Introduction
Physics Today paper
Semiconductor thermistors
Doped semiconductor thermistors --- in particular, ones made from ion-implanted silicon and neutron transmutation doped (NTD) germanium --- were the first thermometers used in x-ray calorimeters. An advantage of using silicon is the existence of well-established micro-machining techniques for fabricating devices in which the active pixels and their links to the heat sink form a monolithic array. NTD germanium's chief advantages are the reproducibility and uniformity of its doping density.
In doped semiconductor thermistors, electricity is conducted by thermally activated charge carriers that hop between localized impurity states through a mechanism called variable range hopping (VRH).9 In VRH, the average hopping distance increases as the temperature is lowered. It does so because an electron tends to tunnel farther to a site requiring less change in energy than to a nearby site with a difference in energy that is large compared with that available from the phonon spectrum.
In doped crystalline semiconductors, which have a Coulomb gap
in their density of states, VRH produces the resistance law
![R=R<SUB>0</SUB>exp[Sqrt(T<SUB>0</SUB>/T)],](resistance_law.gif)
which results in a logarithmic sensitivity given by
. 
is a constant that depends sensitively on the doping density,
increasing from 0K at the metal insulator transition to about
120K at half the critical impurity concentration. 
depends on the resistor geometry and only weakly on doping density.
Calorimetrists originally hoped they could use semiconductor thermistors to build a soft x-ray detector with an energy resolution of better than 1eV (full width half maximum, FWHM). They based their expectations on being able to operate the calorimeter at temperatures below 0.1K and on early estimates that the calorimeter's heat capacity could be kept below 0.01 pJ/K. However, those estimates assumed that all noise sources, apart from phonon noise and Johnson noise, were negligible and that resistance was a function of temperature alone.
In fact, actual devices have proved to suffer from nonideal
effects that have invalidated these assumptions. For
instance, resistance decreases with increasing bias power ---
even at fixed temperature (nonohmic
behavior).10
And at low frequencies, excess 1/f noise is
present.11
These two effects limit
how small the thermistor can be made, and, hence, its
contribution to the heat capacity, since both effects worsen
as the thermistor's size is reduced. They also worsen with
decreasing doping density, which limits .
Given these effects, the limiting resolution has not yet been determined.
The resolution is not the only property affected; the nonohmic behavior also limits practical device speeds, since higher thermal conductance values require that more power be dissipated in the thermometer to reach the optimal bias temperature.
For a practical semiconductor calorimeter, the total heat capacity should be kept below 0.1 pJ/K, yet the detection area should be about 0.1 to 1 mm2. As a result, the selection of an absorber is highly constrained, as follows:
Although the search for the optimal absorber for use with semiconductor thermistors is still under way, the best performance so far has been obtained with absorbers made from mercury telluride and tin. Using NTD germanium thermistors and tin absorbers, Ettore Fiorini and his group at the University of Milan currently hold the resolution record for semiconductor calorimeters of 5 eV FWHM at 6 keV. (Fiorini and Tapio Niinikoski pioneered quantum calorimetry in Europe independently of, but at the same time as, Moseley in the US.) Eric Silver and his coworkers at the Harvard-Smithsonian Astrophysical Observatory are achieving comparable resolution with similar kinds of detectors.
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