EWLT EW
6.2 Microwave Kinetic Inductance Detectors
6.2.1 Physics Overview
The Cooper pairs in a superconductor don’t scatter or dissipate energy as they travel, causing the resistance (atT = 0 K) of a superconductor to be zero. These Cooper- pairs do, however, have mass and inertia. As a result, if an AC electric field is applied to a superconductor, the induced super-current will lag in phase behind the field, as it takes time for the field to accelerate the pairs. This implies that super- conductors have a non-zero (and purely imaginary) impedance even atT =0 K. As opposed to a normal inductor, which accomplishes a phase lag by conservatively storing energy in a magnetic field, this phase lag is accomplished by conserva- tively storing kinetic energy in the Cooper-pair condensate. This phenomena is aptly called “kinetic inductance”. If an incident particle (such as an IR photon or a phonon) has an energy greater than the Cooper-pair binding energy, it can interact and create two quasiparticles. Unlike the condensate, the quasiparticles do scat- ter, which has two effects. First, it increases the resistance, or energy dissipation, of the superconductor. Because they scatter and lose kinetic energy, quasiparticle also do not contribute to the overall kinetic inductance of the metal. This decreases the mass of of the super-current condensate, which increases the kinetic energy re- quired to maintain a given super-current and subsequently (and somewhat counter- intuitively) increases the kinetic inductance. An LC resonant circuit that is built out of such a superconductor will then experience changes in resonant frequency, phase, and quality factor from the splitting of Cooper-pairs inside of its inductive
Figure 6.2: Physics of MKID response: a) Incident particles such as photons (or- ange) or phonons (pink) with energies> 2∆can interact with a cooper pair (black) and create two quasiparticles (green). b) a schematic of the MKID circuit. LKi is the variable kinetic inductance whileLmis the normal magnetic inductance. Cooper pair creation causes in increase both in dissipation as well as inductance in the cir- cuit. c) Magnitude of the complex transmission through the coupled feedline vs frequency. Response of an MKID is shown both before (black solid) and after (red dotted) the creation of a population of quasiparticles. d) Same as in c) but depicting the phase response of the MKID [109] via [49]
element. For a graphical depiction please see 6.2.
Different superconducting materials exhibit this effect to varying degrees. We can characterize this by splitting the inductance L of a superconducting inductor into two pieces
L =LKinetic+LMagnetic (6.1)
LMagnetic is the component of the inductance that stores its energy in magnetic fields and is only dependent on the physical geometry of the inductive element. LKineticis the kinetic inductance and is defined as the component that can be changed by the
Quantitatively, how can we connect this change in quasiparticle density in our MKID to a change we can measure? As mentioned in the previous section, MKID’s act as notch filters, and can be probed by measuring the complex transmission as a function of frequency S21(f) through a feedline that is coupled to the resonator.
Following [110, 49] we can write the this as
SMKID21 (f)=1− Q/QC
1+2ιQf−f
0
f0
(6.3)
withιas the imaginary unit5. The resonator’s response is described in terms of its quality factor Qand resonant frequency f0 = 1/√
LC. As a concept, quality factor describes the energy loss from our resonant circuit and here we have broken it into two components:
1 Q = 1
QI
+ 1 QC
(6.4) where the coupling quality factorQC is due energy lost from coupling to the feed- line (which is the basis of our measurement), and the internal quality factorQI de- scribes all other energy loss mechanisms. Figure 6.3 shows a few representations of equation 6.3, in the complex plane, as well as its polar parameters (magnitude and phase) vs frequency. Our MKIDs operate in the coupling-quality-factor dominated regime where Q1
C Q1
I andQc ≈ Q.
It should be noted that equation 6.3, describes the complex transmission at the resonator. In general, if we were to measure the transmission at a later point along
5Althoughiis typically used in most math and physics texts to denote the imaginary unit, elec- trical engineers typically favor the use of the letter jto prevent a collision with the letter that is used to denote currentI. Seeing as the current density is represented in all fields with the letterJI find this very unsatisfactory. As a result I will use the Greek letterι.
