Thermal calorimeters for high resolution X-ray spectroscopy

Nuclear Instruments and Methods in Physics Research A326 (1993) 157-165 North-Holland

NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A

Thermal calorimeters for high resolution X-ray spectroscopy D. McCammon, W. Cui, M. Juda, J . Morgenthaler and J. Zhang Physics Department, University of Wisconsin, Madison, WI 53706, USA

R.L. Kelley, S.S . Holt, G.M . Madejski, S .H. Moseley and A.E. Szymkowiak NASA / Goddard Space Flight Center, Greenbelt, MD 20771, USA

Thermal detection of individual X-ray photons by small (0 .5X0 .5 mm) calorimeters has been used to achieve an energy resolution as \good as 7 .5 eV FWHM for 6 keV X-rays. Such detectors should have interesting applications in X-ray astronomy as well as laboratory spectroscopy, and they promise a high tolerance for embedded sources. Ideally, it should be possible to improve the resolution greatly by making smaller detectors or operating them at lower temperatures than the 50-100 mK currently used. However, there appear to be fairly fundamental limitations when semiconductor thermistors are used as the thermometer . When trying to achieve energy resolution of 0.1% or better, fluctuations in the thermalization efficiency of the detector must also be considered, and this places additional restrictions on suitable detector materials.

1. Introduction Thermal calorimetry has a long history in nuclear physics, where it was used to measure the integrated energy of various radioactive decays, and the advantages of low temperature operation had been realized at least since 1935, when a small calorimeter was operated at 50 mK to increase its sensitivity [1]. However, the first suggestion that the heat from a single event could be detected appears to have been in 1976 [2], and the first quantitative calculations of the remarkable sensitivity that can be obtained (along with the first experimental efforts toward realizing this sensitivity) were published in 1984 [3-5]. Thermal detectors do not require any particular electron transport properties and thus offer the possibility of making particle detectors in large sizes from a wide variety of materials. This and their unique sensitivity to non-ionizing events make them attractive candidates for the detection of exotic particles and rare decays [3,6,7] . Such applications are treated by E. Fiorini elsewhere in this volume, and we will limit the following discussion to small calorimeters for high resolution spectroscopy . As shown below, these devices can in principle achieve resolutions up to two orders of magnitude better than conventional solid state detectors over much of the 0.1-50 keV energy range . While good energy resolution has obvious applications to laboratory spectroscopy [8,9], we use our own interests in astronomical X-ray observations as an ex-

ample. The energy range between 0.1 and 10 keV is rich in atomic lines of the astrophysically important elements, but grating and Bragg crystal instruments are too inefficient to observe any but the brightest sources in the sky, and the resolution of solid state detectors is not good enough for detailed spectroscopic studies. Fig. 1 shows the energy spectrum of the X-ray diffuse background between 0.1 and 1 keV. This was taken with a proportional counter (with a beryllium filter to help isolate photons below 0 .11 keV from those at higher energies), but over this energy range the resolution of solid state detectors is not much better . These X-rays are thought to be thermal emission from interstellar gas with temperatures around 1-3 X 10 6 K. Fig. 2 shows the results of a theoretical model of the emission from such gas, folded through the response of a spectrometer with 5 eV FWHM resolution . Fig. 2a assumes that there is only galactic thermal emission over this energy range, while fig. 2b includes an extragalactic component that provides the continuum seen between the lines above 0.5 keV. It can be seen that there is a wealth of information in such a spectrum : the thermal nature of the flux can be confirmed, relative abundances of elements and their various ionization states determined, temperature structure measured, and the intensity of the extragalactic component unambiguously separated from galactic thermal emission . Unfortunately, no such data exist, but we are working on a sounding rocket payload that will employ an array of small calorimeters to obtain a spectrum like this.

0168-9002/93/$06 .00 © 1993 - Elsevier Science Publishers B .V . All rights reserved

