2.2 Responsivity
2.2.10 Nonuniform Absorption
But even after adding the gold wirebonds, the frequency and dissipation of the resonators as a function of temperature and loading exhibited behavior that could not be explained by the full instrument model without including an effect akin to substrate heating. This is illustrated in Figure 2.10. At low temperatures (250 mK<Tbath<300 mK) and under both 77 K and 293 K loads the measured dissipation is constant as a function of temperature. The model without substrate heating is unable to replicate this behavior, it actually predicts a decrease in the dissipation with increasing temperature. This is because under the model the number of optically generated quasi-particles is much greater than the number of thermally generated quasi-particles at low temperatures. So as the temperature increases, the total number of quasi-particles remains roughly constant, but the factor that converts quasi-particles to complex conductivityκ1decreases.
Not until one reaches a temperature at which the thermally generated quasi-particles become significant does the dissipation begin to increase. We do not observe this curvature, the measured dissipation and the frequency are flat at low temperatures for nearly all of the resonators. In order to replicate this behavior the model without substrate heating moves toward unphysical parameter values — particularly very low values ofτmax— and even then it yields poor fits.
We include heating in our model to explain this behavior. However, our interpretation is not that the substrate is sitting at an elevated temperature, but rather the quasi-particles. That is, we assume that under heavy optical loading, the quasi-particles are out of equilibrium with the lattice phonons. We further assume that the quasi-particles can be described by a Fermi-Dirac distribution at an effective temperatureT, which is set by the thermal conductance formula
T=
Tbathn +Pe
g 1/n
=
Tbathn +ηePopt
g 1/n
, (2.111)
where we have made the reasonable assumption that the power maintaining the elevated quasi-particle tem- perature is proportional to the absorbed optical power,Pe=ηePopt. This adds two additional parametersnand η˜e=ηe/gto the model.
68
= p
ReZ0
a(0) exp( l) sinh(+ (l z)).
(13) We may now calculate the power dissipated per unit length of the line:
dP(z)
dz = |I(z)|2ReZ
= Pinc4ReZ
ReZ0 |exp( l) sinh(+2 (l z))|2 .
= (z)Pinc . (14)
As an example, the normalized absorption profile (z)/l is plotted below in Fig. 1, assuming microstrip parameters that are typical of our MKID designs, and a length l= 1 mm.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.5 1 1.5 2 2.5 3 3.5
Normalized Position
Power dissipated per unit normalized length
Figure 1: Photon absorption profile in a 1 mm long Nb/SiO
2/Al microstrip line at 250 GHz.
The layer thicknesses are taken to be 2000 ˚ A, 4000 ˚ A, and 400 ˚ A, for the Niobium, SiO
2, and Al, respectively. For this thickness, the aluminum resistivity and sheet resistance are calculated to be 2.2 µ⌦ cm and 0.55 ⌦, respectively. The exponential decay of the incident wave is clearly seen, with small standing–wave ripples superimposed.
3
Figure 2.11: Power absorption profileψ(z)for a 1 mm long Nb/SiO2/Al microstrip line at 250 GHz. Taken from Zmuidzinas [114].
effects of diffusion. We briefly outline this procedure below following the analysis of Golwala [98].
Letzdenote position along the length of the Al section, so that 0≤z≤l. Power from the antenna is incident on the microstrip fromz=0 and the microstrip terminates in an open circuit atz=l. Define the power absorption profileψ(z)as the fractional power absorbed per unit length
dP
dz =ψ(z)Popt,ant,
Z l 0
ψ(z)dz=1. (2.112) Note that not all power incident from the microstrip will be absorbed by the Al section. In the definitions above we have lumped the absorption efficiencyηabs into the overall optical efficiencyηopt,ant presented in Equation (2.105) forPopt,ant. Zmuidzinas [114] derives an analytical expression forψ(z), assuming perfect reflection atz=land neglecting impedance mismatch at the microstrip and aluminum interface. We use this analytical expression and a measurement of the Al sheet resistance at 4 K to determineψ(z). An example power absorption profile is shown in Figure 2.11.
