MKID Responsivity

KIPM-DETECTOR DEVELOPMENT

4.1 Overview of MKIDs

4.1.4 MKID Responsivity

In this section, we will consider the response of an MKID to energy depositions.

This is required to predict the energy resolution of the MKID (Section 4.2.1) and eventually the energy resolution of a full KIPM dark-matter detector (Section 4.2.2).

As discussed in Section 4.1.2, Cooper-pair breaking in an MKID will lower the res- onant frequency and increase dissipation. An increase in dissipation will decrease the internal quality factor of the MKID. Specifically, the quality factor is inversely proportional to the dissipation (Eq. 4.11). Thus, to understand an MKID-based detector’s sensitivity to energy depositions, it is necessary to understand how trans- mission changes with respect to changes in𝜔_𝑟and 1/𝑄_𝑖(Eqs. 4.12 and 4.14). These results can be derived starting with Eq. 4.9.

𝜕

𝜕 𝜔_𝑟 𝑆₂₁ =

−2𝑗

𝑄²

𝑟

𝑄𝑐

𝜔 𝜔²_𝑟

1+2𝑗 𝑄_𝑟 𝜔−𝜔_𝑟

𝜔_𝑟

2 (4.12)

→

𝜕

𝜕 𝜔_𝑟 𝑆₂₁

=

2^𝑄_𝑄^2𝑟

𝑐

𝜔 𝜔²_𝑟

1+4𝑄_𝑟² _𝜔−𝜔

𝑟

𝜔𝑟

2 (4.13)

𝜕

𝜕1/𝑄_𝑖 𝑆₂₁ =

𝑄²

𝑟

𝑄𝑐

1+2𝑗 𝑄_{𝑟 𝜔−𝜔}

𝑟

𝜔𝑟

2 (4.14)

→

𝜕

𝜕1/𝑄_𝑖 𝑆₂₁

=

𝑄²

𝑟

𝑄𝑐

1+4𝑄²_{𝑟 𝜔−𝜔}

𝑟

𝜔_𝑟

2 (4.15)

With these results, we see that the absolute dissipation and frequency sensitivities are maximized on resonance (𝜔 =𝜔_𝑟)2. This is why MKIDs are generally read out

2The𝜔dependence of the numerator in Eq. 4.13 is insignificant to the sensitivity optimization for𝑄_𝑟 ≫1.

𝜕

𝜕 𝜔_𝑟𝑆₂₁

𝜕

𝜕1/𝑄𝑖

𝑆₂₁

=−2𝑗 𝜔 𝜔²

𝑟

(4.16) The next step towards understanding the transmission response to an energy deposi- tion is to understand how𝜔_𝑟and 1/𝑄_𝑖change with respect to changes in Cooper-pair density. By convention, this is usually done using the density of electron quasipar- ticles (𝑛_{𝑞 𝑝}) as the variable of interest. Each conduction electron in the metal exists as either a quasiparticle or half of a Cooper pair, so the two densities are entirely dependent (Eq. 4.17).

𝑛_e=𝑛_{𝑞 𝑝}+2𝑛Cooper-pairs

𝜕 𝑛_{𝑞 𝑝} =−2𝜕 𝑛Cooper-pairs

(4.17)

We can differentiate our definitions in equations 4.6 and 4.11 to begin the derivation.

It is important to remember that the kinetic inductance fraction (𝛼) is not constant but rather a function of 𝐿_𝐾. If 𝛼is treated as a constant during differentiation, the resulting𝑑𝐿_𝐾/𝑑𝑛_{𝑞 𝑝}terms in Eqs. 4.18 and 4.19 will be wrong by a factor of𝛼.

𝑑 𝜔_𝑟 𝑑𝑛_{𝑞 𝑝}

=−𝛼 𝜔_𝑟 2𝐿_𝐾

𝑑𝐿_𝐾

𝑑𝑛_{𝑞 𝑝} (4.18)

𝑑1/𝑄_𝑖 𝑑𝑛_{𝑞 𝑝}

=𝛼 1 𝜔_𝑟𝐿_𝐾

𝑑𝑅 𝑑𝑛_{𝑞 𝑝}

−𝛼 1/𝑄_𝑖

2𝐿_𝐾 𝑑𝐿_𝐾

𝑑𝑛_{𝑞 𝑝} (4.19)

We now need to understand how changing quasiparticle density affects kinetic in- ductance (𝐿_𝐾) and parasitic resistance (𝑅). The complexity of real MKIDs prevents us from using Eq. 4.3 to derive either. Instead, the result is found using BCS and Mattis-Bardeen theory. In Chapter 2 of his thesis, Jiansong Gao uses both to de- rive the relationship between quasiparticle density and a superconductor’s complex impedance for various limits on film characteristics [65]. The film-characteristic

dependence is encoded in the factor 𝛾. In Chapter 2 of Seth Siegel’s thesis, he rewrote Gao’s result in terms of surface resistance 𝑅_𝑆 and surface inductance 𝐿_𝑆 (Eqs. 4.20 and 4.21).

