A Thermal Model of Low-Temperature Light Detectors for Neutrinoless Double Beta Decay Experiments

Abstract

We report a detector model study for light detectors having Ge and Si wafers as absorbers with a metallic magnetic calorimeter (MMC) readout. The model explains the heat flow processes between the thermal components in the detector system, including athermal and thermal phonon transfers and electronic heat diffusion. The temperature dependence of the thermal conductance values was in good agreement with their expectations. The analysis also resulted in finding the characteristic time constants of the athermal phonons for direct absorption in the phonon collector film and for the downconversion to thermal phonons of the absorber wafers. It is a complete detector model to be applied for various detector variations such as the type and dimensions of the wafer absorber.

1 Introduction

AMoRE is an international project to search for neutrinoless double beta decay ($0\nu \beta \beta$) of $^{100}$Mo [1]. The project utilizes scintillating molybdate crystals containing enriched $^{100}$Mo elements based on the sensor technology of magnetic microcalorimeters (MMCs) for simultaneous phonon and scintillation detection at mK temperatures [2,3,4]. The project completed its pilot phase as AMoRE-Pilot with 6 crystal modules with 12 MMC channels for heat and light detection [5, 6]. AMoRE-I, the updated version with 18 molybdate crystals, began data-taking with 18 heat channels and 18 light channels [7]. The collaboration also started to build successive phases with approximately 700–1000 modules as its practical goal in the next 3 years. The same number of heat and light channels are to be constructed for the next phases.

This experimental study is motivated to optimize light detectors for the AMoRE project. The MMC-based light detectors are composed of a Si or Ge wafer. We investigated which absorber material is better at providing larger signals for light detection. Moreover, this work establishes a complete heat flow model between the thermal components of light detectors. The model is tuned to explain the amplitude and the time constants of the light detector signals.

Fig. 1

A 3-D drawing design of the light detectors (top). A two-inch wafer of Ge or Si serves as the light absorber of calorimetric detection to detect scintillation light from a target crystal of $\hbox {CaMoO}_4$ or $\hbox {Li}_2\hbox {MoO}_4$ (not shown). A cross-sectional view is shown for the copper holder. Three circular phonon collector films are noticeable on the wafer. A picture of a light detector is shown on the back side (bottom left). The wafer has a $\hbox {SiO}_2$ layer of antireflection coating on this side. A zoomed picture of the front side is also shown (bottom right). The gold wires thermally connect the gold films and the MMC sensor glued at the center

2 Experiment

We investigated the characteristics of two light detectors, one made with a Ge absorber and the other with a Si absorber. The Ge and Si wafers are two-inches wide with thicknesses of 0.5 mm and 0.28 mm, respectively. Both wafers were double-side polished. They had an antireflection coating of a 90 nm thick $\hbox {SiO}_2$ layer on a side to maximize absorption for scintillation light from $\hbox {CaMoO}_4$ and $\hbox {Li}_2\hbox {MoO}_4$ crystals with their maximum strengths near 540 nm wavelength.

Figure 1 shows the design and pictures of the light detectors. The absorber wafer was fixed to the copper holder with four Teflon clamps, as shown in the bottom left picture. This is the side on which the wafer has the antireflection coating facing a target crystal of $^{100}$Mo basedscintillators for the $0\nu \beta \beta$ experiment. On the other side of the wafer, three circular gold films (5/300 nm Ti/Au) were evaporated to serve as a phonon collector. Five gold bonding wires were attached to each 3 mm diameter gold film, and an MMC sensor was located in the middle of the copper holder. The gold wires with a diameter of 25.4 $\upmu \text {m}$ were annealed at 460 $^\circ$C for 10 h. Their residual resistance ratio (RRR) was approximately 170. The average length of the wires was approximately 8 mm. The MMC sensors used in the two light detectors were fabricated in a batch. The magnetization and heat capacity of an MMC sensor made in the same batch were investigated in our previous report [4]. Similar to other detector setups with MMC readouts [2, 3], the MMC sensor measures the temperature change of the absorber wafer resulted from an energy input, i.e., absorption of scintillation light in the $0\nu \beta \beta$ application with light detectors. The MMC was thermally connected to the copper sample holder with another gold wire in 25.4 $\upmu \text {m}$ diameter with RRR = 30.