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2. Basic physics and performance considerations A calorimeter consists of an absorber to stop and contain the event to be measured, an attached thermometer to measure its temperature rise, and a weak thermal link connecting the absorber to a heat sink so that it can cool off again after an event has been registered. Random movement of energy carriers through the link produces random fluctuations in the energy content of the absorber, producing a background noise against which the increase in energy produced by an absorbed event must be measured . An elementary result of classical statistical mechanics gives the mean square magnitude of these fluctuations [10] : ~~E Z ) = k B T0 C,), where To is the temperature of the heat sink and Co is the heat capacity of the absorber at temperature To . (A crude calculation of Poisson fluctuations on the mean number of energy carriers in the absorber, assuming they have average energy k BTii , gives the same result .) Note that the magnitude of this "thermodynamic noise" is entirely independent of the thermal conductivity of the link, G, To get some idea of the scale of these fluctuations, a piece of silicon 0.5 mm square and 25 lam thick - a suitable size for an absorber in the focal plane array for an X-ray telescope - has a heat capacity of 4 x 10 - ' 5 J K- ' at 100 mK, giving 0 Erms - 0.2 cV . Since the theoretical resolution limit for a conventional solid state detector at 6 keV is about 110 eV FWHM, energy measurements approaching this fluctuation limit are an attractive proposition . To determine the ultimate energy resolution that can be achieved in the presence of these fluctuations, we look at their power spectrum, which does depend

on the conductance Gii of the thermal link . The spectrum is flat below a corner frequency wo equal to the reciprocal of the thermal time constant CoGO ', and falls as f- ' at higher frequencies . The signal pulse from an instantaneously deposited energy will have a sharp rise in temperature followed by an exponential decay with this same time constant . The power spectrum of this pulse has the same shape as that of the thermodynamic fluctuations, as shown in fig. 3. Considering only these quantities, the signal to noise ratio is independent of frequency, and any desired energy resolution could be obtained by using an arbitrarily large bandwidth. The usable bandwidth will be limited by other considerations, the most basic of which is thermometer noise. As a simple example, we take an ideal resistive thermometer, with resistance R and logarithmic sensitivity a --_ d log(R)/d log(T) . The maximum usable bandwidth is determined by r, the ratio of the fluctuation noise at low frequencies to the Johnson noise of the thermometer. It can be shown that r 2 is proportional to a 2b, where b = (T - TO)ITO is the fractional temperature rise produced by the power used to read out the thermometer resistance . The value of a is limited by the thermometer technology, but b clearly has an optimum value: a very small readout power will give a small signal voltage relative to the fixed Johnson noise voltage, while a large readout power will raise the temperature significantly and greatly increase the level of the thermodynamic fluctuations . The exact calculation is complicated by the temperature rise, b : the fluctuations must be recalculated for the case where there is a temperature gradient along the link, and electrothermal feedback caused by variations in bias power during the

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Energy (keV) Fig. 1. Proportional counter pulse height spectrum of the astrophysical soft X-ray diffuse background . The counter included a beryllium K-edge filter to help separate photons below 0 111 keV from those at higher energies.

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thermal link, and the temperature dependence of the heat capacity . This result is still independent of the thermal conductivity of the link and of the thermometer resistance . As shown in fig. 4, 6 is primarily a function of a that depends very weakly on (3 and y, which are respectively the power-law indices for the temperature dependence of G and C. Semiconductor thermistors operating in the hopping conductivity regime have practical a's in the range of 3-8 . Over this range, 6 drops from - 3 to - 1.5, and it scales as a -) at lower sensitivities, where the thermometer noise

signal pulse changes both signal and noise spectra. This problem has been solved exactly for the assumptions used so far [4,11,12], and the net detector noise at optimum bias power (b - 0.12 for most cases) and optimum signal filtering is given by C0)0 s . (2) DETInfi = ~( kT(,

The quantity in parentheses is just the rms magnitude of the energy fluctuations in the absorber when no bias power is supplied, and 6 is a dimensionless factor containing all information about the thermistor, the 600 (a)

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Fig 5 Layout of a 1 x 12 monolithic silicon microcalorimeter array. Each of the twelve elements is 1/4 mm wide by 1 mm long and 15 Wm thick . The supporting legs are - 15 Wm x 15 Wm and 2 .5 mm long .

D. McCammon et al. / High resolution X-ray spectroscopy

o, and as a -05 at higher values of a, where the improvement in resolution comes only from the increase in the usable bandwidth. To this approximation, then, the speed (thermal time constant) of the calorimeter is still arbitrary, and the thermometer resistance can be chosen according to the capabilities of the technology and the requirements of available amplifiers . A silicon junction field effect transistor (JFET) operating at - 130 K with a source impedance > 10 8 SZ has a noise temperature less than 5 mK, and therefore adds no significant noise to a detector operated at any higher temperature . Doped semiconductor thermistors can be fabricated readily in this resistance range, and the size can easily be made small enough that they contribute negliglbly to the heat capacity of the detector . At this point, it seems we have all the pieces necessary to maintain an energy resolution better than 1 eV, provided only that we can find a way of assembling the whole thing without greatly increasing its heat capacity. (This last point is not entirely trivial, since a drop of epoxy 25 wm in diameter has a larger heat capacity at 100 mK than the entire detector .) Other thermometer technologies could apparently provide even better resolution . A suitable supercon-

