2.2 Responsivity
2.2.2 Generation-Recombination Equation
At zero temperature all of the conduction electrons in a superconductor are found in Cooper pairs, a state in which two electrons are bound together by the electron-phonon interaction [84]. The binding energy of a Cooper pair is equal to 2∆≈3.52kBTc where ∆ is the gap energy for a single excitation and Tc is the critical temperature of the superconductor. The binding energies of aluminum and niobium — the two superconductors that we employ in the MUSIC detectors — are approximately 0.36 meV and 2.80 meV, respectively. Although the binding is weak, it prevents inelastic scattering of the Cooper pairs with ions in
(a) (b) (c)
Figure 2.1: Diagram of the processes considered in the generation-recombination equation. (a). Thermal generation: a phonon with energy Ω≥2∆ breaks apart a Cooper pair and produces two quasi-particles.
(b). Optical generation: a photon with energyhν≥2∆breaks apart a Cooper pair and produces two quasi- particles. (c). Recombination: two quasi-particles combine to form a Cooper pair and emit a phonon with energyΩ≥2∆.
the lattice, resulting in the characteristic zero resistance.
Even at zero temperature superconductors have nonzero AC impedance. If an AC electromagnetic field is applied to the superconductor, then the Cooper pairs will oscillate with the field creating an AC current.
Since the Cooper pairs have mass it will take the field some finite time to accelerate them. This results in a phase-lag between the current and the electric field that has a mathematical form equivalent to an inductance.
We call it kinetic inductance because the inductive energy is stored in the kinetic energy of the Cooper pairs.
Note that kinetic inductance only occurs with high mobility charge carriers. Unbound electrons do not have a significant kinetic inductance (at microwave frequencies) because they scatter so often that energy cannot be stored in their motion.
At finite temperatures some fraction of the conduction electrons are thermally excited from the Cooper pair state. These unbound electrons, known as quasi-particles, can inelastically scatter with ions in the lattice.
We refer to the process by which Cooper pairs are thermally excited into quasi-particlesthermal generation.
In this process a lattice vibration (phonon) with energy greater than the binding energy breaks apart a Cooper pair creating two quasi-particles. The reverse process in which two quasi-particles combine to form a Cooper pair and emit a phonon can also occur. We refer to this as recombination. Finally, photons with energy hν >2∆ can break apart Cooper pairs, resulting in the optical generationof quasi-particles. The three processes of thermal generation, optical generation, and recombination are illustrated in Figure 2.1
The first step in modeling the propagation of signal through the MKID is to relate incoming optical power to the quasi-particle density of the superconducting thin film. This can be done using the generation- recombination equation, which is simply the statement that over a given amount of time the change in the number of quasi-particles is equal to the number that were generated thermally plus the number that were generated optically minus the number that recombined. Writing this in differential form we have
∂nqp
∂t =ΓG,th+ΓG,opt−ΓR, (2.10)
whereΓdenotes the rate at which a particular process occurs in units of number of quasi-particles per unit
volume per unit time. In a steady state,∂nqp/∂t=0 and Equation (2.10) becomes
ΓG,th+ΓG,opt=ΓR. (2.11)
We will now calculate the rate at which these three processes occur. The optical generation rate is just
ΓG,opt= 2 V
ηphPopt
2∆ =ηphPopt
V∆ , (2.12)
whereηph is the efficiency with which photons are converted to quasi-particles, Popt is the optical power incident on the MKID from the antenna, andV is the volume of the absorbing section. The factor of 2 in the numerator accounts for the fact that two quasi-particles are created for every broken Cooper pair. The efficiency factorηph is intrinsic to pair-breaking detectors and will depend primarily on the frequency of the incident power. Consider the absorption of a single photon of frequencyνmm. If the energy of the photon is exactly equal to the binding energy hνmm=2∆, then the entirety of the photon’s energy goes into the creation of quasi-particles andηph=1. Forhνmm>2∆, only 2∆will go into the creation of quasi-particles and the remainder will go into the kinetic energy of those quasi-particles. Hence, we expect a monotonic decrease inηphbetweenhνmm=2∆andhνmm=4∆, at which point it should reachηph≈50%. Forhνmm>4∆
the efficiency is expected to level off or slightly increase. This is because the quasi-particles created by the photon will relax to energiesE∼∆through inelastic scattering with the lattice, which results in phonon emission. The timescale for relaxationτcascade'1−10 ns is much shorter than the timescale for recombination τqp[85, 86]. Whenhνmm>4∆the phonons produced via relaxation can have energyΩ≥2∆, and therefore can break Cooper pairs creating secondary quasi-particles. In the limit thathνmm4∆the energy will cascade down through multiple pair breakings. Guruswamy et al. [87] performed a proper simulation ofηph(νmm)for Al superconductors over a wide range of film thicknesses. They measure overall trends withνmmthat are consistent with the somewhat simplistic explanation that was just given. They confirmed previous results that showedηph≈0.58 in the limit of thick films and highly energetic photons hνmm4∆[88, 85]. However, they found that the loss of nonequilibrium phonons to the substrate becomes significant for thin films, and that this degrades the pair breaking efficiency relative to the thick film value. For the MUSIC detectors with film thicknessd≈50 nm and observing bandshνmm= [3.3,4.9, 6.2, 7.5]×∆, their simulations predict that ηph= [0.60, 0.50, 0.46, 0.43].
