r − ν ) for ν < ν r ∆ ˜ T qp ( ν − ν r ) for ν > ν r . (21)(21) shows that the spectrum of changes in the temperature appears in the resonatorcurrent in two sidebands around ν r : i.e. the device functions as a parametricupconverter.It follows from 2 V ( t ) = V ( t ) − Z I ( t ) that the current perturbation ∆ I ( t ) producesan additive change of ∆ V ( t ) = − Z ∆ I ( t ) / V ( t ). (15) can thenbe used to be used to determine the corresponding changes ∆ I and ∆ Q in the outputsof the I/Q mixer: (cid:40) ∆ ˜ I ( f )∆ ˜ Q ( f ) (cid:41) = (cid:40) ˜ F I ( f )˜ F Q ( f ) (cid:41) · ∆ ˜ T qp ( f ) , (22)where (cid:40) ˜ F I ( f )˜ F Q ( f ) (cid:41) = 132 (cid:40) ˜ R I ( f ) − i ˜ R Q ( f ) (cid:41) × (cid:34) V ˜ R ( ν r + f )[1 − S ( ν r + f, T )]1 − S ( ν r , T ) [ ∂ T qp S ] ν r ,T + (cid:40) + − (cid:41) V ∗ ˜ R ∗ ( ν r − f )[1 − S ∗ ( ν r − f, T )]1 − S ∗ ( ν r , T ) [ ∂ T qp S ∗ ] ν r ,T (cid:35) . (23)It has been assumed that f (cid:28) ν r , so that ν r /ν ≈
1. According to (22), theoutputs of the mixer contain spectral information about the variation in quasiparticletemperature. The transfer function ˜ F ( f ) takes into account (i) parametric upconversionof the variations into sidebands of the readout frequency through the modulation of theresonance feature, as described by the terms involving S , (ii) the passage to the mixer lectrothermal Model of Kinetic Inductance Detectors R ( ν ), and (iii) the subsequent down-conversion in themixer through ˜ R I ( ν ) and ˜ R Q ( ν ). Thus, if the optical power, q S , changes, the thermalcircuit can be analysed to give the change in T qp , which in turn gives the I/Q outputsthrough (22). A change in applied readout power is unique in that it affects the outputs of the I/Qmixer in two ways: (i) there is a differential change in quasiparticle temperature throughheating; (ii) the transmitted output changes directly. (22) accounts for the former, butit is necessary to derive a separate expression for the latter.Assume that the readout tone is amplitude modulated by a function m ( t ): V ( t ) = [1 + m ( t )] (cid:60) [ V e πiν r t ] , (24)where the bandwidth of m ( t ) is smaller than ν r . This modulation can be representedby adding a term ∆ V ( t ) to the readout signal having the spectrum∆ ˜ V ( ν ) = V (cid:40) ˜ m ∗ ( ν r − ν ) for ν < ν r (LSB)˜ m ( ν − ν r ) for ν ≥ ν r (USB), . (25)(8) and (15) can now be used to calculate the corresponding change in the resonatoroutput voltage and I/Q mixer outputs respectively: (cid:40) ∆ ˜ I ( f )∆ ˜ Q ( f ) (cid:41) = (cid:40) ˜ H I ( f )˜ H Q ( f ) (cid:41) ˜ m ( f ) , (26)where the transfer functions are given by (cid:40) ˜ H I ( f )˜ H Q ( f ) (cid:41) = 132 (cid:40) ˜ R I ( f ) − i ˜ R Q ( f ) (cid:41) × (cid:20) V ˜ R ( ν r + f ) S ( ν r + f, T )+ (cid:26) + − (cid:27) V ∗ ˜ R ∗ ( ν r − f ) S ∗ ( ν r − f, T ) (cid:21) . (27)For comparison with experiment, it is more convenient to work in terms of the change inincident readout power ∆ P R rather than the modulation function. The two are relatedby ∆ ˜ P R ( f ) = 2 P R ˜ m ( f ) , (28)where P R is the quiescent readout power applied to the KID. Combining the results of Sections 3.2 and 3.3, the overall response at the outputs of theI/Q mixer are∆ ˜ I ( f ) = ˜ F I ( f ) ∆ ˜ T qp ( f ) + ˜ H I ( f ) ∆ ˜ P R ( f )2 P R , (29) lectrothermal Model of Kinetic Inductance Detectors
11∆ ˜ Q ( f ) = ˜ F Q ( f ) ∆ ˜ T qp ( f ) + ˜ H Q ( f ) ∆ ˜ P R ( f )2 P R . where the first term accounts for a change in the effective quasiparticle temperatureand the second term for a change in the applied readout power. If the readout power isheld constant, only the first term in each expression is required; if the readout power ischanged, but there is no internal heating, only the second term in each expressionis required; if the readout power is changed and there is internal heating, the fullexpressions are needed.