ductor biased to the center of its transition can have a sensitivity a > 1000 . Since the energy resolution goes like a -05 for large a, this should give at least ten times better resolution than a semiconductor thermometer, but there are several practical difficulties that have so far prevented the realization of such gains. One of these is simply that it is much harder to match an amplifier to this type of thermometer. A more fundamental problem is that the improvement in resolution can be realized only if the signal bandwidth is increased by the same factor as a has been increased . This quickly runs into the finite time required to thermalize the energy of an event, so large increases in thermometer sensitivity can be exploited only by making the detectors very slow . In principle, a resistive thermometer is a bad choice for reading out a calorimeter . First, power must be dissipated to read it out, raising the temperature of the detector, and the resistance contributes Johnson noise. A temperature-sensitive reactance has neither of these drawbacks, and it is easily shown that a real capacitive or inductive thermometer has an "effective a"= Q d log(Z)/d log(T) . With Q's possibly as high as 10 5, the potential gain in resolution is very large, and attempts are being made to develop both types of reactive readout [13,14]. A very large bandwidth relative to

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the thermal time constant is still required to realize this gain, however. 3. Current state of development Silicon has excellent mechanical and thermal properties, and many techniques for working with it have been developed for the integrated circuit industry . We have been fabricating monolithic calorimeters from a single piece of silicon, thus solving most of the assembly problem. A small part of the volume of the detector can be doped by ion implantation to form the thermometer. This doped material has a very high specific heat compared to pure silicon, but its volume can be made as small as desired. Fig. 5 shows one such design . The starting material is a standard silicon wafer with very light doping . Ion implants form the doped thermometers and the degenerate conductive tracks leading to them . The entire area of the array is thinned to about 15 win from the back side, then the thinned silicon is masked and etched through from the front side to form the individual absorbers, leaving the long thin silicon legs attached to the surrounding wafer as the mechanical supports and thermal links. The conductivity of these links is about 2 x 10 -11 WK - 1 at 100 mK, giving a thermal time constant of a few milliseconds . The detectors are biased through a large load resistor, and the voltage across the thermometer is measured with a JFET input amplifier. The inset in fig. 6 shows the voltage pulse produced by the absorption of a single 6 kcV X-ray from a 55 Fe source. These pulses are filtered and sent to a multichannel analyzer, giving the pulse height spectrum shown in this figure . The two peaks at the right are due to the Mn Ka and Kß X-rays produced by the source . Fig. 7 shows an expanded view of the Mn Ka peak, which is almost

Deconvolved FWHM = 7.3 ± 0.4 eV

Fig. 7. Pulse height spectrum of the Mn Ka line at 5.89 keV .

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Fig. 8. Resistance vs temperature behavior for an ion-im planted silicon thermistor . The straight line is the function R = R,) exp(d / T) ° 5, with d = 9.2 K. resolved into the Ku,-Ka, doublet. The deconvolved detector resolution is about 7.3 eV FWHM, while the total system noise (including thermodynamic fluctuations) is about 4.5 eV FWHM . The difference is due largely to nonlinearity of the detector, which in turn is due to the nonlinear R(T) function of the thermometer, and to a lesser extent to the increase of detector heat capacity with temperature following absorption of an X-ray. 4. Thermometer problems Conduction in doped silicon and germanium thermometers in the 50 mK to 4 K temperature range seems to be dominated by the variable range hopping mechanism with Coulomb gap [15] . As shown in fig. 8, the ion-implanted devices show the expected R = R, exp(d/T) o5 dependence over a wide range of temperature and resistance . Nuclear transmutation doped (NTD) germanium shows the same behavior, but most melt-doped germanium and silicon do not, and instead the resistance curves downward to a greater or lesser extent at low temperatures . This downward curvature is consistent with the existence of small-scale inhomogeneities in the doping density in the melt-doped material . The characteristic slope (4) 0 5 is a sensitive function of the doping density, and if parallel paths exist in the thermometer with different densities, the conduction will be more and more dominated by the more heavily doped paths as the temperature is decreased. The resistance values plotted in fig. 8 were all measured in the limit of very small dissipated power (< 10 -14 W at the lower temperatures). If a finite power is applied, the resistance curve will flatten out at low temperatures, eventually becoming almost independent of temperature (hardly a desirable condition

