7.2 Phonon Detector Performance
7.2.3 Thermalization of Non-Thermal Phonons
In order to further optimize the detectors in the future it is necessary to understand the process of the thermalization of non-thermal phonons in the crystal. As discussed in section 4.2.5, two processes are competing - the thermalization in the absorber (de-scribed by the time constantτcrystal) and the thermalization due to absorption in the thermometer film (described by the time constantτf ilm). These two time constants can-not be measured directly. Instead only the total life time of non-thermal phonons τn, which is the rise time of both components in a calorimetric thermometer, can be deter-mined from measured pulses via the pulse model fit. As a reminder, the time constants for thermalization determine the life-time of non-thermal phonons in the following way
Figure 7.3: The black pulse with a deposited energy of 5.9 keV (55Mn Kα) is measured with the detector TUM56-f in cryostat 1 in Munich. It is compared to a 5.9 keV pulse measured in the CRESST setup with the detector Lise, the phonon detector with the lowest energy threshold in CRESST-II phase 2 (red). The amplitude of the pulse of TUM56-f is larger by a factor of
∼17 compared to the amplitude of the pulse of Lise. This illustrates the improvement of the optimized CRESST-III phonon detector.
(a) TUM26-b (b) TUM56-f
Figure 7.4:Distribution of the life-time of non-thermal phononsτn from a parametric fit to a set of pulses of the phonon detectors TUM26-b (a) and TUM56-f (b). The resulting distribution ofτn is fitted with a Gaussian function (red) to determine the mean. The values obtained for τn are in agreement to the values determined from the fit of the pulse model to the template pulse (see figure 7.1).
(see section 4.2):
1
τn = 1
τf ilm + 1
τcrystal (7.1)
To derive the parameters of the thermalization an exact knowledge ofτn is required.
However, the value of τn which is determined from a fit of the pulse model to the template pulse (see figure 7.1) might be distorted as many pulses are summed for the creation of the template pulse. Thus, the pulse model was individually fitted to a set of pulses in the linear range of the respective detector. From these individual fits, the mean for τn is determined by a fit of a Gaussian function to the obtained distribution (see figure 7.4). Values of τn = 0.97 ms for TUM26-b and τn = 1.30 ms for TUM56-f are determined. They agree within 2 % with the ones determined from the fit of the template pulse. Hence, the error due to the creation of the template is negligible.
The two detectors TUM26-b and TUM56-f both have a crystal of the same size. For this reason, it is expected that the thermalization in the crystal has the same time constantτcrystal (see equation 4.16). The area of the thermometer differs by a factor of
∼2. As the time constant for the thermalization in the filmτf ilm inversely scales with the thermometer areaAt(see equation 4.17), it is expected to be smaller by a factor of two for the detector with a larger TES (TUM26-b) than for the one with the smaller TES (TUM56-f). In total also the life-time of non-thermal phononsτn is expected to be smaller for the detector with the large TES structure. However, the difference of the value depends on the relation ofτf ilm and τcrystal (see equation 7.1). For the two detectors measured, as expected the value ofτn is smaller for TUM26-b by a factor of
∼1.3.
In order to determine the thermalization time constantsτin andτcrystal two different methods are performed and explained in the following.
Method 1: Direct calculation of τf ilm With the first method all time constants are determined individually for each detector. As discussed in section 4.2.5 the time
detector TUM26-b TUM56-f
At[mm2] 36.08 18.57
Method 1 Method 2 Method 1 Method 2
τn[ms] 0.9681 1.297
τf ilm [ms] 2.410 2.017 4.683 4.034
τcrystal [ms] 1.603 1.843 1.734 1.843
ε [%] 40.0 47.7 27.0 31.3
Table 7.1: Time constants determined with two different methods for the detectors TUM26-b and TUM56-f. Method 1 relies on the direct calculation of the time constantτf ilm. In method 2 two different phonon detectors with crystals of the same size and differently sized TES structures are compared. The time constantτn is determined from the fit of the pulse model to pulses in the linear range. For details see text.
constant for the thermalization in the thermometer film τf ilm can be calculated directly according to equation 4.17. However, this relies on the assumption that longitudinal non-thermal phonons are completely absorbed in the tungsten film, while transverse phonons do not interact at all. In [93] it is shown that this is a good assumption for the transmission of non-thermal phonons from silicon to an iridium/gold film. However, up to now this has not been confirmed for the transmission from CaWO4 into a tungsten film.
