Mattis-Bardeen Theory

2.2 Responsivity

2.2.3 Mattis-Bardeen Theory

over the decay due toτmaxthis becomes

n_qp≈ ηphP_opt

RV∆ 1/2

forn_qpn_qp,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+τ_max⁻¹), (2.23)

where we have evaluated the partial derivative using Equation (2.20). It is traditional to then define 1

τ_qp^{eff =2Rnqp+} 1 τmax

. (2.24)

We callτ_qp^effthe 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=τ_qp^effηphδPopt

V∆ . (2.25)

We will discuss the effective quasi-particle lifetime further in Section 2.3.2.1.

the following integrals hold:

σ1=2σn

¯ hω

Z _∞

∆

[f(E)−f(E+h¯ω)] E²+∆²+h¯ωE r

(E²−∆²)h

(E+h¯ω)²−∆² i

dE (2.26)

σ2= σ_n h¯ω

Z ∆

∆−¯hω

[1−2f(E+h¯ω)] E²+∆²+h¯ωE r

(∆²−E²)h

(E+h¯ω)²−∆² i

dE. (2.27)

Here∆is the gap energy,σnis the normal state conductivity, and f(E)is the distribution function for quasi- particles. Assuming that the quasi-particles are in thermal equilibrium, f(E)is given by the Fermi-Dirac distribution

f(E) = 1 1+exp_E−µ∗

k_BT

. (2.28)

The chemical potentialµ^∗in Equation (2.28) is used to account for optically generated quasi-particles [95].

Essentially we are assuming that the quasi-particles are in thermal equilibrium with the lattice, but are not in chemical equilibrium. The assumption of thermal equilibrium should be valid even for optically generated quasi-particles because the timescale for an excited quasi-particle to relax through scattering processes is much shorter than timescale for recombination. We will return to this assumption in Section 2.2.9. The density of excited states in a superconductor is given by

N_s(E) = N₀E

√ E²−∆²

, (2.29)

whereN0is the single-spin density of electron states at the Fermi energy level. Equation (2.29) implies that energies below the gap|E|<∆are forbidden and that there is an increase in the density of states just above the gap. This occurs because the total number of available energy states is constant, and the states that are below the gap in a normal metal are shifted out. The quasi-particle density is obtained by integrating the product of f(E)andN_s(E)over all energies

nqp=4N₀ Z _∞

∆

E f(E)

√ E²−∆²

dE. (2.30)

The final equation required is the self-consistent integral for the band gap 1

N0VBCS

= Z hω_{¯ D}

∆

1−2f(E)

√ E²−∆²

dE , (2.31)

whereωDis the Debye frequency andVBCSis the interaction strength in the BCS model [84]. The limit of weak electron-phonon coupling holds for aluminum so thatk_BT_ch¯ωD. By examining Equation (2.31) in

the limit thatT →0 one can derive the expression

∆₀−∆

∆0

=2 Z ∞

∆

f(E)

√ E²−∆²

dE, (2.32)

which relates the gap at temperatureT<T_cto the gap at zero temperature∆0=∆(T=0). At finite temper- atures the quasi-particle population suppresses the gap relative to the zero temperature value.

In the limit thatk_BT ∆, ¯hω∆, and exp(−(E−µ^∗)/(k_BT))1 the integrals expressed in Equa- tions (2.30), (2.32), (2.26), and (2.27) can be simplified to the following analytical formulae:

nqp=2N₀p

2πk_BT∆exp

−∆−µ^∗ kBT

(2.33)

∆

∆0

=1−

r2πk_BT

∆ exp

−∆−µ^∗ k_BT

=1− n_qp

2N0∆ (2.34)

σ1

σn

= 4∆

¯ hω^exp

−∆−µ^∗ kBT

sinh(ξ)K₀(ξ) (2.35)

σ₂ σn

=π∆ h¯ω

1−2 exp

−∆−µ^∗ k_BT

exp(−ξ)I₀(ξ)

, (2.36)

whereξ =h¯ω/2k_BT is a dimensionless frequency andI₀andK₀are the zero’th order modified Bessel func- tions of the first and second kind, respectively. We now evaluate the three conditions to determine if the analytical formula are appropriate for the MKIDs in MUSIC. We are interested in examining the behavior of our detectors over a range of temperatures fromTlb=200 mK toTub=500 mK and under optical loads that have temperatures up toTload,ub=300 K when referred to the cryostat window.