𝑅_𝑆−𝑅_𝑆,0 𝜔 𝐿_𝑆,0

=|𝛾|𝜅₁(𝑇 , 𝜔,Δ₀)𝑛_{𝑞 𝑝} (4.20)

𝐿_𝑆−𝐿_𝑆,0 𝐿_𝑆,0

=|𝛾|𝜅₂(𝑇 , 𝜔,Δ₀)𝑛_{𝑞 𝑝} (4.21) One implication of the Mattis-Bardeen result is the exponential suppression of magnetic fields inside the superconductor, leading to a concentration of current density near the conductor surface. This effect is why the resistance and inductance above are labeled as "surface" values. The label is not meant to indicate that the resistance, inductance, or current are constrained precisely to the (infinitely thin) conductor surface. The resultant surface inductance originates from the kinetic- inductance effect and does not include magnetic inductance (originating from the device geometry). Therefore, 𝐿_𝑆 = 𝐿_𝐾. The resultant surface resistance originates from quasiparticle scattering. Since quasiparticle scattering is the only source of dissipation, 𝑅_𝑆 = 𝑅. Siegel’s results are described with respect to resistance and inductance at zero temperature with no external excitations (including photons or phonons). I have labeled such values with a "0" subscript. In Siegel’s thesis, these values are labeled "dark," because photons are the typical excitation of concern for telescopes. In the 0 (or dark) state, no quasiparticles will exist to dissipate energy, so𝑅_𝑆,0=0.

𝑅 𝜔 𝐿_{𝐾 ,0}

=|𝛾|𝜅₁(𝑇 , 𝜔,Δ₀)𝑛_{𝑞 𝑝} (4.22)

𝐿_𝐾 −𝐿_{𝐾 ,0} 𝐿_{𝐾 ,0}

= |𝛾|𝜅₂(𝑇 , 𝜔,Δ₀)𝑛_{𝑞 𝑝} (4.23) The kappa functions (𝜅₁and𝜅₂) encode the relationship between complex conduc- tivity (𝜎) and quasiparticle density in a superconductor. Both are defined in Gao’s thesis [65]. The functions are dependent on the temperature of the quasiparticle system. We typically assume quasiparticle temperature is equivalent to the fridge temperature (𝑇), but exposure to persistent backgrounds such as RF photons can cause the effective quasiparticle temperature to be elevated.

The complex conductivity is fully defined in integral form (Eqs. 4.26 – 4.28) in Gao’s thesis [65]. It is a function of the normal-state conductivity (𝜎_𝑛), superconducting energy gap (Δ), temperature (𝑇), readout frequency (𝜔), and effective chemical potential for quasiparticle (𝜇^∗).

𝜎 =𝜎₁+ 𝑗 𝜎₂ (4.26)

𝜎₁ = 2𝜎_𝑛 ℏ𝜔

∫ ∞ Δ

[𝑓 (𝐸;𝜇^∗, 𝑇) − 𝑓 (𝐸 +ℏ𝜔;𝜇^∗, 𝑇)] 𝐸²+Δ²+ℏ𝜔 𝐸

√

𝐸²−Δ²

√︃

(𝐸+ℏ𝜔)²−Δ²

𝑑𝐸 (4.27)

𝜎₂ = 𝜎_𝑛 ℏ𝜔

∫ Δ Δ−ℏ𝜔

[1−2𝑓 (𝐸 +ℏ𝜔;𝜇^∗, 𝑇)] 𝐸²+Δ²+ℏ𝜔 𝐸

√

Δ²−𝐸²

√︃

(𝐸 +ℏ𝜔)²−Δ²

𝑑𝐸 (4.28)

The effective chemical potential is defined by requiring the integral of the Fermi- Dirac distribution with the superconducting density of states to give the correct quasiparticle density. The superconducting density of states is normalized to 𝑁₀ (the density of states for electrons of a single spin at the Fermi energy level).