4. Operating point
To use the expressions derived in Section 3 it is necessary to know the quiescent valueof the effective quasiparticle temperature, T qp , which is not the same as the bathtemperature because of readout-power heating. The analysis presented here is generic,and applies to all electrothermal models of the kind shown in Figure 2. Let T , U and q denote state vectors of temperature, internal energy and net input power foreach of the elements shown in Figure 2. For example, the first elements of the statevectors correspond, respectively, to the temperature, internal energy, and net inputpower q R + q S − q I , for the quasiparticle system. The second elements of the statevectors correspond to the phonon system of the superconductor. The net power flowinginto each element depends on a number of external parameters – such as the signalpower, substrate heater power, readout power, etc. – and for brevity we shall collectthese variables together into a single vector denoted v .If the external parameters are held constant at v , T evolves from a given startingpoint towards a steady-state value of T for which q ( T , v ) = , (30)henceforth referred to as the operating point. In previous papers [12, 13], we havediscussed the solution of (30) and its consequences for device behaviour, and we shallnot repeat this work here. What is found, for example, is that high readout powerheating causes variation in T across a frequency sweep, accounting for the distortionof the resonance curve observed in experiments.There can in fact be multiple values of T for which (30) is satisfied, correspondingto different possible states of the resonator. However, for these states to correspond torealizable operating points they must also be dynamically stable against perturbations.The existence of two or more stable states leads to the hysteretic resonance curves seenexperimentally. In [12, 13] we considered the stability criterion for a system with asingle heat capacity, and here we present the generalisation to systems with several heatcapacities.Consider what happens if T is perturbed instantaneously by ∆ T away from T ,say by noise. Conservation of energy requires ∂ t ∆ U ( t ) = − G · ∆ T ( t ) , (31) lectrothermal Model of Kinetic Inductance Detectors ∂ t ∆ T ( t ) = − C − · G · ∆ T ( t ) . (32) G is a thermal-conductance matrix with elements defined by { G } mn = − (cid:18) ∂q m ∂T n (cid:19) T , v , (33)while C is a heat capacity matrix with elements { C } mn = (cid:18) ∂U m ∂T n (cid:19) T , v . (34)For clarity, we emphasise q m , U m and T m are respectively the net input power, internalenergy and temperature of the m th system, corresponding to the m th entries of q , U and T . Both partial derivatives are evaluated at the system temperatures T at theoperating point and for the assumed values, v , of the external parameters. Because theinternal energy of each element depends only on its own temperature, C is diagonal. Thesize of the perturbation of the system away from equilibrium is measured by | ∆ T ( t ) | ,which from (31) evolves in time according to ∂ t | ∆ T ( t ) | = − T ( t ) † · S · ∆ T ( t ) , (35)where S is the matrix S = 12 (cid:110) C − · G + [ C − · G ] † (cid:111) . (36)(35) implies that provided x † · S · x > x , any perturbation decreases in magnitude with time, and the state is stable.If (37) is not satisfied, noise processes will inevitably excite perturbations that grow intime and drive T into a different state. The possible operating points of a device forgiven v can thus be found by checking the roots of (30) against the stability conditionof (37). (37) corresponds to demanding S is positive definite, which can be testednumerically by checking that its eigenvalues are all non-zero and positive.It is useful to consider the physical processes included in G , as the concepts arecentral to the paper. The classical definition of thermal conductance is the constant ofproportionality relating differential changes in heat flow down a thermal link to changesin the temperature of its ends. These thermal conductances, derived from the variousheat-flow equations listed in Section 2, make up the majority of the elements in G .However, G also contains effective conductance terms not derived from thermal links.For example, changes in the quasiparticle temperature affect the resonance curve andtherefore the readout power dissipated in the device, which contributes to q . Thisprocess contributes an effective thermal conductance in G , which links the quasiparticlesystem to a hypothetical bath. The sign and magnitude of this effective conductancedepend on the readout power and frequency, providing a mechanism for electrothermalfeedback. Both types of mechanism are intrinsic in the definition of (33). lectrothermal Model of Kinetic Inductance Detectors U and q have been assumed to be instantaneously related to temperature. Thisquasistatic approach is sufficient for first-order stability analysis, but we shall reconsiderthe assumption regarding q when discussing small-signal response.
