2.2 Responsivity
2.2.9 Substrate Heating and Nonequilibrium Dynamics
As mentioned in Section 2.2.8, radiation can heat the silicon substrate, elevating its temperature above that of the copper focal plane unit. LetTsubdenote the temperature of the substrate andTbaththe temperature of the
focal plane unit as measured by the Array GRT. The power that must be absorbed by the substrate in order to maintain the elevated temperature is given by a thermal conductance function of the form
Psub=g(Tsubn−Tbathn), (2.109) wheregis the thermal coupling constant andnis the thermal coupling power-law exponent, the exact value of which will depend on the coupling model. We can solve this equation to obtain an expression for the temperature of the substrate
Tsub=
Tbathn +Psub
g 1/n
= Tbathn +P˜sub
1/n
, (2.110)
where we have defined ˜Psub=Psub/gsince the expression depends only on the ratio ofPsubandg.
Substrate heating was substantial in several generations of engineering grade arrays, dominating the nonantenna response measured by the dark resonators. In order to model the substrate heating we used Equation (2.110) and assumed that the temperatureT appearing in the Mattis-Bardeen equations was equal toTsub. This added 1+Nloadparameters to the overall model since a different value ofPsub was required for each loading condition under consideration.
Eventually the problem of substrate heating was remedied by connecting (via gold wirebonds) a border of gold metal film on the detector tiles to the copper on the focal plane unit, thereby improving the thermal conductance. After adding the gold wirebonds, measurements of the resonant frequency as a function of tem- perature of a set of all Nb test resonators located on an engineering grade array placed strong upper limits on the temperature differenceTsub−Tbath.3 mK for all operating conditions of interest. To briefly summarize this measurement: since the critical temperature of Nb is 9.2 K, the number of thermally and optically generated quasi-particles is negligible for the test resonators at the bath temperatures and optical wavelengths at which we operate. Their resonant frequency is entirely set by the temperature-dependent resonant response of a thin layer of two-level systems (TLS) on the substrate. TLS are discussed in greater detail in Section 2.3.2.2, suffice it to say that the temperature dependence of the TLS induced resonant frequency shift is theoretically understood, is approximately linear with a positive slope (see Equation (2.179)), and is observed in the test resonators. Therefore, we can effectively use the test resonators as “TLS thermometers”. We found that their resonant frequencies did not change when we switched from a LN2blackbody load to a room-temperature blackbody load at the cryostat window. If there was substrate heating, the resonant frequency would increase between the LN2and room temperature load. We use this fact to place an upper limit of 1.5 mK on the differ- ence in substrate temperature between the LN2and room-temperature load. If we then make the reasonable assumption that the change in substrate temperature between a 0 K and LN2load is not larger than the change between a LN2and room temperature load, we obtain the quotedTsub−Tbath.3 mK.
0 1 2 3 4 5 6 7 8
1/Qi[x10−5]
2 2.5 3 3.5 4 4.5 5 5.5
ακ1[x10−9 µm3 ]
200 250 300 350 400 450
−6
−4
−2 0 2 4 6
Tb a t h [ mK ]
NormalizedResiduals
200 250 300 350 400 450
200 250 300 350 400 450
Tb a t h [ mK ]
T[mK]
Dark 77K Load 293K Load
No Heating Heating − 77K Heating − 293K
Figure 2.10: Experimental evidence for quasi-particle heating. Top Left: The dissipation 1/Qias a function of bath temperature and loading for a typical MUSIC detector. Black denotes negligible optical loading. Blue (red) denote a 77K (293K) beam-filling, black-body load in front of the cryostat window. The circles are mea- sured data points and the lines are best-fit models. Error bars are less than the radius of the circle. The dashed line is a model without heating and the solid line is a model with heating.Bottom Left:Normalized residuals (i.e.,(data−model)/error) for the best-fit models shown in the plot above. Unfilled circles correspond to the model without heating and filled circles correspond to the model with heating. Top Right: The factor that converts quasi-particle density to dissipation as a function of temperature. The dashed pink line corresponds to the model without heating and the solid blue and red lines correspond to the model with heating. Bottom Right:The temperature as a function of bath temperature. The legend is the same as in the plot above.
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.