𝑛_{𝑞 𝑝} =4𝑁₀

∫ ∞ Δ

𝑓 (𝐸;𝜇^∗, 𝑇) 𝐸 𝑑𝐸

√

𝐸²−Δ²

(4.29) Eqs. 4.27 – 4.29 all use the well-known Fermi-Dirac distribution (Eq. 4.30).

𝑓 (𝐸;𝜇^∗, 𝑇) = 1 1+𝑒

𝐸−𝜇∗

𝑘𝐵𝑇

(4.30) Approximate forms for both kappa functions are also calculated in Gao’s and Siegel’s theses [65, 66]. These forms are functions of readout frequency through the factor 𝜉 (Eq. 4.33). They use the zeroth-order modified Bessel functions of first and second kind (𝐼₀ and 𝐾₀, respectively). The approximations below are specifically

for changes in conductivity due to athermal Cooper-pair-breaking events. Thermal changes will also affect quasiparticle number and conductivity but with a different approximate 𝜅. Gao showed that the athermal and thermal 𝜅 are very similar for sub-Kelvin temperatures in Al [65].

𝜅₁(𝑇 , 𝜔,Δ₀) = 1 𝜋 𝑁₀Δ₀

√︂ 2Δ₀ 𝜋 𝑘_𝐵𝑇

sinh(𝜉)𝐾₀(𝜉) (4.31)

𝜅₂(𝑇 , 𝜔,Δ₀) = 1 2𝑁₀Δ₀

"

1+

√︂ 2Δ0

𝜋 𝑘_𝐵𝑇

𝑒^−𝜉𝐼₀(𝜉)

#

(4.32)

𝜉 = ℏ𝜔 2𝑘_𝐵𝑇

(4.33) The approximate forms are appropriate whenℏ𝜔 ≪ Δ, 𝑘_𝐵𝑇 ≪ Δ, and𝑒⁻

Δ−𝜇∗ 𝑘𝐵𝑇 ≪ 1.

Understanding the different film-characteristic assumptions encoded in𝛾requires a few additional definitions:

• The Mattis-Bardeen kernel is the kernel giving the current density (normal and supercurrent) from the vector potential. It has a characteristic decay length equivalent to the minimum of the coherence length (𝜉₀) and the electron mean free path (ℓ_𝑒).

• The coherence length can be thought of as the average distance between electrons in a Cooper pair or the average size of a Cooper pair. In either case, it is the characteristic length scale of the Cooper pairs.

• The effective penetration depth (𝜆_eff) is the characteristic decay length of the penetrating magnetic field.

The commonly used combinations of film-characteristic assumptions are listed be- low:

• Thick-film, extreme anomalous limit: The film is thicker than ℓ_𝑒. This condition is referred to as thethick-film limit. Additionally, the decay length of the Mattis-Bardeen kernel is much greater than𝜆_effbecause both𝜉₀andℓ_𝑒 are much greater than 𝜆_eff. This decay-length condition is referred to as the

"extreme anomalous limit."𝛾 is equivalent to−1/3.

• Thin-film, local limit: The film thickness is less than the unobstructed ℓ_𝑒, limiting the actual mean free path to be equivalent to the film thickness. This condition is referred to as thethin-film limit. Additionally, the decay length is in agreement with thelocal limit. With this combination of assumptions, the field never decays to zero and the conductor has uniform current density.

𝛾 is equivalent to−1.

The summary of film-characteristic assumptions and associated𝛾values are shown in Eq. 4.34.

𝛾 =

















−1/3 Thick-film, extreme anomalous limit

−1/2 Thick-film, local limit

−1 Thin-film, local limit

(4.34)

We can now differentiate Eqs. 4.22 and 4.23 to learn how the inductance and resistance change with quasiparticle density. The integral form of 𝜅 only weakly depends on 𝑛_{𝑞 𝑝} through the factor 𝜇^∗. The approximate forms (Eqs. 4.31 and 4.32) do not depend on 𝑛_{𝑞 𝑝} at all, so we do not consider that dependence in the differentiation.

𝑑𝑅 𝑑𝑛_{𝑞 𝑝}

=𝜔 𝐿_{𝐾 ,0}|𝛾|𝜅₁(𝑇 , 𝜔,Δ₀) (4.35)

𝑑𝐿_𝐾 𝑑𝑛_{𝑞 𝑝}

=𝐿_{𝐾 ,0}|𝛾|𝜅₂(𝑇 , 𝜔,Δ₀) (4.36) Now we can combine Eqs. 4.12, 4.14, 4.18, 4.19, 4.36, and 4.35 to write down the total response to changes in quasiparticle density.