5. Small-signal response
Now consider how changes in external parameters, such as optical signal power,calibration heater power, readout power, etc., lead to changes in the quasiparticletemperature. The result will be a full small-signal model of device behaviour.
In the small-signal limit, the change ∆ T ( t ) in the system temperatures, in response toa perturbation ∆ v ( t ) in external parameters, can be found by linearising (31) aroundthe operating point T = T and v = v . To first order, the internal energies of theelements vary as U ( t ) = U + (cid:90) ∞ (cid:60) (cid:104) C · ∆ ˜ T ( f ) e πift (cid:105) df, (38)where U is the value of U at the operating point. In general, some of the heat capacitiesmay be frequency dependent, for example in the case of TLSs, but for brevity we shallnot include this possibility here. Similarly, the net energy flows into the elements varyas ∆ q ( t ) ≈ (cid:90) ∞ (cid:60) (cid:104)(cid:104) − G · ∆ ˜ T ( f ) + R · ∆˜ v ( f ) (cid:105) e πift (cid:105) df, (39)where the thermal conductance matrix G is calculated using (33) and the parameterresponsivity matrix R is defined as { R } mn = (cid:18) ∂q m ∂v n (cid:19) T , v , (40)where v m is the m th external parameter, i.e. the m th entry of v . The elements of R give the rates of change of the power entering an element with respect to variationsin external sources. In (38)–(40), we have again assumed the quasistatic limit where G , C and R do not depend on f . In other words, changes in ∆ ˜ T ( f ) and changes in∆˜ v ( f ) occur on time scales that are long compared with the bandwidths of the processesdescribed by G , C and R . This assumption may not always be appropriate in the caseof readout power heating, as will be discussed in the next section.Substituting (38) and (39) into (31) and solving, we find∆ ˜ T ( f ) = (cid:8) [ G + 2 πif C ] − · R (cid:9) · ∆˜ v ( f ) , (41)= K ( f ) · ∆˜ v ( f )which describes the small-signal thermal response to variations in the independentvariables. (41) can then be used with (26) to calculate the changes in the outputs lectrothermal Model of Kinetic Inductance Detectors A small-signal model using the quasistatic forms for G , C and R is sufficient whenthermal processes limit the response time of the device. However in some cases it maybe the electrical response time of the resonator that is the constraint. For example, a5 GHz resonator with a total Q-factor in excess of 10 has a response time of the order of100 µs , which is comparable with typical quasiparticle lifetimes [20]. It is then necessaryto take into account the resonator dynamics and (39) becomes∆ q ( t ) ≈ (cid:90) ∞ (cid:60) [[ − G ( f ) · ∆ ˜ T ( f ) + R ( f ) · ∆˜ v ( f )] e πift ] df, (42)where G and R are now assumed to depend on the frequency f of the perturbations.Most of the terms in G and R can be still be calculated using the quasistatic forms of(33) and (40), but those associated with the readout power heating, q R , must be treateddifferently. In the quasistatic limit these are given by G R = − (cid:18) ∂q R ∂T qp (cid:19) T , v = − P R ∂ [1 − | S ( ν r , T qp ) | − | S ( ν r , T qp ) | ] ∂T qp , (43)and R R = (cid:18) ∂q R ∂P R (cid:19) T , v = 1 − | S ( ν r , T qp ) | − | S ( ν r , T qp ) | . (44)In this section we will develop replacement expressions that take full account of thefinite response time of the resonator. In what follows, we will suppress the temperaturedependence of the S -parameters for notational convenience and the operating-pointvalues should be assumed.First consider G R . The instantaneous power dissipated in the resonator by thereadout signal is given by q R ( t ) = R ( t ) I ( t ). When the resistance of the equivalentcircuit is perturbed, the current also changes and q R ( t ) = [ R + ∆ R ( t )] [ I ( t ) + ∆ I ( t )] ,and so to first-order∆ q R ( t ) ≈ ∆ R ( t ) I ( t ) + 2 R I ( t )∆ I ( t ) . (45)Substituting the time-dependent quantities in terms of their spectra, and ignoringfluctuations in power that occur at frequencies greater than the readout frequency gives∆ q R ( t ) ≈ | I | (cid:90) ν r (cid:60) (cid:104) ∆ ˜ R ( f ) e πift (cid:105) df (46)+ R (cid:90) ν r (cid:60) (cid:104) I ∆ ˜ I ∗ ( ν r − f ) e πift (cid:105) df lectrothermal Model of Kinetic Inductance Detectors R (cid:90) ν r (cid:60) (cid:104) I ∗ ∆ ˜ I ( ν r + f ) e πift (cid:105) df. The first term arises from the time-varying resistance modulating the average value ofthe readout current, and it is not limited by the bandwidth of the resonator. The secondand third terms correspond, respectively, to upper and lower sidebands resulting fromintermodulation between the steady state and perturbation currents.