This present experiment was carried out to examine the two light detectors in a wet dilution refrigerator. Each detector setup had a $^{55}$Fe source to test the resolution for 6 keV inputs. In addition, a $^{241}$Am source was used to measure 60 keV gamma-ray activity in both detectors with Si and Ge wafers. The source events were measured at three different temperatures of 10 mK, 20 mK and 30 mK. Similar light detectors with Ge wafers with MMC readouts were first developed in [8] and used in AMoRE-Pilot [5, 6, 9]. In AMoRE-I, the current version of the light detectors with Si wafers was added based on this work [7].

Fig. 2

The heat (energy) flow model of the light detectors. The initial temperature increases in the absorber wafer with a heat capacity $C_\text {a}$ and the phonon collector films with a heat capacity $C_\text {f}$ originate from the energy deposit, $\dot{Q}_\text {a}$ and $\dot{Q}_\text {f}$, from athermal phonons, respectively. The heat eventually flows to the bath via the thermal components. The MMC sensor measures the temperature change of $C_m$

3 Detector Model

We set a heat-flow model composed of four thermal components for the light detectors, as illustrated in Fig. 2. The subthermal systems were connected to each other with thermal conductance.

The absorber of a Si or Ge wafer is considered a lattice system with its heat capacity $C_\text {a}$ described by the Debye model of the phonon system. The three gold films on the wafer are counted as an independent thermal system of electrons with heat capacity $C_\text {f}$. The thermal conductance $G_\text {af}$ between the absorber wafer and the gold films is attributed to the phonon conductance $G_K$ between the two different media [10] and the thermal conductance $G_\text {ep}$ of the energy transfer via electron–phonon interactions in the metal films [11]. Simply, $G_\text {af}$ is the result of $G_K$ and $G_\text {ep}$ connected in series as $1/G_\text {af} = 1/G_K + 1/G_\text {ep}$.

The thermal connection between the gold films and the MMC was made with 15 annealed gold wires. This heat flow can be explained by two steps of the heat diffusion of conduction electrons in the films and the wires. First, the heat flow includes the lateral thermal conductance $G_\text {fw}$ in the gold films toward the bonding spots of the gold wires as two-dimensional heat diffusion via conduction electrons of the gold films. Then, the heat flows along the gold wires that have their own specific heat. In the thermal model, we assume a thermal component with heat capacity $C_\text {w}$ for half of the heat capacity of the gold wires. This thermal component of the gold wire is thermally attached to the gold films on one side with a thermal conductance of $G_\text {fw}$ and to the MMC temperature sensor on the other side with a thermal conductance $G_{wm}$. $G_{wm}$ corresponds to the thermal conductance along the gold wires. The remaining half of the gold wire heat capacity is added to the MMC heat capacity $C_m$. A similar approximation was made for the heat flow along a bar conductor channel [12]. The thermal conductance $G_{mb}$ of another gold wire acts as the major cooling path between the MMC sensor and the thermal bath of the copper sample holder. Last, the holding structure for the absorber wafer results in a cooling path with a thermal conductance $G_\text {ab}$ between the wafer and the bath.

When particle absorption takes place in the light detector, we consider the initial energy transfer into the detector system caused by athermal phonons with their energy around 20–50 K [13, 14]. These athermal phonons travel around the detector material, and eventually convert into a thermal phonon distribution of the wafer lattice via surface, isotope, and impurity scatterings.

In our detector model, the temperature of the lattice system increases from the thermalization process of athermal phonons denoted as $\dot{Q}_\text {a}$. Moreover, athermal phonons that reach the interface between the gold film and the substrate have some chance to scatter an electron transferring its energy to the electrons in the film in a short time. This direct heat transfer from athermal phonons to the gold phonon-collector film is considered athermal heat transfer denoted as $\dot{Q}_\text {f}$, as indicated in Fig. 2.