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for a thermometer) . It cannot be determined experimentally whether this is an electric field or a power density effect, but we have compared our observed behavior with the quantitative predictions of published nonlinearity models, and none of them fit the data over a wide range of temperatures or fields . Fig. 9 shows the power density required to significantly reduce the resistance (and temperature sensitivity) of doped semiconductor thermometers as a function of the low-power sensitivity for various temperatures . It can be seen that the allowable power dissipation drops rapidly with decreasing temperature, and is smaller for thermometers with higher sensitivity. This power handling capability is very similar for N- and P-type silicon and does not seem to be sensitive to the compensation (ratio of minority to majority impurity density) in the 5-50% range . NTD germanium comes out about 50 times lower than silicon in PIV under the same conditions, but it also has about 50 times lower impurity heat capacity per unit volume . The power-handling capabilities are therefore about the same for Si and Ge thermometers with similar sensitivities and heat capacities . Melt-doped material usually shows a much smaller power-handling capacity . This is again consistent with the presence of small-scale doping nonuniformities, since the fraction of the volume containing the most heavily doped paths will dissipate all the power at low temperatures . Since the optimum bias level for a detector requires enough power to raise the absorber temperature by about 12%, it is clear that there is a lower limit on the size of the thermometer that can be used, and therefore on its contribution to the total heat capacity . The ultimate consequence of this is a limit on the speed of the detector . If the conductance of the thermal link is increased to shorten the thermal time constant, it

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Frequency (Hz) Fig. 10 . Measured noise spectral density for a silicon thermistor thermally sunk to a 343 mK cold plate. Bias currents were 0, 0.2, 0.4, and 0.8 nA . The drop at high frequencies is due to the 10 pF stray capacitance . The arrow indicates the calculated Johnson noise level.

requires proportionately more bias power to produce the optimum temperature rise . The volume and heat capacity of the thermometer must be increased by the same factor to avoid a major loss in thermometer sensitivity. Eventually, the thermometer dominates the total heat capacity, G and C increase together, and the thermal time constant r - CG -1 reaches a limiting value. For operation at 100 mK, this lower limit on the time constant seems to be - 1 ms, and it gets longer at lower temperatures . Another effect that limits the performance of these thermometers is shown in fig. 10 . Noise measurements on a thermometer directly mounted on a heat sink show just the expected Johnson noise at zero bias, but have an additional 1/f component when bias is applied . This excess noise is not caused by the thermometer contacts ; it appears fundamental to the hopping conduction process, and scales inversely with the square root of the thermometer volume . NTD germanium shows very similar noise levels when the factor of 50 discussed above is taken into account. We do not yet have enough data to quantify how this noise scales under various conditions, but it does seem to get worse rapidly for higher thermometer sensitivity and lower temperatures . There has been some theoretical work done on excess noise in hopping conductivity [16,17], but it is not easy to make quantitative predictions from this work, nor is it obvious that it applies in our range of operating conditions . From our limited data, it appears that the fractional decrease in detector resolution due to this excess noise should become smaller as the size and heat capacity of the thermometer are increased, reaching some limiting value once the thermometer heat capacity dominates the total. IV . THERMAL

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5. Absorber problems The only functions required of the absorber are to contain the event and to turn its energy into heat, but the latter is not so simple a process as it sounds, and for high resolution detectors with E/DE = 1000 or more the material selection is much more constrained than one might have imagined . Problems can be divided into two categories : (a) conversion of part of the energy into phonons is delayed for times long compared to the thermal integration time of the detector, and (b) the energy distribution of the phonons remains non-thermal for times longer than the detector time constant. As an example of delayed conversion, about 30% of the energy of an X-ray absorbed in silicon will go initially into producing electron-hole pairs. In the absence of a collecting field, these quickly become localized on traps at impurity or lattice imperfection sites. Since the temperature is too low to excite them out of these traps or to produce any mobile carriers to recombine with them, the lifetimes for most of the trapped carriers will be very long. Statistical fluctuations in the amount of trapped charge from one event to the next will limit the resolution to only slightly better than that of a conventional silicon detector, and positional variations will make things even worse. Insulators behave like semiconductors with wide band gaps . Normal metals seem free of metastable states, but their electronic heat capacity decreases only linearly with T and is very high relative to that of a dielectric at low temperatures . Most of our work has been with narrow-gap semiconductors and semimetals, where the amount of energy tied up by trapped carriers can be neglected, and with superconductors, whose conduction electrons are almost entirely paired at temperatures at least a factor of 8 below the transition temperature, and therefore do not contribute significantly to the heat capacity . Superconductors have their own metastable state problem, however. About half the X-ray energy initially goes into breaking up Cooper pairs, and some means must be found to ensure that the resulting quasiparticles free electrons) recombine sufficiently quickly. The phonon spectral problems exist because the major processes that degrade the high energy phonons produced by an absorbed X-ray freeze out long before equilibrium with the lattice temperature is reached. Highly nonthermal phonon energy distributions can therefore persist for a long time . The situation is complicated, because we do not understand the effects of the phonon spectrum on the resistance of the thermometer. The thermometer may also play an important part in degrading the phonon energy ; otherwise, surface interactions seem to be the only significant process. We have observed that when the absorber is acous-

