Physical Principles of Low Temperature Detectors:

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1 Physical Principles of Low Temperature Detectors: Ultimate Performance Limits and Current Detector Capabilities C. Enss 1 and D. McCammon 2 1 Universität Heidelberg, Kirchhoff-Institut für Physik, INF 227, D Heidelberg, Germany 2 Physics Department, University of Wisconsin, 1150 University Ave., Madison, WI 53706, USA The rapid development of low temperature thermal detectors since the early1980s has resulted in remarkable improvements in the sensitivity and precision of many types of measurements. We will discuss the operating principles of these detectors in the most general possible terms. We will try to show how the physics of some popular thermometer systems introduces performance limits, and how the different figures of merit and optimizations that result affect various applications. PACS numbers: 07.85, Mc, V, Bx 1. INTRODUCTION Thermal detection of radiation, or more general the calorimetric detection of energetic particles, has a long history in physics. It is based on the first law of thermodynamics. For example, more than a century ago in 1880 Langley developed a resistive bolometer to investigate the infrared radiation from the sun. 1 Some years later in 1903 Curie and Laborde were the first ones to detect radioactive particles via their heat production. 2 The idea of low temperature detectors was born in 1935 when Simon suggested that the sensitivity of thermal detectors could be enhanced considerably by operating them at cryogenic temperatures. 3 A few years later this idea was taken up and a superconducting calorimeter was constructed capable of detecting individual alpha particles. 4 However, for most of the last century the calorimetric detection of particles remained sporadic. This changed considerably in 1984 when novel low temperature particle detectors were proposed for several important applications in nuclear physics, astronomy, and astrophysics. 5,6,7 From that point on there has been a constantly growing interest in the development and use of cryogenic particle detectors which has led to a large variety of detector types and an enormous range of applications. Among the most prominent applications today we note high resolution X-ray astronomy, material analysis, neutrino physics, Lamb shift experiments, nuclear waste determination and the search for dark matter in form of WIMPs. This is far from a complete list, but indicates how broad the spectrum is. The driving force for the development of cryogenic particle

2 C. Enss and D. McCammon detectors is applications that cannot be performed otherwise. As we will see, low temperature particle detectors have unique advantages over other types of detectors. In this article we will give first a brief overview of the general concepts of the detectors used today. After that we will restrict our discussion to calorimetric detectors such as semiconductor thermistors, superconducting transition edge sensors, and metallic magnetic calorimeters. 2. TYPES OF DETECTORS As pointed out before, many different types of detectors are being developed today. We can distinguish roughly between non-equilibrium and equilibrium detectors. The prototype of the latter is a classical calorimeter that consists of an absorber suited for the particle to detect which is weakly thermally coupled to a heat bath. The temperature of the absorber is monitored by a sensitive thermometer. The temperature rise δt resulting from the absorption of a particle with energy E is given by δt = E /C tot, where C tot denotes the total heat capacity of absorber and thermometer. For a true equilibrium detector the thermalization among all possible degrees of freedom in the absorber/thermometer systems is required, which means thermal time constant should be long compared to the internal equilibration time. For many real devices the contribution of several different degrees of freedom and their interaction among each other might present a very complex an interesting solid stated physics problem, which makes it hard to understand the detector response. The three most popular thermometers used today for such near equilibrium calorimeters are highly doped semiconductors, superconductors operated at the superconducting to normal transition, and metallic paramagnets. We will discuss the physics of those thermometers in more detail in later chapters of this article. Non-equilibrium detectors include kinetic inductance detectors (KIDs), 8 superconducting tunnel junctions (STJs) 9 and superheated superconducting granules 10. Both KIDs and STJs are based on Cooper pair breaking and the production of quasiparticles (~free electrons) in a superconducting material. While in case of KIDs the enhancement of quasiparticles is determined by the change of the kinetic inductance of the material, the transport of quasiparticles is required for STJs, because the tunneling current is used as the signal. To read out a KID the superconducting absorber material is part of a tank circuit and the frequency shift is used as a measure for the change in the kinetic inductance. Because of this feature KIDs are well suited for frequency domain multiplexing. Detectors based on superheated superconducting granules have been developed primarily for investigations of solar neutrinos. The basic idea is that small superconducting grains are heated slightly above the equilibrium superconducting transition temperature but remain in the superconducting state. An incoming particle of sufficient energy will force a transition into the normal state and this will be observed by the change in the magnetic

