2.4.2.1 RelatingδZs to δσ
MKIDs operate on a principle that the surface impedance Zs of a superconducting film changes when photons break Cooper pairs and generate quasiparticles (QPs)[14, 16]. The responsivity of MKIDs is related todZs/dnqp, wherenqp is the QP density.
It follows from the discussion in the previous section that the change in Zs is related to the change inσby
δZs
Zs
=γδσ σ, γ =
−1/2 Thick film, local limit
−1/3 Thick film, extreme anomalous limit
−1 Thin film, local limit
. (2.80)
Thus,dσ/dnqp becomes an important quantity in discussing responsivity of MKIDs.
2.4.2.2 Effective chemical potential µ∗
One straightforward way of calculatingdσ/dnqpis throughdσ/dnqp=∂n∂σ(Tqp(T)/∂T)/∂T (the ratio between the change in the conductivity and the change in the quasiparticle density, both caused by a change
in the bath temperature) from Eq. 2.70, Eq. 2.71, and nqp= 4N0
Z ∞ 0
1
1 +ekTE dǫ, ǫ=p
E2−∆2 (2.81)
where N0 is the single spin density of states. The result from such a calculation givesδσ due to a change in thermal QP density from a change in bath temperature, which does not directly apply to excess QPs from pair breaking. To account for excess QPs, we adopt Owen and Scalapino’s treatment[49] and introduce an effective chemical potentialµ∗to the Fermi distribution function
f(E;µ∗, T) = 1
1 +eE−µkT∗ . (2.82)
Physically, Eq. 2.82 treats the QPs as a Fermi gas with a thermal equilibrium distribution char- acterized by the chemical potentialµ∗ and the temperatureT. This assumption is valid because at low temperatures, phonons with energy less than 2∆ (under-gap phonons) are much more abundant than the phonons with energy larger than 2∆ (over-gap phonons); therefore the time scale τl for excess QPs to thermalize with the lattice (phonon) temperatureT (assisted by under-gap phonons) is much shorter than the time scaleτqpfor excess QPs to recombine (assisted by over-gap phonons);
as a result, during the timeτl< t < τqp, the QPs may be described using the Fermi function given by Eq. 2.82.
With the introduction of µ∗, the total QP density nqp (including both thermal and excess QPs), the superconducting gap ∆, and the complex conductivity σ can be rederived by substi- tutingf(E;µ∗, T) forf(E;T) in the corresponding BCS formula and the Mattis-Bardeen formula.
The relevant equations now modify to
nqp = 4N0
Z ∞ 0
1
1 +eE−µkT∗ dǫ, ǫ=p
E2−∆2 (2.83)
1 N0V =
Z ~ωc
0
tanhE2kT−µ∗
E dǫ (2.84)
σ1
σn
= 2
~ω Z ∞
∆
[f(E;µ∗, T)−f(E+~ω;µ∗, T)](E2+ ∆2+~ωE)
√E2−∆2p
(E+~ω)2−∆2 dE + 1
~ω Z −∆
∆−~ω
[1−2f(E+~ω;µ∗, T)](E2+ ∆2+~ωE)
√E2−∆2p
(E+~ω)2−∆2 dE (2.85)
σ2
σn = 1
~ω Z ∆
∆−~ω
[1−2f(E+~ω;µ∗, T)](E2+ ∆2+~ωE)
√∆2−E2p
(E+~ω)2−∆2 dE. (2.86)
2.4.2.3 Approximate formulas of ∆,nqp, σ, anddσ/dnqp for both cases Under the condition that ~ω ≪ ∆ (Cond. 1), kT ≪ ∆ (Cond. 2) and e−E−µ
∗
kT ≪ 1 (Cond. 3), Eq. 2.83–2.86 have the following analytical approximate formula[50]:
nqp = 2N0
√2πkT∆e−∆−µ
∗
kT (2.87)
∆
∆0
= 1−
r2πkT
∆ e−∆−µ
∗
kT = 1− nqp
2N0∆ (2.88)
σ1
σn
= 4∆
~ωe−∆−µ
∗
kT sinh(ξ)K0(ξ), ξ= ~ω
2kT (2.89)
σ2
σn
= π∆
~ω[1−2e−∆−µ
∗
kT e−ξI0(ξ)] (2.90)
whereIn,Kn are thenth order modified Bessel function of the first and second kind, respectively The first two conditions (Cond. 1 and Cond. 2) are apparently satisfied by a typical Al MKID withTc= 1.2 K and microwave frequencyω/2πbelow 10 GHz. Meanwhile, the QP density due to pair breaking from a photon with energyhνis estimated bynqp≈ ∆Vhν. Assuming a sensing volume V ∼3µm×0.2µm×100µm(center strip width×film thickness×quasiparticle diffusion length) and takingT=0.1 K, N0= 1.72×1010 µm−3eV−1, and ∆ = 0.18 meV for Al,e−E−µ
∗
kT is estimated from Eq. 2.87 to be 0.1 for a 6 keV photon and 1.4×10−5for a 1 eV photon, both much less than 1. Thus for Al MKIDs up to X-ray band, the third condition (Cond. 3) is also satisfied.