(9), (21) and (10), together with the result Z R ( ν ) = S ( ν )2 [1 − S ( ν )] Z , (47)can be used to rewrite (46) in terms of S -parameters and temperature perturbations.Casting the result into the form of (42), it can be shown, after some tedious algebra,that G R ( f ) = − P R (cid:26) | S ( ν r ) | + αS ( ν r + f ) S ( ν r ) ∂S ( ν r ) ∂T qp + | S ( ν r ) | + αS ∗ ( ν r − f )[ S ∗ ( ν r )] ∂S ∗ ( ν r ) ∂T qp (cid:27) , (48)where α = 1 − | S ( ν r ) | − | S ( ν r ) | . Although (48) is complicated, it is easy to evaluatenumerically, and it accounts for the frequency dependence of the absorbed readout powerwhen the quasiparticle temperature is modulated. Notice that the terms S ( ν r + f ) and S ∗ ( ν r − f ), which involve the readout frequency, may be different when the modulationfrequency is comparable with the bandwidth of the resonator. In particular, G R canhave an imaginary component, leading to the possibility of an oscillatory feature in thethermal response as described by (41) and indeed the outputs of the I/Q mixer given by(29). This feature corresponds to the exchange of energy stored in the electromagneticfields of the resonator and energy stored in the heat capacity of the thermal elements.If f is sufficiently small such that S ( ν r ± f ) ≈ S ( ν r ), (48) gives G R ( f → ≈ − P R (cid:60) (cid:26) [1 − S ∗ ( ν r )] ∂S ( ν r ) ∂T qp (cid:27) , (49)which is identical to the quasistatic limit found by evaluating the partial derivative in(43). In this case the response function is real, and oscillatory interactions between theelectrical and thermal systems do not occur. In the case where f exceeds the bandwidthof the resonator S ( ν r + f ) → S ∗ ( ν r − f ) →
0, because the resonator is an opencircuit out of band. (48) then becomes G R ( f → ∞ ) ≈ − P R (cid:60) (cid:26)(cid:20) S ∗ ( ν r ) − S ( ν r ) − (cid:21) ∂S ( ν r ) ∂T qp (cid:27) . (50)This term comes from the first term in (46) and it remains even for very high modulationfrequencies, showing that feedback can occur for modulation frequencies much greaterthan the line width of the resonator. Physically, the effect is due to the resistance, andtherefore the power, being modulated by the quasiparticle temperature, even though lectrothermal Model of Kinetic Inductance Detectors R R ( f ), which describes how the dissipated power responds to changesin the applied readout power. In this case, the equivalent resistance of the resonator isconstant at R , and (46) immediately gives the dissipated power as∆ q R ( t ) ≈ R (cid:90) ν r (cid:60) (cid:104) I ∆ ˜ I ∗ ( ν r − f ) e πift (cid:105) df (51)+ R (cid:90) ν r (cid:60) (cid:104) I ∗ ∆ ˜ I ( ν r + f ) e πift (cid:105) df. Using (9), (25) and (47) one finds that∆ q R ( t ) = 4 P r (cid:60) (cid:2) S ( ν r ) − | S ( ν r ) | (cid:3) (52) × (cid:90) ν r (cid:60) (cid:26)(cid:20) S ( ν r + f ) S ( ν r ) + S ∗ ( ν r − f ) S ∗ ( ν r ) (cid:21) ˜ m ( f ) e πift (cid:27) df, and using (28) gives∆ q R ( t ) = 12 (cid:2) − | S ( ν r ) | − | S ( ν r ) | (cid:3) × (cid:90) ν r (cid:60) (cid:26)(cid:20) S ( ν r + f ) S ( ν r ) + S ∗ ( ν r − f ) S ∗ ( ν r ) (cid:21) ∆ P r ( f ) e πift (cid:27) df, (53)and then according to (42) R R ( f ) = 12 (cid:2) − | S ( ν r ) | − | S ( ν r ) | (cid:3) × (cid:20) S ( ν r + f ) S ( ν r ) + S ∗ ( ν r − f ) S ∗ ( ν r ) (cid:21) . (54)In the quasistatic limit, f →
0, (44) is recovered. In the case where ν r ± f moves outsideof the resonance, R R ( f ) →