3 Physical Principles of Low Temperature Detectors susceptibility. Although the basic idea seemed simple, the realization of such detectors turned out to be very challenging, and today this detector principle is not much in use. Other types of detectors are based on the unique properties of liquid helium. For both superfluid 3 He and superfluid 4 He, detection schemes have been worked out and demonstrated. We will not discuss those here in detail but refer to references 11 and IDEAL THERMAL CALORIMETRY To see what unique characteristics of thermal detectors are important, we can look at the limitations of detectors based on the collection of charge and light. We will see that thermal calorimetry offers several advantages that may be important for different applications, but the following discussion is biased toward the potential for much improved energy resolution. Other properties and applications to power detection (bolometers) will be pointed out as they come up. The fundamental limitation on energy resolution in a charge detector such as a high-purity germanium diode is that only part of the incident energy goes into ionization. There are fluctuations in the division of this energy between the detection channel, which is free electron-hole pairs, and other channels, mainly lattice excitations called phonons. As shown in figure 1a, this is a non-equilibrium detector and the energy in the detection channel must be measured quickly before it decays further, mostly into additional phonons. This places stringent requirements on the material of which the detector is made: it must have excellent charge transport properties to allow the free charge that is produced to be collected before it decays. Currently only very pure silicon and germanium have sufficiently good charge transport to reach the limit of statistical fluctuations in the branching ratio, while a few other materials come close for small devices. Figure 1b shows the importance of equilibrium. The small system (thermometer) contains only a small fraction β of the total system energy, so relative statistical fluctuations in the estimate of total system energy N are large: 1/!N. But these are reduced by the square root of the number of independent estimates obtained during the measurement, or a factor of!! meas., so for sufficiently good thermometer coupling (small τ), the total energy can be measured with arbitrary accuracy. Phonon-mediated detectors can also be non-equilibrium. For example, the dark matter detectors for the CDMS experiment use phonon sensors on the surface of large Germanium and Silicon disks. These have superconducting collectors that are sensitive only to phonons with E >> kt. They therefore have similar statistical limitations to ionization detectors (but with much larger N, they are effectively limited instead by considerations such as positional uniformity of collection efficiency).

4 C. Enss and D. McCammon Fig. 1. a) Energy channels in a silicon solid state detector. Statistical fluctuations in the initial ionization-phonon branching ratio fundamentally limit the energy resolution of this type of detector, even if the energy in the charge channel is measured perfectly before any of it decays. For silicon this limit is about 118 ev FWHM at 6 kev. b) Location channels. The thermometer contains only a small fraction of the system energy. But in equilibrium, a short τ allows many independent samples and an accurate estimate of the total system energy. The random transport of thermal energy over the link to the heat sink as shown in figure 2 produces fluctuations in the energy content of the detector that can be thought of as N fluctuations in the number of energy carriers, where each has an energy ~kt. These provide a fundamental background noise against which the energy increase due to the signal must be measured, but as shown in figure 3 do not by themselves limit the accuracy. This can be seen more quantitatively in figure 4, which shows the power spectrum of an exponential pulse and the power density spectrum of the thermodynamic fluctuation noise (TFN). Both of these are the same shape with a single-pole roll off at f c =G/(2πC), so the signal-to-noise ratio is the same in all frequency bins of equal width. Each bin provides an estimate of the signal amplitude, and as discussed below, under rather general assumptions the noise in different bins is uncorrelated. This gives the result shown in figure 4, where the r.m.s. uncertainty in the event energy, ΔE, goes to zero as the measurement bandwidth Δf is made arbitrarily large. This also shows that the scale of the TFN is important, but not the whole story. All good things must come to an end, of course, and figure 4b shows the most usual fates of this optimistic infinite bandwidth scenario. Either there is an additional source of noise (often from the thermometer) that becomes larger than the TFN at some point, or there is something that limits the risetime of the signal, introducing another pole that makes the signal spectrum fall off faster than the TFN. In either case, the signal to noise ratio degrades rapidly above this point and limits the useful bandwidth to some