Based on the model, a set of differential equations can be established to represent the heat flow between the thermal components as

$$\begin{aligned} C_\text {a} \dot{T}_\text {a}= & {} -G_\text {af} (T_\text {a}-T_\text {f} ) -G_\text {ab} (T_\text {a}-T_\text {b} ) + \dot{Q}_\text {a}(t) \end{aligned}$$

(1)

$$\begin{aligned} C_\text {f} \dot{T}_\text {f}= & {} -G_\text {af} (T_\text {f}-T_\text {a} ) - G_\text {fw} (T_\text {f}-T_\text {w} ) + \dot{Q}_\text {f}(t)\end{aligned}$$

(2)

$$\begin{aligned} C_w \dot{T}_w= & {} -G_\text {fw} (T_w-T_f ) - G_{wm} (T_w-T_m ) \end{aligned}$$

(3)

$$\begin{aligned} C_m \dot{T}_m= & {} -G_{wm} (T_m-T_w ) - G_{mB} (T_m - T_B) \end{aligned}$$

(4)

where $T_i$ indicates the temperature of each component, $\dot{Q}_\text {f}(t) = \varepsilon \cdot \frac{E}{\tau } \cdot e^{ -t/\tau }$ is the direct absorption of athermal phonons in the phonon collector film, $\dot{Q}_{a}(t) = (1 - \varepsilon ) \cdot \frac{E}{\tau } \cdot e^{ -t/\tau }$ indicates the thermalization of athermal phonons into the thermal phonon system, $\tau$ is the mean lifetime of athermal phonons, and $\varepsilon$ is the portion of the direct absorption into the film. Note that $\int _0^\infty (\dot{Q}_\text {a} + \dot{Q}_\text {f} ) dt = E$. where E is energy incident on the absorber.

For given values of the heat capacities, thermal conductances, time constants, and input energy, all the temperature responses of $T_i$ can be numerically found by solving the combined differential equations. In this study, we set the heat capacity values as the calculated values using the geometries of the thermal components and the literature values of the Debye temperatures of the wafer materials and the Sommerfeld constant of gold. The thermal conductance values and the time constants were determined as fitting parameters of a numerical solution of the model equations in comparison to the measured signals. The parameters are to be compared with the expected values for the thermal conductances and the time constants.

Fig. 3

Fit results of the conductance parameters for the light detector with Ge and Si absorbers at 10, 20 and 30 mK. The guidelines indicate their tendency proportional to $T^{4}$, T and $T^{3}$ in left, middle and right plots, respectively

4 Results and Discussion

The energy spectra measured with the two detectors showed clear peaks for 6 keV and 60 keV events from the sources. The 6 keV events appeared in the spectra as two distinct peaks for the $K_\alpha$ and $K_\beta$-lines. The Si detector showed a full width half maximum (FWHM) resolution of 174 eV at 10 mK. Similar resolution was obtained at other temperatures of 20 mK and 30 mK. Moreover, the pulse height of the Si detector was 10–20% larger than that of Ge detector under the same conditions.

We used 60 keV signals as the reference signals to investigate unknown conductance parameters in the model. We used a nonlinear regression method for the least squared difference between the calculated and measured pulses. As the solution of the differential equation results in a signal pulse of the temperature change in the unit of K, the measured signals were converted in the temperature unit based on the DC calibration measurement with a large temperature variation. For both detectors with Ge and Si absorbers, the fit results showed good agreement; the residuals of the difference are less than 1.5 % of root mean squares relative to the pulse height.

Figure 3 shows the thermal conductance values found from the analysis for all the datasets. The conductance values seem to be temperature dependent. Among the 5 parameters the $G_\text {af}$ values show the strongest dependence on the temperature, and seem to be proportional to $T^4$. As shown in the data points and their guide lines in the left plot, the $G_\text {af}$ value at 10 mK and 30 mK differ by two orders of magnitude, and little difference is found between the conductance values for Ge and Si. It agrees with our understanding of $G_\text {af}$ where $G_\text {ep}$ becomes the major bottleneck of the heat flow in the temperature region [15]. The $T^4$ dependence is a distinct characteristic of the thermal conductance associated with electron–phonon interactions.