With the calculated values ofτf ilm and the values for the life-time of non-thermal phonons τn, the time constant τcrystal can be determined with equation 7.1. All resulting values are shown in table 7.1 for both detectors as method 1. With this method for the time constant τf ilm values of 2.4 ms (for TUM26-b) and 4.6 ms (for TUM56-f) are determined. Values of τcrystal = 1.6 ms for TUM26-b and τcrystal = 1.7 ms for TUM26-b are derived. As the two values agree well, this hints that the assumption for the absorption probability of non-thermal phonons in the tungsten film is valid. This corresponds to a fraction of non-thermal phonons absorbed in the thermometer of ε = 40.0 % for TUM26-b and ε = 37.0 % for TUM26-b.
As method 1 relies on certain assumptions, which might not be fulfilled for the investigated detectors a second method is performed.
Method 2: Comparison of two crystals In contrast to the first method, the second me-thod needs inputs from both measurements and compares them. It exploits the relations of the time constantsτf ilm and τcrystal expected from the geometry of the crystals and the thermometers. As mentioned above,τf ilm is expected to be twice as long in the detector TUM56-f with the smaller thermometer compared to TUM26-b. Additionally,τcrystal is expected to be the same in both detectors.
Combining this, the two time constants (τf ilmandτcrystal) can be determined with equation 7.1 from the life time of non-thermal phononsτn of both measurements.
The resulting values are also shown in table 7.1. A value of τcrystal = 1.8 ms is determined for both crystals. For the time constant τf ilm a value of 2.0 ms for TUM26-b and a value of 4.0 ms for TUM56-f are resulting. This corresponds to a
fraction of non-thermal phonons absorbed in the thermometer of ε= 47.7 % for TUM26-b andε= 31.3 % for TUM26-b.
The values for the time constants determined with both methods differ by∼10-20 % but are roughly consistent. This is an indication that the assumptions made for the direct calculation of the time constantτf ilmare valid for the transmission of non-thermal phonons into a tungsten film.
One uncertainty which influences both methods in the same way is the usage of phonon collectors in the thermometer structures described here. The energy transport through the phonon collectors might be delayed compared to the absorption of non-thermal phonons directly in the tungsten film and, thus, might influence the rise time of the pulse. In [106] the time constant for this process was found to be O(10µs) (for an aluminum film with a length of 4 mm) and, thus, short compared to the pulse rise time. Moreover, this uncertainty affects both methods in the same way. The absolute values can be distorted due to these uncertainties but it is valid to compare them. To investigate the parameters in a further measurement the usage of thermometers without phonon collectors would be beneficial.
The model for the time constants τf ilm and τcrystal allows to calculate the expected time constants for the large crystals operated in CRESST-II phase 2 . The time constant τf ilm can be directly calculated. According to equation 4.17 it is τf ilm ≈23 ms. From the known relation ofτcrystal ∝Va/Aa and the values of the time constant determined for the small crystal, a value ofτcrystal ≈5 ms is expected for the large crystals. With these values, a life-time of non-thermal phonons ofτn= 4.1 ms is expected (see equation 7.1).
However, CRESST-II phonon detectors with a bolometric TES evaporated directly on the large crystal feature a life time of non-thermal phonons of τn ≈ 12 ms. Due to equation 7.1 τn must always be shorter than each of τf ilm and τcrystal. Thus, the expected value for τf ilm of 23 ms fits well to the measured value of τn. However, the expected value forτcrystal ≈5 ms is too small.
There are several possible explanations for this discrepancy:
The crystal surface might influence the thermalization of non-thermal phonons in the crystal and, thus, the value ofτcrystal. While on the large crystals operated in CRESST-II usually only one side is roughened and the other sides are polished, on the small crystal five sides are roughened. A dedicated measurement with crystals of the same geometry with differently treated surfaces could investigate this influence.
As mentioned above, the influence of the phonon collectors increases the uncer-tainty of the method. As the bolometric TES of the large crystal does not feature phonon collectors, it might not be valid to transfer the time constants.
The expected dependence ofτcrystal on the absorber volumeVa and the absorber areaAa(see equation 4.16) might not be valid. The thermalization of non-thermal phonons in the crystal could to be influenced by additional effects. For example the geometry of the crystal plays a role as it influences the mean scattering length of non-thermal phonons in the crystal. This is investigated further in [102] but the determined results cannot explain the discrepancies found here.