5 Physical Principles of Low Temperature Detectors Random transport of energy between heat sink and detector over thermal link G produces fluctuations in the energy content of C. The magnitude of these can easily be calculated from the fundamental assumption and definitions of statistical mechanics:!e rms = kt 2 C Fig. 2. Fundamental thermodynamic fluctuation noise. Fig. 3. Signal can still be measured to high accuracy in presence of thermodynamic fluctuation noise by looking at the corners. Here the signal amplitude is equal to the r.m.s. value of the fluctuations. effective f max that can be substituted for Δf. Since the resolution depends only on the ratio f c/ /f max, something can be gained by making the detector as slow as the application can withstand. Detector design usually ends up a complicated tradeoff of internal time constants, thermometer noise, heat capacity, and operating temperature. The problem of how to filter the signal to get the best possible estimate of the its amplitude in the presence of an arbitrary noise spectrum can be solved quite generally for the linear small-signal case if we assume that the signal pulse shape and the noise power spectrum are both fixed and known. The noise in different time bins is correlated, except for the special case of white noise, but is uncorrelated in frequency space so long as it is "stationary," which means that its statistical properties do not change with the signal. This minimal requirement should be met by almost any linear system. We can then readily construct the optimal filter in the frequency domain, given that in the i th frequency bin the expected noise is n i and the

6 C. Enss and D. McCammon a) b) #!E = 2"f c& % ( $!f ' 1/2 k T 2 C Fig. 4. a) Constant ratio of signal to TFN allows signal amplitude to be measured with arbitrarily high accuracy as the bandwidth Δf goes to infinity. b) Usual limitation is that either an additional noise source or an additional pole in the signal response causes the signal to total noise ratio to drop above some frequency. signal amplitude is s i. We need to choose a set of bin weights w i to maximize the signal to noise ratio, given that! " E = " w i s and!e $ i rms = &# w i n i % i =1 i =1 ( ) 2 ' ) ( 1 2. (1) Taking the ratio of these and maximizing it by differentiating with respect to any w i gives w i = s i n i 2 (within a common constant). (2) The s i are the Fourier transform of S(t) and are in general complex. The noise in ΔE rms has random phase, so we are free to choose the phase of the w i to maximize the signal sum. This is done by taking w i = s ˆ i n 2 i, which makes all the terms real and gives filtered pulse that is symmetrical and peaks at t = 0 in the time domain. The derivation of this Wiener filter is discussed in more detail in reference 13. Changes in heat capacity, thermal conductivities, and thermometer sensitivity with temperature all tend to make these devices very non-linear, so the small signal requirement for application of this linear theory may be considerably smaller than the signals encountered in some applications. This can introduce a host of complications and new considerations. The optimal filtering issue for this case has been investigated with some generality by Fixsen et al. 14,15

7 Physical Principles of Low Temperature Detectors 4. DETECTORS WITH IDEAL MAGNETIC THERMOMETERS In this section we will briefly discuss the question of what fundamentally limits the performance of a magnetic calorimeter. We will consider it as an ideal calorimeter in the sense that the amplifier noise and the magnetic Johnson noise are neglected. These will be present in any real device, but the magnitude of their contribution to the noise is not fundamental and depends on experimental details such as the geometry of the calorimeter setup. Although it might be hard in some configurations to find a setup in which these non-fundamental noises sources are negligible, in principle no limit results from them. The fundamental limitation of an ideal magnetic calorimeter originates from the finite coupling between absorber and thermometer, which limits the usable bandwidth. The fluctuations in the energy content of the calorimeter are minimized when the heat capacity of the absorber and the thermometer are equal. We note that this is also the condition for maximum signal, thus minimizing effects of amplifier noise. In this limit the rms value of the fluctuations of the energy content is given by &( 0 rms 2! # = kt C $ % ( 1 " ' E, (3) where! 0 and! 1 are the time constants for the coupling of absorber to thermometer and absorber to heat bath, respectively, and C is the absorber heat capacity DETECTORS WITH IDEAL THERMISTORS A thermistor is any device whose resistance is a function of temperature. Using one as the thermometer for a sensitive calorimeter or bolometer might seem like a really bad idea: the bias power required to make resistance changes visible raises the temperature of the detector and increases the TFN, and since the resistor is a dissipative element it introduces unavoidable Johnson noise. However, this category includes the commonly-used doped semiconductor thermometers as well as the very promising superconducting transition edge sensors (TESs), both saved by high sensitivity, convenience of use, and the availability of amplifiers with electrical noise temperatures well below usual detector temperatures. The Johnson noise produces a flat noise term that must be added to the TFN as shown in figure 4a, and the useful bandwidth f max is closely related to the frequency where they intersect. However, the TFN is fixed in temperature units, while the Johnson noise is fixed in voltage or current. This introduces the concept of an optimum bias: if the readout current is very small, the resistance fluctuations due to TFN produce very small voltages and are completely dominated by the fixed Johnson noise, while a high bias will increase the temperature and therefore the magnitude of the

