# Development of Cryogenic Detectors for Neutrinoless Double Beta Decay Searches with CUORE and CUPID
## 1 Introduction  
Particle detectors exploit several mechanism to convert an energy deposition from ionizing radiation into a measurable quantity. The most widely used technologies measure either the ionization produced in the detector volume or scintillation signals excited in the radiation-absorbing material by the interacting particles. However, the energy detectable via these channels is only a small fraction of the total energy deposit, as the largest part is converted into lattice vibrations (i.e., heat) and escapes detection. A possible solution to increase the sensitivity consists in using a detector able to measure the vibration quanta (i.e., phonons) produced by an interaction: a cryogenic calorimeter. In a very simplified model, a cryogenic calorimeter can be sketched as an absorber, usually a diamagnetic crystal, attached to a phonon sensor. When a particle releases energy in the absorber, its temperature increases and phonons are produced in its lattice. The phonons propagate in the absorber and are converted into a measurable signal by the sensor, acting as a thermometer. Each energy deposit in the absorber (\(E_{\rm Dep}\)) induces a temperature increase (\(\Delta T\)), given by \(\Delta T=E_{\rm Dep}/\rm C\), where \(\rm C\) is the absorber capacity. To increase the sensitivity of this measurement method, \(\rm C\) has to be minimized. This goal is achieved by keeping the absorber at a temperature of \(\sim\)10 mK by means of a thermal machine, usually a dilution refrigerator. In these conditions, in fact, \(\rm C\propto(T/T_{\rm D})^{3}\), where \(T_{\rm D}\) is the Debye temperature of the material. The higher \(T_{\rm D}\) is, the higher will the temperature rise due to a given energy deposit in the crystal, and thus the detector sensitivity. The energy resolution of such detectors is their main characteristic, and it is usually below 0.1% at \(\sim\)MeV energies. This characteristic is determined by the detector technology itself, and is rather consistent with different absorbers. The resolution is only limited by the thermal noise and by eventual thermalization imperfections. Since the average energyof a phonon is \(\sim\)\(\mu\)eV, the average number of phonons created by a \(\sim\)MeV interaction is extremely large, allowing only small statistical fluctuation of the information carriers. Moreover, different materials can be used as absorbers, providing the possibility to investigate different processes through different strategies. The main downsides of these technique are the inability to natively measure different de-excitations, for example heat and scintillation light, and the need for high-power cooling devices to maintain the cryogenic temperatures. The former features translate into an inability to identify interacting particles producing undesired backgrounds, while the latter strongly limits the possibility to operate large detector masses with an easily scalable architecture. Cryogenic calorimeters are widely used to search for Dark Matter direct interactions by the SuperCDMS [1], EDELWEISS [2] and CRESST [3,4] collaborations, and Neutrinoless Double Beta Decay by the Cryogenic Underground Observatory for Rare Events (CUORE) [  
This extremely rare process is of great interest since, if detected, it would simultaneously prove the existence of lepton number-violating processes, state the Majorana nature of neutrinos and provide a value to their absolute mass scale [14; 15; 16]. The experimental search for \(0\
u\beta\beta\) relies on the detection of the two electrons emitted in the process. Since the recoil energy of the nucleus is negligible, the two electrons carry a total kinetic energy equal to \(Q_{\beta\beta}\). The expected signature is therefore a peak in the sum energy spectrum of the electrons centered around the \(Q_{\beta\beta}\). The energy region for this search is usually referred to as the region of interest (ROI), and its width is a multiple of the full width at half maximum (FWHM) energy resolution of the used detector, evaluated at the \(Q_{\beta\beta}\).