Introduction
Physics Today paper
Transition-edge sensors
Superconducting transition-edge sensors (TES) operate in the
narrow region between the normal and superconducting states
of a metal film. (See the second example in
figure 4.) For a
TES thermometer, the logarithmic sensitivity 
can be more than
an order of magnitude higher than for a practical semiconductor thermistor.
There are two principal means of developing a voltage (and
hence a resistance) across an electrically biased
superconducting film undergoing its phase transition. If the
superconductor is nonuniform, or has a temperature gradient
due to self-heating, the resistance is determined by the size
of a normal region that grows with temperature. A voltage can
also be generated across a superconductor by magnetic flux
flow across the film. Depending on the physical parameters of
the film, this voltage generation can be characterized by the
nucleation of flux channels, called phase slip lines, over
which the superconducting order parameter slips in increments
of 2
.
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The energy resolution of a calorimeter is determined by the
ratio of the signal to the phonon noise and by the useful
bandwidth. The use of a thermometer with a higher
increases
the useful bandwidth by raising both the signal and the
phonon noise above the level of white noise, thereby
improving the energy resolution for a fixed heat capacity.
(Recall figure 3 above.)
In the small-signal limit, for the same heat capacity, TES calorimeters therefore significantly outperform semiconductor calorimeters in energy resolution. For x-ray applications, however, the heat capacity of a calorimeter is constrained by the onset of nonlinearity in the detector response to energetic photons.
In both TES and semiconductor thermistors, the sensitivity
depends nonlinearly on temperature. For TES calorimeters,
however, sensitivity falls abruptly once the normal state has
been reached. To keep the temperature excursion due to the
absorption of an x-ray photon from exceeding the dynamic
range of the detector, the heat capacity C of a TES calorimeter
must be increased by a factor proportional toalpha. Since, for
large values of , the resolution scales as
, the fundamental
limits on the energy resolution of a TES x-ray calorimeter
are similar to the original predictions made for
semiconductor calorimeters.14
In fact, the main advantage of a TES calorimeter lies in such practical issues as the design flexibility that a larger heat capacity budget allows. TES calorimeters can use materials with high specific heats, such as normal metals, that thermalize deposited energy quickly and efficiently, increasing the probability of actually achieving the predicted resolution.
The concept of optimal bias acquires an interesting twist when applied to a TES calorimeter. Clearly, for a particular superconductor, the bias temperature must lie within the superconducting transition. With the use of the proximity effect, however, it is possible to engineer a TES thermometer with a critical temperature Tc at any convenient temperature. To achieve this feat, a bilayer is made by combining a thin superconducting film that has a Tc much higher than is useful with a thin film of a normal metal to push that Tc down. The value of Tc is determined by the thicknesses of the two layers, the properties of the two films, and the resistance of the interface.
Given a practical refrigerator temperature and the ability to obtain
a particular value of at any Tc,
the choice of Tc for the highest energy resolution
is determined by the same bias optimization as for any resistive device.
In designing a TES calorimeter for an expected maximum incident
photon energy, we choose a value for the heat capacity C that
avoids large-signal nonlinearity, yet fully uses the linear
part of the dynamic range. It works out that, for a given
and a maximum photon energy Emax, we choose C
such that C is proportional to
Emax/Tc. This choice
counteracts the usual heat capacity penalty for operating at
a higher bias, and it results in an optimal bias temperature
that is higher than that for the semiconductor devices and in
much slower degradation in resolution with temperature above
that optimum. The much larger scale of the heat capacity
makes it practical to design a TES calorimeter that has a
lower heat capacity, but is biased at a higher temperature.
What is there to gain from designing a TES with a higher bias
temperature? Operating in such a way increases the magnitude
of the electrothermal feedback (already substantial because
of the large ) because more
bias power is required,
which increases the dynamic range of the fed-back reduction in
power. With negative ETF, the TES will self-regulate within
its transition, and increasing the bias temperature relative
to that of the heat sink improves the stability of the bias
point.
Increasing the ETF also makes the temperature recover faster, and it is useful to do that until the corner frequency of the thermal response matches the useful bandwidth of the device. The temperature can then be recovered in a time close to the limiting thermalization and diffusion times, thereby raising the maximum x-ray flux that can be spectroscopically analyzed before the onset of pileup. (Pileup is what happens when photons arrive at a detector faster than they can be individually read out.)
The fabrication of TES x-ray calorimeters by one of us (Irwin) and the National Institute of Standards and Technology group with energy resolution (FWHM) of 2.4 eV at 1.5 keV and 4.5 eV at 6 keV and at count rates higher than 400 counts per second is a significant advance for high-throughput x-ray spectroscopy. These results have been obtained with aluminum-silver and molybdenum-copper proximity-effect TES thermometers fabricated on silicon nitride membranes.
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