8 C. Enss and D. McCammon Fig. 5. Energy resolution in units of the thermodynamic fluctuation noise (TFN) as a function of temperature rise due to bias power. Alpha is the logarithmic temperature sensitivity, and γ refers to the temperature sensitivity of the total heat capacity: C = C 0 ( T T 0 )!. Optimum bias power results in a 10 20% temperature rise for γ = 1 3. TFN much faster than it reduces the effective Johnson noise. High thermometer sensitivity improves things for free, with f max directly proportional to the logarithmic sensitivity! " d log R d logt. The optimum bias depends somewhat on how fast the detector heat capacity increases with temperature, but when expressed as a fractional temperature rise of the detector above the bath temperature is largely independent of everything else, as shown in figure 5. For α not too small, the optimum bias always results in a 10 20% temperature rise and resolution!e RMS " 4 # k B T 0 2 C 0, (4) where C 0 is the heat capacity at the heatsink temperature, T 0. Since the readout bias power is dissipated in the detector, it becomes part of the input signal power. Changes in the output signal (current or voltage) therefore produce changes in the total input signal unless the bias circuit resistance is just equal to the thermistor resistance. This electrothermal feedback can be positive or negative (having the same or opposite sign as the input signal that produced it) depending on the relative magnitudes of the thermistor and bias circuit resistances and the sign of dr/dt. It has the largest effect at low frequencies, and none in the limit of high frequencies where the thermal time constant allows no temperature change. Feedback in a linear system can markedly change the frequency

9 Physical Principles of Low Temperature Detectors Frequency-domain filtered pulse:!! s ˆ " i 2 n s i = s i i " i =1 i =1 n i 2 2 Fig. 6. The optimally filtered pulse in frequency space is just the square of the signal to noise ratio. Increasing the thermometer sensitivity increases the s/n ratio at high frequencies and makes the filtered pulse faster. Turning on a strong negative electrothermal feedback suppresses low frequencies at the detector, but has no effect on the s/n ratio nor the output of the optimal filter. response, but it has no effect on the signal to noise ratio for any noise source within or in front of the feedback loop. The output of an optimal filter is therefore unaffected by feedback, as shown in figure 6. However, as pointed out by K. Irwin 17, negative electrothermal feedback is of great importance for stabilizing high-gain detectors and increasing the count rate capabilities of non-linear systems, while positive feedback has been used to improve s/n ratios when limited by noise of the following amplifier. The very small bias resistor values needed for negative feedback with positive temperature coefficient thermistors require the use of current readout, while voltage readout is advantageous with the large resistors used with negative coefficient thermistors. Equations for current output can conveniently be converted to voltage and vice versa using the circuit dual theorems that arise from the symmetry of Kirchoff s laws. Swapping voltage with current, resistance with conductance, and series inductance with parallel capacitance results in another set of correct equations for the altered circuit. This is explained in more detail in ref. 13, or in any linear circuits textbook. Measurement of the complex admittance or impedance of a detector can provide much useful information on its time constants. 18,19 For a simple detector with a positive (negative) temperature coefficient thermistor, the low-frequency admittance di/dv (impedance dv/di) will be much reduced from the total conductance I/V (resistance V/I). As the frequency is increased, the admittance (impedance) increases and lags in phase, tracing a semicircle in the complex plane and becoming entirely real and equal to the total conductance (resistance) in the high frequency limit. With more complicated internal structure, the behavior is more complex, but can be modeled and fit to determine internal time constants, as shown in figure 7. Current (voltage) dependence of the conductance (resistance) is directly revealed by a high-frequency admittance (impedance) that is less than the total conductance (resistance).