tically well-coupled to the thermometer, there is an initial response to an absorbed X-ray that decays with a characteristic time much shorter than the thermal time constant of the detector . The amplitude of this initial response is highly variable from pulse to pulse, and is larger for events that occur closer to the thermometer. This behavior is consistent with a direct interaction of high energy phonons with the the thermometer, where the probability of having these phonons reach the thermometer before they are degraded in energy is higher for the closer events . We expect that there would be significant statistical fluctuations in this part of the signal even if the geometry were held constant, due the the relatively small number of high energy phonons. More experimental work is needed to find out what is actually going on here, and to determine the optimum design strategy for the detector as well as the effect that phonon energy spectrum fluctuations might ultimately have on energy resolution . 6. Conclusions Current results for small cryogenic calorimeters show that it is possible to get a factor of 20 or more improvement in resolution over conventional solid state detectors in the soft X-ray range, an increase that already offers revolutionary benefits to some areas of spectroscopy . The basic theory of these devices indicates that it should be possible to make detectors that are even better and faster, and that work at higher energies . However, significant improvements in resolution, particularly at higher energies, will require better understanding of the physics of the phonon system and its interaction with the thermometer. Much work needs to be done on the selection of materials and structures for the absorber that minimize the formation of metastable energy trapping states while maintaining low specific heats. This is particularly true for high energy detectors, where more massive absorbers are needed to contain the event, and where statistical fluctuations in any trapped energy represent a larger absolute effect on the resolution . Finally, significant improvements in speed from the current 1-10 ms range will probably require a different thermometer technology. References [1] F Simon. Nature 135 (1935) 673. [2] T.O . Nimikoski, Liquid and Solid Helium (Halsted, New York, 1975) p. 145. [3] E. Fiorini and T.O . Nunikoski, Nucl . Instr. and Meth . 224 (1984) 83 .

D McCammon et al. / High resolution X-ray spectroscopy [4] S .H . Moseley, J .C. Mather and D . McCammon, J . Appl . Phys . 56 (1984) 1257 . [5] D . McCammon, S .H . Moseley, J .C. Mather and R .F. Mushotzky, J . Appl . Phys. 56 (1984) 1263 . [6] B . Sadoulet, IEEE Trans . Nucl . Sci . NS-36 (1988) 543. [7] E . Fiorim, Physica B167 (1991) 388 . [8] D . McCammon, M . Juda, D .D . Reeder, R .L . Kelley, S .H . Moseley and A .E Szymkowiak, Neutrino Mass and Low Energy Weak Interactions : Telemark, 1984, eds. V. Barger and D . Cline (World Scientific, Singapore, 1985) p 329 .

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[9] C .K. Stahle, Ph .D . Thesis, Stanford University (1991) . [10] F . Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965) pp. 213, 242 . [ll] J .C . Mather, Appl . Opt. 21 (1982) 1125 . [12] J .C . Mather, Appl . Opt. 23 (1984) 584 . [13] D .G . McDonald, Appl . Phys . Lett . 50 (1987) 775 . [14] E.H . Silver et al ., Nucl . Instr. and Meth . A277 (1989) 657 . [15] A .L. Efros and V .I . Shklovskii, J . Phys . C8 (1975) L49. [16] B .I . Shklovskii, Solid State Commun . 33 (1980) 273 . [17] Sh .M . Kogan and B .I . Shklovskil, Sov . Phys . Semicond . 15 (1981) 605 .

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