k_BT ∆

The critical temperature of aluminum isT_c=1.2 K. ThereforeTlb=0.17T_candTub=0.42T_c. Over this range of temperatures∆does not differ from∆₀by more than 5% so we may write∆=∆₀=1.76k_BT_c. We then have thatk_BTlb/∆₀=0.10 andk_BTub/∆₀=0.25.

h¯ω∆

The gap energy of aluminum is∆0=0.18 meV, which corresponds to a frequency of 45 GHz. The MKIDs have resonant frequencies between 3−4 GHz. Therefore, ¯hω/∆₀=0.08.

exp(−(E−µ^∗)/(k_BT))1

Since we are always consideringE≥∆in the integrals above, we can use exp(−(∆−µ^∗)/(k_BT))as an upper bound. This quantity can be approximated by Equation (2.33) with the quasi-particle densitynqp

inferred from Equation (2.22) for our expected optical loading. We are interested in obtaining an upper bound so we assume that the detectors are 100% efficient and have a bandwidth of 45 GHz. ThenT_load,ub results in 185 pW of absorbed power. AssumingN₀=1.07×10²⁹ J⁻¹µm⁻³andR=9.4µm³s⁻¹, we obtain exp(−(∆−µ^∗)/(k_BT))≤0.02.

The second and third conditions are certainly satisfied. The first condition will begin to break down as the

Figure 2.2: The ratio of ^{∂ σ(T,nqp)}

∂nqp calculated using Equations (2.40)−(2.41) to ^{∂ σ(T)/∂T}

∂nqp(T)/∂T calculated using Equations (2.37)−(2.39) as a function of temperature. Blue (red) denote the real (imaginary) part of the complex conductivity.

temperature approachesTub. This exercise suggests that the analytical formula provide a good approximation for the behavior of our detectors over all operating conditions of interest, but to obtain accurate conclusions the full integrals should be used, especially at large temperatures.

We are interested in two scenarios. The first is adarkscenario in which the number of optically generated quasi-particles is negligible. This can be accessed experimentally by placing an aluminum cover over the detector array and cooling it down to sub-Kelvin temperatures. In this case we can setµ^∗=0 and obtain explicit expressions for all quantities of interest in terms of the temperature

nqp,th(T) =2N₀p

2πk_BT∆₀exp

−∆0

k_BT

(2.37) σ₁^dark(T)

σ_n ⁼ 4∆₀

¯ hω ^exp

− ∆0

k_BT

sinh(ξ)K₀(ξ) (2.38)

σ₂^dark(T) σn

=π∆₀ h¯ω

"

1− s

2πk_BT

∆0

exp

− ∆₀ k_BT

−2 exp

− ∆₀ k_BT

exp(−ξ)I₀(ξ)

#

. (2.39)

The second scenario is where the detectors are illuminated and the number of optically generated quasi- particles is significant compared to the number of thermally generated quasi-particles. In this scenario we write Equations (2.26) and (2.27) in terms ofnqpin order to suppress the dependence on the chemical potential.

This yields

σ1(T,nqp) σ_n ⁼

1 N₀h¯ω

r 2∆₀

πk_BT sinh(ξ)K₀(ξ)n_qp (2.40)

σ2(T,n_qp) σ_n ⁼

π∆0

¯ hω ⁻

π

2N0h¯ω ¹⁺ r 2∆₀

πk_BT exp(−ξ)I₀(ξ)

!

n_qp. (2.41)