The thermal conductance values found for $G_\text {fw}$, $G_{wm}$, and $G_{mb}$ showed weak dependence on the temperature change as shown in the middle plot of Fig. 3. The values for the Ge and Si cases follow a similar behavior with some differences in factors less than 2. Because $G_\text {fw}$, $G_{wm}$, and $G_{mb}$ represent the conductance parameters for corresponding materials made of gold, they should be related to the thermal diffusion of the electrons in the metal. We added guide lines proportional to T corresponding to the conductance values. At low temperatures, the thermal conductance G of a metal is estimated by the Wiedemann–Franz (WF) law as $G^\textrm{WF} = L_0 T/R$, where $L_0$ is the Lorentz number and R is the resistance of the metallic conductor along the heat flow. $G_\text {fw}$ considers the heat flow of the films in the lateral direction between the gold films themselves and the bonding wires with half of their total heat capacity, as discussed above.

The values of $G_{mb}$ found from the analysis agree with the WF expectation marked as a solid line in the middle plot. However, the other two parameters were found to be 2–3 greater values from the WF expectation lines calculated from their expected electrical resistances considering the number, length, and RRR of the wires. We attribute these excess values of the thermal conductance to the overestimation of the heat capacity of $C_\text {w}$ that was set to half for all wires. If a smaller portion of the wires is considered to contribute to $C_\text {w}$, the fitting results for $G_\text {fw}$ and $G_{wb}$ should be reduced accordingly. For instance, instead of the 50/50 assumption of the heat capacity distribution of the gold wires to $C_\text {w}$ and $C_m$, letting 20/80 distribution of the heat capacity, $G_\text {fw}$ and $G_{wm}$ values found in the model study may agree better with the expected values from the Wiedemann–Franz law. This is our next step of the model study.

The thermal conductance labeled $G_\text {ab}$ represents the thermal connection of the absorber wafer to the copper holder via Teflon contacts. Although the $G_\text {ab}$ values found from the model fit roughly follow the $T^{2.5}$ dependence, we added guide lines of the $T^3$ dependence for Ge and Si detectors independently in the right plot because $G_\text {ab}$ originates from the phonon transfer mechanisms through the interface between different dielectric media and in the dielectric Teflon. The $G_\text {ab}$ values for the Ge detector were 3.4 times greater than the $G_\text {ab}$ values for the Si detector on average. We interpret this difference from the clamping force difference for the wafers with different thicknesses. Using the same holder design and materials, the Teflon clamps pushed the 0.5 mm thick Ge wafer stronger than the 0.28 mm thick Si wafer against the copper holder. The $G_\text {ab}$ in the detector model influences the falling part of the signals together with $G_{mb}$, with little effect on the pulse height and the rise time of the pulses.

Fig. 4

Fit results of the time constant values for the light detector with Ge and Si absorbers at 10, 20 and 30 mK

The time constant $\tau$ of the athermal phonons and the ratio $\varepsilon$ of the direct energy deposit to the phonon collector films are important values responsible for the pulse height and the rise time of the signals. Another set of time constant parameters $\tau _\text {a}$ and $\tau _\text {f}$ can be defined as $1/\tau = 1/\tau _\text {a} + 1/\tau _\text {f}$, $\tau _\text {a} = \tau / (1- \varepsilon )$, and $\tau _\text {f} = \tau / \varepsilon$ to characterize the time constant of the energy deposit in each of the subthermal systems independent of the other term [16]. All the time constant parameters are assumed to be temperature independent. Figure 4 show the fit results of the parameters. $\varepsilon$ values were about 0.5 for all cases, while $\tau$ was 176 $\upmu \text {s}$ and 78 $\upmu \text {s}$ for the Ge and Si wafers, respectively. No apparent temperature dependence resulted in the time constant parameters, as expected.