6. Overall small-signal response (41) can be used to calculate all of the temperature variations in the system when anyof the external parameters is varied. According to (29), it is only necessary to know thechange in quasiparticle temperature to calculate the outputs of the I/Q mixer. Thus, wetake the appropriate row k ( f ) of K ( f ) corresponding to the quasiparticle temperature,and write ∆ ˜ I ( f ) = ˜ F I ( f ) k ( f ) · ∆˜ v ( f ) + ˜ H I ( f ) ∆ ˜ P R ( f )2 P R , (55)∆ ˜ Q ( f ) = ˜ F Q ( f ) k ( f ) · ∆˜ v ( f ) + ˜ H Q ( f ) ∆ ˜ P R ( f )2 P R . (55) is the primary result of this paper, because it provides a full description of thesmall-signal response measured at the mixer outputs. It includes the electrothermalfeedback effects associated with the readout power, and also effects due the potentiallynarrow filtering of high-Q resonators when the readout power is varied. lectrothermal Model of Kinetic Inductance Detectors
7. Noise
Fast temporal fluctuations appear in the outputs of the I/Q mixer as a consequence ofnoise internal to the KID, and noise associated with external sources such as the readoutelectronics [20]. When modelling low-noise detectors, such as bolometers, internal noiseis usually treated by considering the physical origins of each of the microscopic processes,and then incoherently adding the contributions. The overall result usually turns out tobe an instance of the more general fluctuation-dissipation theorem, where the noiseis determined by the mean relaxation behaviour of the complete device rather thanthe exact statistics of the individual processes. For complex electrothermal systems,where thermal fluctuation noise may occur as a consequence of energy being exchangedrandomly between elements, for example through the Kapitza conductance and weakquasiparticle-phonon coupling, it is beneficial to adopt a general approach based onLangevin analysis [25].Our analysis is carried out in two steps: (i) Introduce a set of equivalent noise-power sources that drive energy into each of the elements of the electrothermal model,corresponding to fluctuations in the extensive variables of the system, most notably U . (ii) Use a thermodynamic argument to determine the magnitude of the thermalfluctuations in U , and thereby determine the corresponding magnitudes of the noise-power inputs. The advantage of this approach is that system noise can be calculated forany arbitrary arrangement of elements. Also, our scheme opens the door to more generalnon-equilibrium calculations [26]. By way of verification, it will be shown in Section 8.2that when quasiparticle-phonon coupling is the dominant weak thermal link, the schemereproduces generation-recombination noise without any explicit reference to the physicsof this process.Connect a set of randomly varying power sources to the thermal elements of Figure2. The sources are included mathematically by expressing (31) in terms of the variationsin internal energy around the quiescent value, and adding to it the noise-power vector∆ P N ( t ): ∂ t ∆ U ( t ) = − G · C − · ∆ U ( t ) + ∆ P N ( t ) . (56)Following the usual procedure for Langevin analysis, assume the following: (i) Theadded noise has zero mean, (cid:104) P N ( t ) (cid:105) = , (57)which occurs because if this were not the case the operating point would simply shiftuntil it is satisfied. (ii) The second order correlations satisfy (cid:104) ∆ P N ( t )∆ P † N ( t ) (cid:105) = N δ ( t − t ) , (58)where N is a real-valued symmetric matrix. The physical motivation for this assumptionis that the microscopic processes driving the noise are only correlated only on veryshort time scales. (iii) All higher order correlation functions are zero. More generally,these assumptions can be argued on the basis that thermodynamics predicts that the lectrothermal Model of Kinetic Inductance Detectors U form a Gaussian process, and (i)–(iii) correspond to the noise processwith the fewest free-parameters that can reproduce this effect. If ∆ U has the value∆ U at time t , the general solution of (56) for t > t is∆ U ( t ) = e − ( t − t ) G · C − · ∆ U + (cid:90) tt e − ( t (cid:48) − t ) G · C − · ∆ P N ( t (cid:48) ) dt (cid:48) , (59)where use has been made of the exponential of