The important result here is that the complex conductivity is a linear function of the quasi-particle den- sity. We can take the partial derivative of Equations (2.40) and (2.41) with respect tonqp to determine the conversion factor∂ σ/∂nqpbetween changes in quasi-particle density and changes in the complex conductiv- ity. We can do the same for the dark scenario by evaluating∂ σ/∂nqp= (∂ σ/∂T)/(∂nqp/∂T)using Equa- tions (2.37)−(2.39). The ratio of these two conversion factors as a function of temperature is shown in Figure 2.2 for a typical MUSIC detector. The two are equal to within 20% over the temperature range of interest. This implies that theoretically the change in the complex conductivity that results from the thermal generation of a quasi-particle (through a change in the bath temperature) is approximately equivalent to the change that results from the optical generation of a quasi-particle. This equivalence enables calibration of the optical response by measuring the thermal response in a dark scenario. If this were not the case, the optical response would have to be measured directly using a calibrated black-body source, which is much more challenging. The equivalence between thermally and optically generated quasi-particles was confirmed experimentally in an early version of the MUSIC detectors that consisted of Al CPWs on a sapphire substrate [96]. More recently Janssen et al. [97] performed a rigorous comparison of thermal and optical response in two hybrid NbTiN-Al MKIDs and found them to be equal to within a factor of two. The factor of two discrep- ancies were larger than their measurement uncertainties, and they suspected that the discrepancies were due to the inability to accurately determine∆0andηph, parameters necessary to compare the thermal and optical response within their model. We will have to address similar difficulties with our calibration procedure.

Examining Equations (2.40) and (2.41) we see that the complex conductivity of the superconductor will depend on the normal state conductivityσn=1/ρn. This can be obtained by measuring the sheet resistance of the superconductor aboveT_c. It can also be predicted theoretically. However, as we will see in the following sections it is the fractional change in the complex conductivity — not the absolute value — that is important in predicting detector response. We define this change with respect to the dark value at zero temperature. In the dark scenario, asT→0 the quasi-particle density vanishes exponentially, and as a result so too does the real component of the complex conductivityσ₁^dark(0) =0. The imaginary component exponentially approaches a finite value ofσ₂^dark(0) = (π∆0/¯hω)σ_ndue to the kinetic inductance of the Cooper pairs. Therefore

σ₀=σ₁^dark(0)−jσ₂^dark(0) =−jπ∆₀

h¯ω σ_n. (2.42)

Figure 2.3:Left:Mattis-Bardeen prediction for a typical MUSIC resonator with frequencyfres=3.2 GHz and gap energy∆0=0.21 meV.Right:the ratio of frequency to dissipation response is shown in purple and the angle between the quasi-particle direction and the direction normal to the resonance curve is shown in green.

We can then write

δ σ

|σ| ≡σ−σ0

|σ0| = (κ1+jκ2)nqp, where

κ₁(T,ω,∆₀) = 1 πN₀∆0

r 2∆₀

πk_BT sinh(ξ)K₀(ξ) (2.43)

κ2(T,ω,∆0) = 1 2N₀∆0

"

1+ r 2∆₀

πkBTexp(−ξ)I₀(ξ)

#

. (2.44)

The functionsκ₁andκ₂are plotted as a function of temperature in the left panel of Figure 2.3 for val- ues of ω and∆₀typical of a MUSIC detector. These functions act as the conversion between changes in quasi-particle density and changes in the complex conductivity. We examine the ratio ofκ₂toκ₁in the right panel of Figure 2.3. This corresponds to the ratio of the imaginary and real response to the generation of quasi-particles. The imaginary response is intrinsically larger than the real response, and their ratio increases approximately linearly with temperature. The MUSIC detectors operate at a temperatureT =240 mK where κ2/κ1≈4. We can further simplify the presentation of Equation (2.43) by writing it as

δ σ

|σ| =|κ|e^jΨnqp, (2.45)

withκ= q

κ₁²+κ₂²andΨ=arctan(κ2/κ1). This makes it clear that changes in quasi-particle density will appear in a well defined direction in the complex plane. The angleΨis also shown in the right panel of Figure 2.3. At our operating temperatureΨ(240 mK)'78^◦so that the response is only slightly rotated from the imaginary direction.

We note that the quantityδ σ/|σ|is small, even for quasi-particle densities corresponding to temperatures as large as 0.5Tc. This is due to the suppression of the quasi-particle density by the factor exp(−∆₀/k_BT)≈ exp(−1.76T_c/T). At temperaturesT ≤0.5T_c we have thatκ1n_qp<0.01 andκ2n_qp<0.10 for the typical MUSIC detector [98]. Therefore, going forward we will only consider up to first order inδ σ/|σ|in the development of the model.