10 C. Enss and D. McCammon Fig. 7. Complex impedance or admittance measurements can be fit by models to determine several time constants in a multipart thermal circuit. 6. COMPARISON OF THE RESULTS OF IDEAL RESISTIVE AND MAGNETIC CALORIMETERS If we assume that we have devices with the same heat capacity operated at the same temperature, the ratio of the resolution of ideal magnetic and resistive calorimeters is given by # E # E mag resistive = " 1 (!! ) (5) Practical devices using current materials and technology will generally have alphas between 10 and 100 or a ratio of rise time to fall time in the range 10 4 to This makes (5) of order unity, which means there should not be a big difference in the ultimate resolution one can reach with either technique for a given heat capacity and operational temperature. For magnetic calorimeters there should be the possibility to go to extremely long fall times to obtain ultrahigh resolution. However, this would be suitable only for applications for which very low count rates are tolerable. 7. METALIC MAGNETIC CALORIMETER PHYSICS In case of a magnetic calorimeter the temperature information is obtained from the change of magnetization of a paramagnetic sensor, which is located in a small magnetic field. 20 In principle there is a wide range of paramagnetic materials to choose from. In practice, however, for most applications there are only a few reasonable candidates. Dielectric

11 Physical Principles of Low Temperature Detectors paramagnets are not suitable for most applications because they are inherently slow due to the very long spin lattice relaxation times at low temperatures. Magnetic moments in metals respond orders of magnitude faster due to their coupling to conduction electrons. The downside of using metallic paramagnets is that the electrons increase that heat capacity of the sensor and enhance the interaction between the localized magnetic moments. The latter effect can be minimized by choosing rare earth ions as magnetic impurities, because their magnetic moment is associated with partially filled 4f-shells, which are located deep inside the outer 5s and 6p shells of these ions. In this way the interaction of the magnetic moments with conduction electrons is considerably smaller than in case of elements of the transition metal series. Among the rare earth ions erbium appears to be a particularly good choice, because of its small de Gennes factor. 21 Since most of the experimental work on MMCs has been carried out with sensors of erbium ions in a gold host, we will restrict our discussion of the physics of MMCs to this alloy. The change of the magnetization! M of a Au:Er sensors located in a small magnetic field caused by the absorption of energy δe is given by! M " M! E =, (6) " T where C tot denotes the total heat capacity of the calorimeter. To predict the performance and optimize a MMC we need to calculate the total heat capacity and the temperature dependence of the magnetization. In the crystal field of the Au host Er 3+ can be approximated as two-level system with effective spin ½ having an isotropic g factor of g = 34/5. Before we discuss the contribution of the localized magnetic moments of erbium to the total heat capacity and magnetization we will briefly mention two contributions arising from nuclear magnetic moments. The isotope 167 Er carries a nuclear spin of I = 7/2 and influences the magnetization and the heat capacity by its hyperfine interaction with the 4f electrons. Since the magnetization is reduced and the heat capacity enhanced by this effect in the typical temperature range in which MMCs are operated, their performance is degraded by the presence of 167 Er. The natural abundance of this isotope is 23%. By using isotopically enriched erbium 166 Er or 168 Er, this unwanted effect can be avoided. The other contribution originates from the nuclei of the host material. The 198 Au nuclei have spin I =3/2. The distortion of the fcc lattices by the erbium impurities leads to large electric field gradients and a splitting of the nuclear levels due to the quadrupolar interaction. This results in an additional heat capacity. This contribution can only be avoided by going to other host materials. In order to calculate the contributions of the localized magnetic moments of the Er ions to the magnetization and total heat capacity one has to take the interactions among them into account. The interactions consist of C tot