a matrix [27]. Using (58), the equal-timecorrelation matrix of the fluctuations is (cid:104) ∆ U ( t )∆ U ( t ) † (cid:105) = e − ( t − t ) G · C − · ∆ U ∆ U † · e − ( t − t )[ G · C − ] † + (cid:90) tt e − ( t (cid:48) − t ) G · C − · N · e − ( t (cid:48) − t )[ G · C − ] † dt (cid:48) . (60)The first term describes the relaxation of the system back to equilibrium from thestarting condition, and the second term describes the steady-state fluctuations. In thelimit t >> t , the operator exp[ − ( t − t ) G · C − ] must tend to zero if T is an equilibriumpoint, as discussed in Section 4. Evaluating the integral in (60) then gives (cid:104) ∆ U ( t )∆ U ( t ) † (cid:105) = M , (61)where N = G · C − · M + M · (cid:2) G · C − (cid:3) † . (62)If the two-time correlation function is evaluated, the noise in ∆ U is found to be non-white, and the noise is correlated on timescales determined by the heat capacities andweak thermal links. (62) and (61) provide a way of calculating N from the observedfluctuations in U , but we have said nothing about their magnitudes.A standard result in thermodynamics [28] is that the internal energy of a heatcapacity C in equilibrium with a bath at temperature T exhibits thermal fluctuationsof magnitude (cid:104) ∆ U (cid:105) = k b T C. (63)Strictly, the assumption of thermal equilibrium is not satisfied for an operating pointabove the bath temperature, but this is a standard assumption in bolometer theory.The question of noise in detectors under non-equilibrium conditions is an open one, anda number of different approaches have been proposed [29, 26]. The scheme presentedhere, would lend itself to non-equilibrium calculations.In the spirit of equilibrium thermodynamics, assume that the fluctuations in theenergy of each element are given by (63), but evaluated at the operating temperatureof that element, so (cid:104) ∆ U ( t )∆ U ( t ) † (cid:105) = k b C · T · T , (64)where { T } mn = { T } m δ mn . Inserting (64) into (62), we find N = k b (cid:104) G · T · T + T · T · G † (cid:105) . (65) lectrothermal Model of Kinetic Inductance Detectors G , which is a fluctuation-dissipation relationfor KIDs. (65) is closely related to the usual practice of placing noise sources acrossthermal conductances to represent noise.In order to determine the recorded noise it is necessary to propagate the effects ofthe equivalent noise sources to the outputs of the I/Q-mixer. Comparing (39) and (56),we see that the noise power term must enter (39) in the same way as variations in theexternal parameters term. However, we need the frequency domain representation of∆ P N ( t ). Taking the Fourier transforms of (57) and (58), we obtain (cid:104) ∆ ˜ P N ( f ) (cid:105) = , (66)and (cid:104) ∆ ˜ P N ( f )∆ ˜ P † N ( f ) (cid:105) = 2 N δ ( f − f ) , (67)for single-sided spectra. Replacing R · ∆˜ v ( f ) with ∆ ˜ P N ( f ) in the subsequent analysisin Section 5, we find the corresponding noise signal has the properties (cid:104) ˜ A ( f ) (cid:105) = 0 (68)and (cid:104) ˜ A ( f ) ˜ B ( f ) (cid:105) = 2 ˜ F A ( f ) ˆ x · [ G + 2 πif C ] − · N · (cid:104) G † − πif C (cid:105) − · ˆ x † ˜ F B ( f ) . (69)where A and B can be I or Q and ˆ x is a unit vector with a non-zero entry at the elementof ∆ ˜ T ( f ) corresponding to the temperature of the quasiparticle system.Of particular interest is the Noise Equivalent Power (NEP), which requires explicitknowledge of the how the KID is read out, i.e. whether the transmission phase-shift,amplitude-shift or some weighted linear combination of the two is used. The small-signal transformations discussed in (6) and (69) can be used, respectively, to calculatethe appropriate responsivity and noise, which then allows the calculation of NEP.
8. Comparison with other models
The usual approach to modelling KIDs is to use a simplified quasiparticle-numbermodel to understand small-signal behaviour, and non-linear resonator theory to explainhysteresis [20, 30]. In this section we show how our more general electrothermal modelreproduces the behaviour of these other approaches, under a single framework, in theappropriate limits.