As discussed above, $\tau _\text {a}$ is determined by the frequency of inelastic scattering of the athermal phonons by impurities in the bulk crystal and at the crystal surface. If the surface inelastic scatterings are considered as the major thermalization mechanism of the thermal system of the crystal lattice, $\tau _\text {a}$ should be proportional to a geometrical time constant $\tau ^\textrm{geo}_\text {a}$ characterized as the inverse of the reflection frequency of the phonons at the surface. It can be approximated as $\tau ^\textrm{geo}_\text {a} = 2 V_\text {a}/ (A_\text {a} \cdot v_p)$ where $V_\text {a}$ and $A_\text {a}$ are the volume and the surface area of the wafer, and $v_p$ is the average phonon speed perpendicular to the wafer surface. $\tau _\text {a}$ is simply proportional to their thicknesses when the wafers have similar surface conditions and bulk impurities.

$\tau _\text {a}$ values of about 0.35 ms and 0.16 ms were found from the analysis for Ge and Si wafers with thickness of 0.5 mm and 0.28 mm, respectively. The ratio $\tau / \tau ^\textrm{geo}_\text {a}$ between the measured $\tau$ and the estimated $\tau ^\textrm{geo}_\text {a}$ accounting for the wafer dimensions and the phonon velocities indicates the average number of reflections of the athermal phonons at the surfaces before downconversion to thermal phonons or absorption in the film. Their values were found to be 900 and 1200 for the Ge and Si wafers, respectively.

A similar analogy can be applied to $\tau _\text {f} \propto \tau ^\textrm{geo}_\text {f} = 2 V_\text {a}/(A_\text {f}\cdot v_p)$ where $A_\text {f}$ is the area of the phonon collector film. Then, $\varepsilon$ should be proportional to $A_\text {f}/A_\text {a}$ independent of the wafer thickness and the temperature. As shown in Fig. 4, the parameters found for $\varepsilon$ are close to 0.5 for all the cases. Because both Ge and Si cases share the same values for $A_\text {f}/A_\text {a}$ and the same thickness of the phonon collector film, $\varepsilon$ values should be similar as found from this model study. The ratio $\tau _\text {f} / \tau ^\textrm{geo}_\text {f}$ indicates the chance for the phonons entering the film/substrate interface to be absorbed in the phonon collector rather than reflected at the interface or entering but returning back to the substrate. They were 11% and 8% for the Ge and Si substrates with gold films with the same thickness of 300 nm. Considering the acoustic impedance mismatches, the absorption ratios are in good agreement with each other and with another measurement with the same gold thickness in a sapphire wafer, which resulted in a ratio of 17% with the corrected value of the phonon speed [16].

5 Conclusion

We developed a detector model of a light detector having a Ge or Si wafer in the thermal calorimetric detection of heat/light signals for $0\nu \beta \beta$ experiments. The model includes the dynamics of the initial athermal phonons in the substrates and the phonon collector films. The heat capacities were fixed with the theoretical calculations based on a few assumptions to simplify the model. The values for thermal conductance and the time constants were free to fit between the measured and calculated signals.

The resulting heat flow model agrees well with the expectations of the corresponding thermal conductances between the subthermal components in the detector system. The time constant values provide consistent results with the understanding of the absorption mechanisms of the athermal phonons in the detector components. The model may overestimate the heat capacity of the gold wire as an independent component. However, the overestimation caused a small variation in the pulse shape of the simulation result, and this over estimation can be adjusted in the next step.

This detector model generalizes the light detectors with Ge and Si wafers with an MMC readout. This model has been applied for the light detectors used in the series of AMoRE experiments [3, 7] and other R &Ds with various molybdate crystals [17] and $\hbox {CaF}_2$ crystals [18]. Moreover, we plan to develop light detectors with wafer sizes larger than two-inch diameters. The present detector model provides a straightforward estimation of the signal size and the shape that can be tested in future work.