12 C. Enss and D. McCammon Fig. 8. left: Magnetization of Au:Er with 300 ppm enriched 166 Er as a function of 1/T at different magnetic fields. right: Specific heat of Au:Er with 300 ppm enriched 166 Er as a function of temperature at different magnetic fields. The solid lines are calculated numerically taking into account the RKKY interaction. two mechanisms: the magnetic dipole interaction and the RKKY interaction, which is the indirect exchange interaction via conduction electrons. Numerical methods have been developed to calculate these properties. Figure 8 shows the magnetization and heat capacity of a Au:Er sensor with 300 ppm of erbium in different magnetic fields as a function of temperature. Although the model has only one adjustable parameter (the magnitude of the RKKY interaction) for the whole set of curves, the agreement between calculation and experimental result is excellent. This clearly shows that an optimization of MMCs based on this model should be possible. Indeed, explicit expressions have been derived to optimize the signal of different kinds of MMCs in terms of magnetic field and the detector geometries. 22 In terms of detector design two principally different approaches have been developed. Initially MMCs consisted of a cylindrical sensor which was either placed directly in a SQUID or read out via a flux transformer. In such designs the sensor is located in a homogeneous external magnetic field. More recently it was realized that for many applications a different configuration is favourable. 20,23 Here the sensor material is deposited as a film on top of a meander-shaped superconducting loop in which a large persistent current is injected. This current produces a strongly inhomogeneous field within the volume of the sensor. The deposition of energy in the sensor results in a decreasing magnetic susceptibility and in turn in a change of the inductance of the loop. The meander-shaped loop can be part of a flux transformer to a dc SQUID or the SQUID loop itself can be shaped like a meander. There are several advantages for meander type

13 Physical Principles of Low Temperature Detectors Fig. 9. Examples of possible coupling schemes used for MMC detectors. In (a) to (c) cylindrical sensors are used located in a homogenous external field B. In (d) and (f) a persistent current in a meander type superconducting loop generates an inhomogeneous magnetic field in the volume of the sensor. For more details see reference 20. MMCs. Large filling factors in planar array structures and large area detectors can be designed without a reduction of signal to noise ratio, and since the field is a multipole field of higher order, magnetic cross talk is reduced in neighbouring pixels. Figure 9 shows a several examples of possible MMC coupling schemes. 8. DOPED SEMICONDUCTOR PHYSICS Doped germanium thermometers were first developed by Frank Low as sensitive infrared bolometers. 24 These and the ion-implanted silicon and nuclear transmutation doped germanium thermistors in widespread use today all operate in the variable-range hopping with coulomb gap conduction regime, where the conduction electrons are strongly localized by the random

14 C. Enss and D. McCammon Fig. 10. Phenomenological evidence for a hot electron effect in variable range hopping conductivity in doped silicon. Resistance is assumed to be a function of electron temperature alone. T S is the lattice temperature. disorder of the donor/acceptor locations. This results in a temperature dependence predicted to be R = R 0 exp( T 0 T ) 1 2 where T 0 is controlled by the doping density. This is observed to hold over several decades of resistance, although systematic deviations are generally observed for T 0 /T > ,26 Hopping conduction is expected to be somewhat voltage-dependent, and the effects of this are observed but are usually small over the range of parameter space of interest for low temperature detectors. A much larger and more important effect is observed that has all the characteristics that one would expect of a hot electron system where the bias power is dissipated only in the electrons, which attain a temperature higher than the lattice phonon temperature as shown in figure ,28 Such behavior is expected and observed in metals, but is not consistent with the standard theory of strongly localized electrons. The power dependence of the resistance might accidentally fit the simple power-law thermal conductivity that is derived, but further evidence for hot electrons is shown in figure 11. Here the characteristic time determined from complex impedance measurements is combined with the G derived from the R(P) curves to determine an apparent heat capacity of the electron system. This value is just equal to the separately measured heat capacity of the doped semiconductor system. 26 An excess noise is also observed that over a wide range of temperature and bias is equal to the thermodynamic fluctuations that would be expected between the electron and phonon systems in this model. A good physical theory is lacking, but this simple model appears to correctly predict everything that is relevant to detector performance. It is therefore possible to optimize the thermometer design for a particular application and correctly predict the performance that can be obtained.