The quasiparticle-number model assumes that all of the microscopic systems are at thebath temperature, and there exists an excess of quasiparticles over the expected thermalpopulation [20]. The total number of quasiparticles, N qp , then parameterizes the excited lectrothermal Model of Kinetic Inductance Detectors T qp can vary, and that[ C ∂ t + G ] ∆ T qp ( t ) = ∆ P qp ( t ) , (70)where G is the effective thermal conductance between the quasiparticles and a fictitiousbath at the operating temperature, C is the heat capacity of the quasiparticle systemand ∆ P qp is the change in the net power flow into the quasiparticle system from externalsources. (70) is effectively a treatment of a simple KID as a classical bolometer, wherethe quasiparticle system functions as the isolated heat capacity whose temperature ismonitored.At low reduced temperatures, N qp and U qp , the internal energy of the quasiparticlesystem, are approximately N qp = 2 n V (cid:112) πk b T ∆ g e − ∆ g /k b T (71) U qp = ∆ g N qp , (72)as is shown in Appendix A. The temperature dependence of ∆ g is sufficiently weak thatit can be ignored to first-order. Taking the temperature derivatives of (71) and (72)then gives ∆ N qp = (cid:18) N qp ∆ g k n T (cid:19) ∆ T qp (73) C = N qp ∆ k n T , (74)which relate changes in quasiparticle number to changes in temperature. Substituting(73) and (74) into (70), we recover the usual rate equation for the excess quasiparticlenumber in the quasi-particle number model, (cid:2) ∂ t + τ − (cid:3) ∆ N qp ( t ) = Γ( t ) , (75)where the relaxation time τ and generation rate Γ are related to the thermal parametersby τ = C /G (76)Γ = ∆ P qp ( t ) / ∆ g . (77) N qp is maintained by a balance of generation processes, where a Cooper pair is brokenby a phonon or photon, and loss processes, where quasiparticles recombine into Cooperpairs or are lost from the superconductor. These processes are intrinsically random andso N qp fluctuates [31], leading to so-called Generation-Recombination (GR) noise [20].GR noise arises naturally in our model as the Langevin noise source associated withthe quasiparticle-phonon coupling conductance. Following the methods of Section 7, itcan be accounted for by considering a noise power input ∆ P N in (70), i.e.∆ P qp ( t ) = η ∆ P S ( t ) + ∆ P N ( t ) , (78) lectrothermal Model of Kinetic Inductance Detectors η ∆ P S ( t ) is the optical signal and the noise term satisfies (cid:104) ∆ P N ( t ) (cid:105) = 0 (cid:104) ∆ P N ( t )∆ P N ( t ) (cid:105) = 2 k b T G δ ( t − t ) . (79)The NEP is the square root of the single-sided spectral density of an equivalent white-noise input signal ∆ P S ( t ) that represents the noise generated in the device. It followsfrom (78) and (79) thatNEP = 1 η (cid:113) k b T G . (80)(74) can then be used with (76) to express (80) in the formNEP = 2∆ g η (cid:114) N qp τ . (81)which is the standard result for the GR-noise limited NEP of a KID [11, 20].An alternative line of reasoning is to use (72) to relate changes in U qp to changesin N qp . In conjunction with (74), this allows the conversion of the thermal fluctuationsin the energy in the quasiparticle system, as described by (63), into equivalent numberfluctuations. The result is (cid:104) ∆ N (cid:105) = N qp , (82)which is the standard expression for the mean-square number fluctuations associatedwith GR noise [11]. The only area where the relationship with other models is not entirely clear relatesto the hysteretic switching seen in resonance curves at high readout powers. Inour model, hysteretic switching arises naturally as a consequence of the quasiparticletemperature having two stable states, and the system being driven between thesestates by a dynamical feedback mechanism associated with the absorption of readoutpower. A different, but related, approach is to consider the magnetic-field dependenceof the kinetic inductance [32]. In the context of KIDs, this effect has been treated bySwenson [30] by assuming that the lumped inductance depends on the square of thecurrent: L ( I ) = L [1 + ( I/I ∗ ) + · · · ] . (83)The non-linear terms in (6) result in Duffing-like resonator dynamics, with the possibilityof multiple amplitude states for a given drive signal. Hysteresis results when theresonator switches between these states.Our scheme also predicts that L will depend on I , but because of readout powerheating. Each of the lumped resonator parameters R , L and C depends on T qp , whichin turn depends on the power P dissipated in the quasiparticle system. For a givenparameter X , we may therefore approximate X ≈ X ( T B ) + ∂X∂T qp ∂T qp ∂P P, (84) lectrothermal Model of Kinetic Inductance Detectors T b . P can be expressed as Z R I /
2, where Z R is the real part of the resonator’s impedance at the readout frequency.With this substitution we can obtain an expression for each X of the form of (83),giving an effective I ∗ , which characterizes the current at which nonlinear effects becomeimportant for each of the lumped parameters. Consequently, we expect our model toproduce similar results to that of Swenson when the non-linear terms in the reactiveresponse ( L ) dominate, but to see differences if the non-linearity in the dissipativeresponse ( R ) is also significant.Ultimately, it comes down to whether one thinks that magnetic field or heatinggives rise to the non-linearity that drives resonance curve distortion and bifurcation fora particular device. The physical process must be found through experiment, and acomparison of measured and calculated values of I ∗ may provide a way of distinguishingmechanisms.
