Thermal Couplings

An energy deposition in the absorber is followed by relaxation processes which result in different temperatures of the sub-systems of the calorimeter. These temperature differences lead to a heat flow between the sub-systems until equilibrium is reached once again. For an understanding of the signal it is necessary to understand the thermal couplings between the different sub-systems of the calorimeter.

2A time of order L/v(with the largest crystal dimension Land the sound velocityv) is required to establish a uniform distribution of phonons in the absorber.

Phonon Transmission

As explained before, phonons are distributed over the whole absorber crystal shortly after an interaction. When hitting the absorber-thermometer interface, phonons can be either transmitted into the thermometer or reflected back into the absorber. The transmission probability of acoustic phonons across the absorber-thermometer interface can be calculated within the theory of anisotropic elastic continua.

In [93] a calculation of the energy flux per unit area and unit time across the boundary from material 1 to material 2 ˙Q1→2was performed by summing over all modes and wave vectors of incident phonons. This energy flux is given by:

Q˙1→2 = E

V 1

2hv_⊥αi (4.7)

where hE/Vi is the average energy density of the phonons in the absorber, v⊥ is the phonon group velocity perpendicular to the interface, andαis the transmission probabil-ity. Hence,hv_⊥αidescribes the transmission of the incident phonons over the interface averaged over modes and angles of incidence. As non-thermal and thermal phonons fea-ture the same angular distribution of wave vectors, the transmission through an ideal boundary does not depend on the frequency. Therefore, this equation can be used to describe the energy transmission of both by inserting the appropriate energy density hE/Vi.

Kapitza Conductance

With the energy densityhE/Viof the thermal phonons, the heat boundary conductance GK of thermal phonons, calledKapitza conductance, can be derived using equation 4.7 [93]: where C/V is the heat capacity per unit volume of material 1. In the case of phonon transmission from absorber to thermometer, the heat capacity of the dielectric absorber has a cubic temperature dependence (see equation 4.2). Hence, the Kapitza coupling for the absorber-thermometer interface exhibits the same dependence:

G_K ∝T³. (4.9)

The transmission coefficients for the materials relevant for the CRESST detectors calculated by numerical methods [93] are listed in table 4.3. Anisotropic conditions are assumed and the tungsten films are considered as polycrystalline metals with randomly oriented single crystals.

Electron-Phonon Coupling

During detector operation the thermometer film is stabilized in the superconducting transition, where only few electrons are bound in Cooper pairs. For this reason, phonons basically interact with the remaining free electrons. While the phonon transmission into the metal film does not depend on the frequency, the absorption of phonons in

Transmission G_K/T³ hv_⊥αi η¯

Table 4.3:Kapitza conductanceGK perT³, the transmission of the incident phonons over the interface averaged over modes and angles of incidencehv_⊥αi, and effective absorption η¯listed for different absorber-thermometer interfaces of CRESST detectors. Parameters are calculated for the transmission of phonons across the (001) plane of silicon, the (1¯102) plane of sapphire, and the (001) plane of CaWO4. Tungsten is considered to be polycrystalline with the crystals randomly oriented [93]. Values taken from [92].

the film does. Therefore, the electron-phonon coupling G_ep in the thermometer has to be considered separately for the thermal and the non-thermal phonons. This coupling depends on the mean free path of the phononsl_p, which can be described by the product l_p = q·l_e. Thereby q is the phonon wave vector and l_e is the mean free path of the electrons. The latter is dominated by scattering on crystal defects and impurities and, therefore, is independent of the temperature. Thus, the temperature dependence of the mean free path of phononsl_p is only given by the phonon wave vectorq.

Thermal Phonons For thermal phonons at low temperatures, whereq·l_e 1 applies, it can be shown that the temperature of the electron-phonon coupling Gep can be expressed as³ [99, 100] :

G_ep ∝T⁵. (4.10)

This strong temperature dependence leads to a thermal decoupling of the electron system in the thermometer from the phonon system in the thermometer and, thus, the absorber. Therefore, the thermal signal is strongly suppressed. As the mean free path length of the thermal phonons is much larger than the thermometer thickness, for the absorption of the thermal phonons the thermometer volume and not the thickness is relevant.

Non-thermal Phonons For non-thermal phonons q ·l_e 1 applies and the mean free path of longitudinal and transverse phonons is given in the Pippard model [101].

Longitudinal phonons interact strongly with the electrons and are efficiently absorbed in the tungsten films (thickness of 200 nm), whereas transverse phonons do hardly interact.

Assuming, for non-thermal phonons, that longitudinal phonons are completely absorbed and transverse phonons do not interact, the average absorption probability η¯ can be calculated [93]. Values for ¯η for the materials used in the CRESST detectors are listed in table 4.3.

3This relation is weakened at higher temperatures and, furthermore, by the metal film being in the superconducting transition. However, at the operating temperature of CRESST (∼ 15 mK) the relation in equation 4.10 is usually assumed [93].

Due to the strong interaction among electrons the phonon energy is quickly thermal-ized and distributed in the electron system of the thermometer, whereby the tempera-ture of the electron system is increased.

The before-mentioned decoupling of electrons and thermal phonons (G_ep ∝T⁵) can lead to a significant overheating of the electron system in the thermometer with respect to the absorber. Therefore, the detector sensitivity is determined by the heat capacity of the thermometer film and not by the heat capacity of the absorber. The absorber influences the detector sensitivity rather by its phonon transport properties.

Thermal Coupling to the Heat Bath

The thermal coupling of the thermometer to the heat bathG_ebis realized via a structure of a normal conducting metal, which is either a gold bond wire or a structure of a thin gold film on the substrate. It is dominated by an electron-electron coupling. The coupling strength defines the thermal relaxation time of the thermometerτ =Ce/Geb, where C_e is the heat capacity of the electron system of the thermometer. The heat conductanceG_eb of the structure follows the Wiedemann-Franz law [56]:

G_eb = LT

R , (4.11)

with the residual resistanceR of the gold structure at temperaturesT and the Lorenz number L = 2.45·10⁻⁸W Ω K⁻² [56]. This coupling can be varied over a wide range by adopting the resistance of the gold structure via its geometry.

The absorber is thermally coupled to the heat bath via the coupling G_ab, which has two components. On the one hand, there is a thermal coupling due to the mechanical mounting structures, i.e. the clamps holding the crystal. On the other hand, phonons can be transmitted directly to the gold structures and escape into the heat bath. The coupling between absorber and heat bath is expected to have a temperature dependence ofGab ∼T³ [93].