Now we are ready to derive σand its derivative for the two cases.
Case 1: thermal QPs due to temperature change
In Eq. 2.87–2.90, only two of the four variables ∆, nqp, µ∗, andT are independent. By taking µ∗ and T as independent variables, setting µ∗ = 0, and keeping only the lowest-order terms in Eq. 2.87–2.90, we arrive at the following results
σ1(T)
σn = 4∆0
~ω e−∆0kT sinh(ξ)K0(ξ) (2.91)
σ2(T) σn
= π∆0
~ω [1−
r2πkT
∆0
e−∆0kT −2e−∆0kTe−ξI0(ξ)] (2.92) nqp(T) = 2N0
p2πkT∆0e−∆0kT (2.93)
dσ1
dnqp
= σn
1 N0~ω
r2∆0
πkT sinh(ξ)K0(ξ){
∆0
kT −ξcosh(ξ)sinh(ξ)+ξKK10(ξ)(ξ)
∆0
kT +12 } (2.94)
dσ2
dnqp = σn −π 2N0~ω[1 +
r2∆0
πkTe−ξI0(ξ){
∆0
kT +ξ−ξII10(ξ)(ξ)
∆0
kT +12 }] (2.95)
where the directives are evaluated bydσ/dnqp= ∂n∂σ(T)/∂T
qp(T)/∂T. Case 2: excess QPs due to pair breaking
Using Eq. 2.87 to suppress the explicit dependence of µ∗, taking nqp and T as independent variables and keeping the lowest-order terms in Eq. 2.88–2.90, we arrive at the following result
σ1(nqp, T)
σn = 2∆0
~ω
nqp
N0√
2πkT∆0
sinh(ξ)K0(ξ) (2.96)
σ2(nqp, T) σn
= π∆0
~ω [1− nqp
2N0∆0
(1 + r2∆0
πkTe−ξI0(ξ))] (2.97) dσ1
dnqp
= σn
1 N0~ω
r2∆0
πkT sinh(ξ)K0(ξ) (2.98)
dσ2
dnqp
= −σn
π 2N0~ω[1 +
r2∆0
πkTe−ξI0(ξ)] (2.99)
where the directives are evaluated bydσ/dnqp =∂σ(nqp, T)/∂nqp. In this case, we find that σis a linear function ofnqp and
κ = δσ/|σ| δnqp ≈ 1
πN0
r 2 πkT∆0
sinh(ξ)K0(ξ) +j 1 2N0∆0
[1 + r2∆0
πkTe−ξI0(ξ)]. (2.100) It can be derived from Eq. 2.80 that
δZs
|Zs| =κ|γ|δnqp. (2.101)
2.4.2.4 Equivalence between thermal quasiparticles and excess quasiparticles from pair breaking
0 0.1 0.2 0.3 0.4 0.5
−10
−8
−6
−4
−2 0 2 4 6x 10−6
T (K)
dσ/d n qp dσ1/dn
qp, thermal
dσ2/dn
qp, thermal
dσ1/dnqp, excess dσ2/dnqp, excess σn
Figure 2.10: dσ/dnqp vs. T calculated for two cases. ∂n∂σ1(T)/∂T
qp(T)/∂T and ∂n∂σ2(T)/∂T
qp(T)/∂T are plotted by the upper and lower solid lines, ∂σ1∂n(nqpqp,T) and ∂σ2∂n(nqpqp,T) by dashed lines. Other parameters are f = 6 GHz,N0= 1.72×1010µm−3eV−1, and ∆0= 0.18 meV for Al.
Comparing Eq. 2.94 and Eq. 2.95 to Eq. 2.98 and Eq. 2.99, we find the two cases only differ from
each other by the factors inside the curly brackets, which are found to be close to unity over the temperature and frequency range that MKIDs operate in.
The values of dσ/dnqp of the two cases are evaluated for Al and plotted in Fig. 2.10. We see that the thermal QP curves (solid lines) separates very little from the excess QP curves (dashed lines), which means that adding a thermal quasiparticle (by slightly changing the temperature) and adding a non-thermal quasiparticle (by breaking Cooper pairs) have the same effect on changing the complex conductivity. The equivalence between thermal and excess QPs allows us to use bath temperature sweep to calibrate the responsivity of MKIDs instead of using a external source.