9. Conclusions
We have presented an electro-thermal model of KIDs. Our approach has severaladvantages: (i) It takes into account readout-power heating, and properly accountsfor the fact that the quasiparticle temperature is elevated above that of the bath. (ii)It calculates the small-signal and noise behaviour as perturbations about the actualquasiparticle temperature, rather than assuming that the quasiparticle system is at thebath temperature. (iii) It includes electrothermal feedback caused by readout powermodulation, taking into account the ring-down time of the resonator. (iv) It can beapplied to complicated thermal arrangements, such as resonators on membranes, takinginto account thermal fluctuation noise between the various elements. (v) Generation-recombination noise appears naturally as a thermodynamic fluctuation of the energystored in the quasiparticle system. (v) It can account for the nonlinearities thatgive rise to resonance curve distortion and hysteretic switching, and leads to differentresponsivities and NEPs on the different hysteretic branches. The scheme reduces tothe commonly used quasiparticle-number model in the appropriate limits.
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Phys. Rev.B lectrothermal Model of Kinetic Inductance Detectors Appendix A. Approximate expressions for the quasiparticle density andinternal energy of the quasiparticle system (71) is frequently used to approximate the quasiparticle number density in asuperconductor at low reduced temperature (
T /T C < . N qp of quasiparticles and the total energy U qp of thequasiparticle system are given by N qp = 4 n V (cid:90) ∞ ∆ g E √ E − ∆ f ( E, T ) dE (A.1) U qp = 4 n V (cid:90) ∞ ∆ g E √ E − ∆ f ( E, T ) dE. (A.2)where n is the number density of states at the Fermi surface of the normal state, V is the volume of the superconductor, ∆ g is the superconducting gap energy and f ( E, T ) is the Fermi-Dirac distribution. At low reduced temperatures, ∆ g (cid:29) k b T and f ( E, T ) ≈ exp( − E/k b T ). The substitution E = ∆ cosh u can then be used to bringintegrals (A.1) and (A.2) into the forms N qp = 4 n V K (∆ g /k b T ) ∆ g (A.3) U qp = 2 n V [ K (∆ g /k b T ) + K (∆ g /k b T )] ∆ g , (A.4)where K n ( x ) is the n th modified Bessel function of the second kind. The K n ( x ) have thesame asymptotic form for large x for all n [35], as given by K n ( x ) ≈ (cid:112) π/ x exp( − x ).This approximation can be made in (A.3) and (A.4), in which case (71) and (74) areobtained. Appendix B. Power flow between the quasiparticle and phonon systems
In Section 8, we demonstrated a link between the relaxation time of the excessquasiparticle population and the thermal conductance. Consider the case where thedominant relaxation mechanism is the heat flow q I from the quasiparticles to the phononsystem of the superconductor. The relaxation time will then be τ = τ r /
2, where τ r is therecombination lifetime of a single quasiparticle. The factor of a half arises because twoquasiparticles are lost in each recombination event, so the quasiparticle number decaysat twice the rate of a single quasiparticle considered in isolation. Kaplan [36] has shown τ r can be approximated by τ r = τ √ π (cid:18) k b T C (cid:19) (cid:18) T C T (cid:19) e ∆ /k b T qp , (B.1)where T C is the critical temperature, ∆ is the value of ∆ g at absolute zero and τ is amaterial-dependent lifetime. Values of τ are given in Table I of [36]. Using (76) and(74), we then have that the effective thermal conductance is G = 343 πk b n V ∆ τ ∂∂T (cid:0) T e − /k b T (cid:1) T qp , (B.2) lectrothermal Model of Kinetic Inductance Detectors ≈ k b T C . The total heat flow out of each systemis found by integrating (B.2) with respect to temperature, and the net heat flow followsfrom the difference of the two terms. The integration can be performed trivially and weobtain a result of the form of (3), as ∆ g ≈ ∆ at low reduced temperatures. Further,as we are assuming q I is dominated by recombination processes, η g ( P abs ) ≈ s ≈ πk b n ∆ / τ . Calculation of η g ( P abs ))