15 Physical Principles of Low Temperature Detectors Fig. 11. Further evidence for hot electrons in VRH. left: Panel A shows the apparent G derived from D.C. measurements. Panel B is the time constant deduced from A.C. impedance measurements. Both range over more than two decades. Their product is shown by the open circles in panel C. This can be interpreted as the heat capacity of the electron system, and it agrees within 10% with an independent measurement of the excess heat capacity of identically doped silicon, as shown by the filled circles. right: Measured excess noise agrees well with calculated noise due to TFN between electron and phonon systems over a wide range of temperature and bias power. 9. TES PHYSICS Although this type of thermometer is today the most widely used for cryogenic particle detectors, the complex physics associated with operating these devices at the superconducting phase transition is not very well understood. It is impossible to give an account of all important physics of TES sensors within the framework of this article. We therefore will restrict our discussion to some general aspects and refer the reader for further details to a recent review. 29 For most modern TES detectors low temperature superconductors are used. The superconducting state of those can be described by the BCS theory on a microscopic basis. Near the superconducting transition a phenomenological model the Ginzburg-Landau theory is well suited to describe many aspects of the physics of superconductors. Generally we have to distinguish between superconductors of type I and type II. Whether a film is type I or type II depends on the Ginzburg-Landau parameter κ which is the ratio of the London penetration depth λ L and the coherence length ξ. Practical TES films are usually rather thin and produced by vapor deposition or sputtering and the mean free path of the electrons is only of the order of a few tens or hundreds of nanometers. Since the coherence lengths are typically larger than 1 µm these films are in the dirty limit. This means that

16 C. Enss and D. McCammon Fig. 12 Different designs of TES detectors for different applications. for calculating κ the coherence length is replaced by the mean free path of the electrons. Transition edge sensors with T c below 1 K can be either type I or type II. The physics of the transition, the noise, the sensitivity to magnetic fields, and the magnitude of the critical currents are greatly influenced by the film being type I or type II. TES detectors for X-ray detection are mostly type I superconductors, whereas TES for detection of optical photons are often type II. For practical devices the proximity effect is very important since it allows one to engineer to some extent the transition temperature T c. Proximity multilayers are superconductors made of bilayers of a superconducting and a normal metal which are thinner than the coherence length in the superconducting film. Depending on the thicknesses of the layers, these composite superconductors exhibit a reduced T c but maintain a sharp transition. The proximity effect occurs because the number density of Cooper pairs does not drop sharply to zero normal to the superconductor interface. Another successful method of tuning T c is doping the superconducting material with magnetic impurities. Depending on the application, several different types of TES detectors have been developed. Figure 12 shows three typical designs. In (a) the particles are directly absorbed in the TES film. Typical applications are detectors for optical photons. In case of (b) the TES film is deposited on a Si 3 N 4 membrane and an absorber in electrical contact with the TES film is attached. Such devices are typically used for X-ray detection. Some detectors use a design in which the incoming particles are scattered in large mass absorbers and the resulting high frequency phonons generate quasiparticles in a superconducting film which diffuse to the TES film (c). This scheme is for example applied in some dark matter detectors. One other important aspect of TES detectors is the use of negative electrothermal feedback. 17 For a TES sensor under voltage-bias conditions, the Joule power decreases with increasing temperature and the electrothermal feedback is negative. There are several advantages of

17 Physical Principles of Low Temperature Detectors Simon s wish TES 1.8 ev FWHM Si Thermistor 3.2 ev MMC 2.7 ev 240x240x6.7µm3 410x410x8µm3 180x180x5µm3 Bi/Au HgTe Au Au Fig. 13. Some of the best results for the Mn Kα line at 5.89 kev (l-r: refs 30, 31 and 32). The TES results show that Simon Bandler s wish at LTD-11 in Tokyo paid off. For comparison, this result is shown in the upper right inset in the line profile from a very good (almost theoretically perfect) HP Ge detector. The linewidth for the thermal detectors is dominated by radiation damping of the Mn lines (dashed). negative electrothermal feedback. The device can be self-biased at a temperature within its transition. The feedback speeds up the pulse fall time and increases the useful count rate. Moreover, it increases the stability of the TES and the TES detector becomes self-calibrating. 10. SAMPLE OF CURRENT PERFORMANCE We end with the medley of recent nice results and applications shown in figures 13 16, apologizing in advance for many equally interesting ones we haven t included. Descriptions and references are given in the figure captions. REFERENCES 1. S. P. Langley, The Bolometer and Radiant Energy Proceedings of the American Academy of Arts and Sciences XVI (1881). 2. P. Curie, A. Laborde, Compt. Rend. 136, 673 (1903). 3. F. Simon, Nature 135, 763 (1935). 4. D. Andrews, R. Fowler and M. Williams, Phys. Rev. 76, 154 (1949). 5. E. Fiorini and T. Niinikoski, Nucl. Instr. and Meth. 224, 83 (1984). 6. S. H. Moseley, J. C. Mather, and D. McCammon, J. Appl. Phys. 56, 1257 (1984).