If the distribution of quasi-particles is nonuniform then the quasi-particles will tend to diffuse out of high density regions and into low density regions. This process is described by a diffusion current
J(r,t) =−D∇nqp(r,t), (2.113)
where we assume that the diffusion constantDis independent of position. We can write a general continuity equation to describe the local conservation of quasi-particles
∂nqp
∂t +∇·J=ΓG,th(r,t) +ΓG,opt(r,t)−ΓR(r,t), (2.114) where the now position-dependent optical and thermal generation rates act assourceterms that adds quasi- particles into the system, and the recombination rate acts as asinkterm that removes quasi-particles from the system. The diffusion length is expected to be much greater than either the widthwor thicknessdof the Al section, so it is safe to assume that the quasi-particle density is constant over the cross-sectional area. This reduces Equation (2.114) to the one-dimensional partial differential equation
∂nqp(z,t)
∂t −D∂2nqp(z,t)
∂z2 =ΓG,th(z,t) +ΓG,opt(z,t)−ΓR(z,t). (2.115)
If we consider the steady state behavior where ∂nqp
∂t =0, then Equation (2.115) yields a position-dependent version of the generation-recombination equation
ηphPopt,antψ(z) +ηphpopt,dir
∆
=−DA∂2nqp(z)
∂z2 +RA
n2qp(z)−n2qp,th + A
τmax
[nqp(z)−nqp,th] , (2.116)
wherepopt,dir=Popt,dir/lis the power directly absorbed per unit length andA=wd is thecross-sectional area.
Here we are making the reasonable assumptions that the direct absorption and thermal generation occur uniformly throughout the aluminum section. Givenψ(z)the differential equation above can be solved nu- merically to obtainnqp(z). The boundary conditions for the equation are
∂nqp(z)
∂z z=0,l
=0, (2.117)
which physically correspond to the fact that the quasi-particles cannot diffuse out of the Al section.
It can be shown [103] that a position-dependent quasi-particle distribution is weighted by the square of the resonator current distribution in the determination of the resonator response. For aλ/4 transmission line resonator the current has a standing wave distribution at resonant frequency that is given by I(z) = I(0)cos(πz/2L)whereLis the total length of the transmission line. The current is a maximum at the shorted, inductive end (z=0) and a minimum at the open, capacitive end (z=L). As a result the “effective” quasi-particle density is calculated as
nqp= Rl
0cos2 πz2L nqp(z)dz Rl
0cos2 πz2L
dz . (2.118)
The MUSIC resonators are of a hybrid design in which the inductive portion is a distributed CPW transmis- sion line and the capacitive portion is a nearly lumped element IDC. The current should be approximately
uniform over the length of Al section, since it occupies a small fraction of the total length of the inductive portion at the high current, shorted end. We assume that cos2 2Lπz
≈1 for 0≤z≤l. The mean quasi-particle density over the length of the Al section is then the quantity that determines resonator response. We sep- arate the expression for the position-dependent quasi-particle densitynqp(z)into the mean value nqp and a position-dependent factorφ(z):
nqp(z) =nqpφ(z), 1 l
Z l 0
φ(z)dz=1. (2.119) We can then integrate Equation (2.116) over the length of the Al section to obtain an implicit expression for the mean density
ηph(Popt,ant+Popt,dir)
∆ =−DAnqp
Z l 0
∂2φ(z)
∂z2 dz+RV
"
n2qp l
Z l 0
φ2(z)dz−n2qp,th
# + V
τmax
[nqp−nqp,th]. (2.120)
The diffusion term disappears due to the boundary condition Z l
0
∂2φ(z)
∂z2 dz= ∂ φ(z)
∂z
l 0
=0. (2.121)
If we then introduce the correction factor
ζ=1 l
Z l 0
φ2(z)dz (2.122)
we obtain the relatively simple equation ηphPopt,ant+ηphPopt,dir
∆ =RV ζn2qp−n2qp,th
+ V τmax
(nqp−nqp,th). (2.123)
An explicit expression fornqpcan then be obtained by applying the quadratic formula
nqp=
ηphPopt,ant+ηphPopt,dir
ζRV∆ +1 ζn
2
qp,th+ 1
ζRτmax
nqp,th+ 1
4ζRτmax
1/2
− 1 2ζRτmax
, (2.124)
where again we have discarded the second root since it corresponds to negative values. Comparing to Equa- tion (2.21) we see that, in regards to the steady-state quasi-particle density, the nonuniform absorption essen- tially just increases the recombination constantRby a factorζ. The factor ofζ−1in front of then2qp,th term also means that the influence of the thermal quasi-particle population is suppressed due to the nonuniform absorption.