The recombination rate is defined as
ΓR=−1 V
dNqp
dt
R
. (2.13)
If each individual quasi-particle recombines at a rateτqp−1then the change in the the total number of quasi- particlesNqpper unit time will be
dNqp
dt
R
=−Nqp
τqp
. (2.14)
Solving this equation results in a exponential decay in the total number of quasi-particles with time
Nqp(t)∝exp(−t/τqp). (2.15)
We refer toτqpas the quasi-particle lifetime. It is given by the equation 1
τqp
=Rnqp+ 1 τmax
, (2.16)
where the recombination coefficientRis the recombination rate per unit density of quasi-particles andτmax−1 is the maximum quasi-particle lifetime. Note that the physical mechanism responsible for saturation of the lifetime atτmax is not fully understood, but this behavior is observed experimentally [89, 90, 91]. The expression forΓRis then
ΓR=Rn2qp+ nqp
τmax
. (2.17)
We see that the rate due to pair recombination scales asn2qp. This is exactly what one would expect since two quasi-particles have to “find each other” in order to recombine.
Plugging Equations (2.12) and (2.17) into the generation-recombination equation yields
ΓG,th+ηphPopt
V∆ =Rn2qp+ nqp τmax
. (2.18)
We still need to determine the thermal generation rateΓG,th. Now the total number of quasi-particles is just the sum of the number generated thermally and the number generated optically, i.e.,nqp=nqp, th+nqp, opt. Consider the limit of no optical loading. ThenPopt→0 andnqp, opt→0, giving us
ΓG,th=Rn2qp, th+nqp, th
τmax
. (2.19)
The above expression is useful because the thermal quasi-particle densitynqp, this completely determined by the temperature and gap energy. We will derive an analytical expression for it in the following section.
Inserting Equation (2.19) back into Equation (2.18) yields ηphPopt
V∆ =R n2qp−n2qp, th + 1
τmax
(nqp−nqp, th). (2.20)
We can then use the quadratic formula to obtain an explicit expression for the quasi-particle density
nqp= ηphPopt
RV∆ +n2qp, th+ 1 Rτmax
nqp, th+ 1
4Rτmax
1/2
− 1 2Rτmax
. (2.21)
Here we have discarded the second root since it implies negative values. In the limit that the optical generation of quasi-particles dominates over thermal generation and that the recombination of quasi-particles dominates
over the decay due toτmaxthis becomes
nqp≈ ηphPopt
RV∆ 1/2
fornqpnqp,thandτqpτmax (2.22)
and we find that the quasi-particle density scales as the square root of the incident optical power. Note that in all analysis that follows we use the full expression given in Equation (2.21).
Going forward we will be interested in how small changes in incident optical power — due to scanning across an astronomical source, for example — translate to changes in the quasi-particle density. This is given by
δnqp= ∂nqp
∂Popt
δPopt
= ηphδPopt
V∆(2Rnqp+τmax−1), (2.23)
where we have evaluated the partial derivative using Equation (2.20). It is traditional to then define 1
τqpeff =2Rnqp+ 1 τmax
. (2.24)
We callτqpeffthe effective quasi-particle lifetime. It is the timescale for the decay of a small perturbation to the quasi-particle population. Note that it differs fromτqp only in the factor of two in front of the recombination term. Equation (2.23) can then be written as
δnqp=τqpeffηphδPopt
V∆ . (2.25)
We will discuss the effective quasi-particle lifetime further in Section 2.3.2.1.