𝑑𝑆₂₁ 𝑑𝑛_{𝑞 𝑝}

=

𝛼|𝛾|^𝑄2𝑟

𝑄𝑐

𝜔 𝜔𝑟

𝐿𝐾 ,0

𝐿𝐾

1+2𝑗 𝑄_{𝑟 𝜔−𝜔}

𝑟

𝜔𝑟

2

𝜅₁− 𝜔_𝑟 2𝑄_𝑖𝜔

𝜅₂+ 𝑗 𝜅₂

=

𝛼|𝛾|^𝑄2𝑟

𝑄_𝑐 𝜔 𝜔_𝑟

𝐿_{𝐾 ,0} 𝐿_𝐾

1+2𝑗 𝑄_𝑟 𝜔−𝜔_𝑟

𝜔_𝑟 2

𝜅− 𝜔_𝑟 2𝑄_𝑖𝜔

𝜅₂

(4.37)

Those familiar with MKIDs will recognise Eq. 4.37 once a number of common assumptions are applied. The first assumption is that the MKID is always being read out precisely on resonance (𝜔 = 𝜔_𝑟). This is a logical choice since (as discussed at the beginning of this section) the maximum responsivity is seen on-resonance.

However, the assumption is not entirely realistic, since the exact resonant frequency will change with inductance (Eq. 4.18) during a Cooper-pair breaking event. Some MKID readouts use tone-tracking to read out on resonance during events of any size (improving dynamic range). For small signals, this is unnecessary and the assumption remains reasonable. Assuming𝜔 =𝜔_𝑟 gives Eq. 4.38.

𝑑𝑆₂₁ 𝑑𝑛_{𝑞 𝑝}

=𝛼|𝛾|𝑄²

𝑟

𝑄_𝑐 𝐿_{𝐾 ,0}

𝐿_𝐾

𝜅− 1 2𝑄_𝑖

𝜅₂

(4.38) Under the assumption (generally confirmed by measurement) of a high internal quality factor, the latter𝜅₂term becomes trivial and can be ignored. This gives Eq.

4.39.

𝑑𝑆₂₁ 𝑑𝑛_{𝑞 𝑝}

=𝛼|𝛾|𝑄²

𝑟

𝑄_𝑐 𝐿_{𝐾 ,0}

𝐿_𝐾

𝜅 (4.39)

Lastly, 𝐿_𝐾 is assumed to be approximately equal to 𝐿_{𝐾 ,0}. This can be achieved by operating at temperature significantly below 𝑇_𝑐 and by minimizing external excitations. The latter condition is desirable for low-background searches anyway.

One can still make this assumption at higher temperatures if the resonant frequency is observed to make only a small fractional change from its lowest-temperature value. The resulting Eq. 4.40 is the typically used responsivity for MKIDs.

𝑑𝑆₂₁ 𝑑𝑛_{𝑞 𝑝}

=𝛼|𝛾|𝑄²

𝑟

𝑄_𝑐

𝜅 (4.40)

indicates that a thicker film does not directly improve responsivity.

MKID sensitivity is also very stable. 𝛼, 𝛾, and the quality factors are the result of MKID design and should be stable if the MKID is operated at relatively stable temperature. 𝜅values are also relatively stable with temperature. Calculated values can be seen in Figure 4.1, taken from [66]. Response stability is what makes MKID calibration possible.

Figure 4.1: 𝜅 calculated using Mattis-Bardeen theory for MKIDs with frequency 3.2 GHz,Δ₀= 0.21 meV (typical for aluminum), and various temperatures. Figure is taken from [66]

.

Eq. 4.40 can also be used to approximately convert small event signals from units of transmission (𝛿 𝑆₂₁) to energy in the MKID (𝛿 𝐸). The conversion requires multiplying by the superconducting energy gap (Δ).

𝛿 𝐸 = Δ𝛿𝑛_{𝑞 𝑝} ≈Δ 𝑑𝑛_{𝑞 𝑝} 𝑑𝑆₂₁

𝛿 𝑆₂₁ = 𝑄_𝑐Δ 𝛼|𝛾|𝑄²

𝑟𝜅

𝛿 𝑆₂₁ (4.41)