We now consider the effect of nonuniform absorption on the quasi-static, small-signal response. Let
δPopt,ant and δPopt,dir denote small changes in power absorbed through the antenna and the power directly
absorbed, respectively. This will result in a small position-dependent change in the quasi-particle density
δnqp(z) =nqpδ φ(z). Performing a perturbation analysis of Equation (2.116) yields ηphδPopt,antψ(z) +ηphδPopt,dir/l
∆ =−DAnqp
∂2(δ φ(z))
∂z2 +Anqp
2Rnqpφ(z) + 1 τmax
δ φ(z), (2.125) which is a differential equation similar to Equation (2.116) but now for the small signal response profile δ φ(z). We would like to derive an expression for the mean response over the length of the Al section, since this is the quantity that we actually measure. Integrating both sides of Equation (2.125) from 0 tol and dropping the diffusion term, again due to boundary conditions, yields
ηph(δPopt,ant+δPopt,dir)
∆ =V nqp
1 l
Z l 0
2Rnqpφ(z) + 1 τmax
δ φ(z)dz
=V
2Rnqpζδ+ 1 τmax
nqp
1 l
Z l 0
δ φ(z)dz
=V
2Rnqpζδ+ 1 τmax
δnqp. (2.126)
Here we have defined the mean change in quasi-particle density
δnqp=1 l
Z l 0
δnqp(z)dz
=1 lnqp
Z l 0
δ φ(z)dz (2.127)
and also the correction factor
ζδ = Rl
0φ(z)δ φ(z)dz Rl
0δ φ(z)dz , (2.128)
which is the average value of φ(z) when weighted by the small signal response profile δ φ(z). Equa- tion (2.126) is nonlinear because the factor that scalesδPopttoδnqpdepends onζδ, which depends onδ φ(z), which depends onδPoptthrough the differential Equation (2.125). We would like to find the linear expression that must hold in the limit of very small changes. If we make the reasonable ansatz that the fluctuation in the quasi-particle density at a given position is proportional to the mean quasi-particle density at that position or δ φ(z)∝φ(z)thenζδ =ζ and Equation (2.126) becomes
ηph(δPopt,ant+δPopt,dir)
∆
=V
2ζRnqp+ 1 τmax
δnqp. (2.129)
We recover linearity so that the small signal responsivity only depends on the steady-state properties as de- sired. In addition, a single parameterζcompletely characterizes the effects of nonuniform absorption and dif- fusion. We point out that in order to calculateζ one must numerically solve the differential Equation (2.116)
for the quasi-particle profileφ(z). If we simply redefine the effective quasi-particle lifetime as 1
τqpeff =2ζRnqp+ 1 τmax
(2.130) then we obtain
δnqp=τqpeffηph(δPopt,ant+δPopt,dir)
V∆ , (2.131)
which is the counterpart to Equation (2.25) that accounts for direct absorption, nonuniform absorption, and diffusion.