18 C. Enss and D. McCammon. 25 kev! Fig. 14. Infrared to gamma rays! left: phase-resolved spectrum of the Crab pulsar, with 0.15 ev energy resolution and 300 ns time resolution simultaneously from the near-ir through the UV (Romani et al 2001).33 right: This gamma-ray detector has a resolving power of 4000 probably the highest ever attained with an energy-dispersive instrument (Ullom et al 2007).34 Kurie-Plot: 187 Re Energy [kev] 300 µg AgReO4 Crystal 760 g TeO Crystal Fig. 15. Neutrino physics. left: Calorimetric measurement of beta decay endpoint eliminates many sources of systematic error in neutrino mass determination (Arnobaldi et al, 2003).35 right: Thermal detectors allow new candidates for neutrinoless double beta decay measurements. This detector from Cuoricino is also the world s best alpha spectrometer, with a resolution of 3.2 kev at 5.4 MeV (Arnobaldi et al, 2005).36

19 Physical Principles of Low Temperature Detectors K-capt. L-capt. Fig. 16. Low, medium, and high energy calorimetry. left: Calorimetric measurement of 7 Be decay shows first direct measurement of L-capture decay rate which produces only a 55 ev recoil from the neutrino emission, while K-capture also creates a 55 ev Auger electron (P.A. Voytas et al, 2002). 37 center: MMC measurement of 36 Cl allow precise measurement of this long-lived β-decay (Heidelberg/Saclay Collaboration, 2006). 38 right: 97 MeV FWHM might not sound like great resolution, but it is R ~1000 at 86 GeV! (GSI, 2005) A. Drukier and L. Stodolsky, Phys. Rev. D 30, 2295 (1984) 8. see for example: P. K. Day, H. G. LeDuc, B. A. Mazin, A. Vayonakis, and J. Zmuidzinas, Nature 425, 817 (2003). 9. see for example: P. Lerch and A. Zehnder, in Cryogenic Particle Detection, C. Enss (ed.), Topics in Applied Physics 99, (Springer; Berlin 2005) pp see for example: K. Pretzl, Nucl. Instr. Meth. A 454, 114 (2000). 11. see for example: J. S. Adams, Y. H. Huang, Y. H. Kim, R. E. Lanou, H. J. Maris, and G. M. Seidel, Nucl. Instr. Meth. A 444, 51 (2000). 12. see for example: F. Mayet, D. Santos, G. Perrin, Yu.M. Bunkov and H. Godfrin, Nucl. Instr. Meth. A 455, 554 (2000). 13. D. McCammon, in Cryogenic Particle Detection, C. Enss (ed.), Topics in Applied Physics 99, (Springer; Berlin 2005) pp D.J. Fixsen et al., in Low Temperature Detectors, (F.S. Porter et al. Eds.), Proc LTD-9 (AIP, New York 2002), p D. J. Fixsen, S. H. Moseley, B. Cabrera, E. Figueroa-Feliciano, Nucl. Inst. and Meth. A 520, 555 (2004). 16. A. Fleischmann, Adv. Solid State Phys. 41, 577 (2001). 17. K. Irwin, Appl. Phys. Lett. 66, 1998 (1995). 18. J. E. Vaillancourt, Rev. Sci. Instr.76, (2005). 19. M. A. Lindeman et al., Rev. Sci. Instr.78, (2007).

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