Figure 6.3: Visualization of equation 6.3 in various parameters. Left: TheIQplane generally represents the real and complex components ofSMeasured21 , but in this case we are depicting the real and imaginary components of SMKID21 . Shown is the re- sponse for a resonator with QI = 106 and QC = 1×104 (blue), 2.5×104 (green), 6.3 × 104 (red), 1.5 × 105 (cyan), 4× 105 (magenta), and 106 (yellow). Right:
Magnitude vs frequency (top) and phase vs frequency (bottom) for the same set of resonators.
our transmission line6 timeτafter the resonator it would be
SMeasured21 (l)= ASMKID21 e−2πιfτ (6.5) where Adescribes the gain, or attenuation, the signal experiences during the time period. As a result, if we were to plotSMeasured21 (f) as in figure 6.3 it would be rotated by an angle of 2πfτ. To differentiate it from the complex plane of equation 6.3 the real and imaginary components ofSMeasured21 are usually depicted asI andQrespec- tively7and this rotated plane is called theIQplane.
We can write the surface impedance (Zs) of a superconductor in terms of the surface resistanceRsand the surface inductanceLsas
Ls=Rs+ιXs=Rs+ιωLs =Rs+ιωµ0λeff (6.6) WhereXs, is the surface reactance, andωthe angular frequency. Following [Gao]
6Assuming this is a nicely behaved feedline with no strange impedance mis-matches.
7As a result of this extremely unfortunate choice of notation the lettersIandQare both quite semantically oversubscribed.
impedance will be dominated by the reactance term. In this regime (Rs Xs and T TC) we can use equation 6.3 and find
δS21MKID|f=f0 = Q2 Q2C δ 1
QI
−2ιδf0
f0
!
≈αQ2 Q2C
δZS
|ZS| (6.9)
In the local, thin film limit, the thickness d limits the mean free path l, and both are much smaller than the coherence lengthξ0and London penetration depthλL. In this limit we relate the surface impedance to the complex conductivityσlike
Zs = 1
σd = 1
(σ1+ισ2)d (6.10)
allowing equation 6.9 to be written
δSMKID21 = αQ2 QC2
δσ
|σ| (6.11)
From [111] we can express the components of this complex conductivity (σ = σ1+ισ2) in terms of three integrals
σ1 σn
= 2
~ω Z ∞
∆
(f(E)− f(E+~ω))(E2+ ∆2~ωE2) p(E2+ ∆2)((E+~ω)2−∆2)
dE + 1
~ω Z −∆
∆−~ω
(1−2f(E+~ω))(E2+ ∆2~ωE2) p(E2+ ∆2)((E+~ω)2−∆2)
dE (6.12)
σ2 σn
= 1
~ω Z ∆
max[∆−~ω,−∆]
(1−2f(E+~ω))(E2+ ∆2~ωE2) p(E2+ ∆2)((E+~ω)2−∆2)
dE (6.13)
where the f(E) is the standard Fermi function. In the low frequency (~ω ∆) and low temperature limit (kbT ∆) [Gao et al LTD] derived an approximately
analytic form in terms of the change in quasiparticle densityδnqp. σ1
σn
= 2∆δnqp
~ωN0√
2πkbT∆sinh ~ω 2kbT
!
K0 ~ω 2kbT
!
(6.14) σ2
σn
= π∆
~ω
1− δnqp
2N0∆
1+
s 2∆ πkbTe
−~ω
2kbTI0 ~ω 2kbT
!
(6.15)
Where I0 and K0 are respectively the modified Bessel functions of the first and second kind, and N0 is the single-spin density of states at the Fermi level. This allows us to rewrite equation 6.11 as
δS21 =ακQ2 QC
δnqp (6.16)
where
κ= 1 πN0
r 2
πkbT∆sinh ~ω 2kbT
!
K0 ~ω 2kbT
!
+ι 1 πN0∆0
1+
s 2∆ πkbTe− ~
ω
2kbTI0 ~ω 2kbT
!
(6.17) which finally relates the change in measured complex transmission to the change in the our MKID’s quasiparticle population.