Geometry and Physics of the PL38 and FID38 Simulation

This section is dedicated to building the geometry and defining the electrodes of the PL38 and FID38 design simulations. The description of these detector designs is illustrated with the an-notated schemes4.8for the PL38 and4.9for the FID38. These scheme are used as reference for

Revolu�on Axis Copper Chassis (GND)

Bo�om Electrode (B) Top Electrode (A) Exi�ng Volume NTD

Fiducial Volume

Figure 4.8: Cross-section scheme of the PL38 detector design. The scheme is not to scale and the dimensions and parameters of this design are listed in the table 4.2. The copper chassis is represented by the electrically grounded rectangle. It is separated from the germanium crystal by the vacuum colored light-blue. The two aluminium electrodes are represented in black at the surface of the crystal. The default polarization of PL38 is (VA,VB) = (+1,−¹)V. The colored volumes inside the crystal are drawn from electric field lines with common start and end points.

building the geometry in the FEM software. The schemes are not entirely at scale and are thus accompanied with the lists of the parameter values4.2and4.3 for the PL38 and FID38 designs respectively.

The two PL38 and FID38 detector designs use cylindrical germanium crystals of height HGe = 10 mm and radiusRGe = 15 mm as absorber. The mass of these crystals are about 38 g hence the name of the designs as to differentiate them from more massive 200 g and 800 g EDEL

-WEISSdetectors. Each crystal is surrounded by an hollow cylindrical chassis which is electrically grounded to 0 V. This copper chassis is spaced from the crystal by a distancedCu =3 mm of vac-uum. The vacuum is simulated with a relative electric permittivity ofǫr(Vacuum) = 1 and the germanium crystal possesses a much higher value of ǫr(Ge) = 16.3. Although present in real detectors to maintain the crystal, Teflon clamps are not simulated. Indeed, their influences is considered negligible due to their small size and low permittivity ǫr(^Teflon) = 2.1. Similarly, the real detectors inspired by the PL38 and FID38 design should possess NTD thermal sensors but these thermistances are not simulated. Their predicted gluing locations are annotated on the scheme. These locations are chosen as to reduce the contact of the NTD with the aluminium electrode. For the PL38 design, the NTD is located in the central hole of the top electrode in con-tact with the bare germanium surface. For the FID38 design, the NTD should be glued on top of a veto electrode ring. As such, a majority of the NTD surface should be in contact with the bare germanium surface while perturbing the electric field very locally in the top veto volume. The main difference between the two detector designs concern the number and shape of aluminium electrodes. The PL38 design holds its name from the two full planar electrodes while the FID38 design features four fully inter-digitized electrodes. The simpler PL38 design is presented first

Parameter Symbol Default Value

Ge crystal Height HGe 10 mm

Ge crystal Radius RGe 15 mm

Distance between crystal and copper chassis dCu 3 mm

Electrode Thickness hAl 1 µm

Radius of the central NTD hole rcenter 1.5 mm

Corner length L_lat 2 mm

Main Voltage Bias Vbias 2 V

Symmetric factor of the voltage bias Sbias 0.5 Table 4.2: List and Value of the default parameters for the PL38 design.

A

A B A B A B A

B

C C

C C

C

D D D

D NTD Loca�on Revolu�on Axis

Bulk Volume Top Veto Volume Bo�om Veto Volume Corner Volume Equatorial Volume

D C A B

Top Veto Electrode Top Collect Electrode Bo�om Veto Electrode Bo�om Collect Electrode

Figure 4.9: Cross-section scheme of the FID38 detector design. The scheme is not to scale and the dimensions and parameters of this design are listed in the table4.3. The copper chassis is represented by the electrically grounded rectangle. It is separated from the germanium crystal by the vacuum colored light-blue. The aluminium pads and rings are represented in black at the surface of the crystal and labeled according to their electrode of attribution. The default polar-ization of FID38 is(VA,VB,VC,VD) = (−^0.25,+1,+0.25,−¹)V. The colored volumes inside the crystal are drawn from electric field lines with common start and end points.

Parameter Symbol Default Value

Ge crystal Height HGe 10 mm

Ge crystal Radius RGe 15 mm

Distance between crystal and copper chassis dCu 3 mm

Electrode Thickness h_Al 1 µm

Electrode Width wAl 80 µm

Radius of the innermost planar electrode rcenter 0.25 mm Width of bare Ge crystal on corners w_bare 0.3 mm Width of the outermost veto electrode wouter 0.08 mm

Number of planar electrodes nplan 7

Number of lateral electrodes nlat 2

Interdistance of Planar electrodes dplan 1.98 mm Interdistance of Lateral electrodes dlat 2.40 mm

Equatorial distance deq 2 mm

Main Voltage Bias Vbias 2 V

Ratio Veto/Main voltage bias Rveto −^0.25

Symmetric factor of the voltage bias Sbias 0.5 Table 4.3: List and Value of the default parameters for the FID38 design.

followed by the description of the more complex FID38 design.

The ionization channel of the PL38 detector design consists in two collecting electrodes read-ily visible of the scheme4.8. Each electrode is denominated by its location on the top or bottom surface but for more clarification we can define the following convention for the electrode de-nomination and indexes in this work:

{^Top,Bottom} ⇔ {^A,B} ⇔ {^{1, 2}} ^(4.33) Each electrode is basically a continuous aluminium deposit on the full surface with some exten-sion on the lateral surface. The extenexten-sion length on the lateral surface is notedLlatwith a default value of 2 mm. While the bottom electrodeBcovers the full bottom surface of the crystal, the top electrodeAfeatures a central hole. This hole is meant to provide the NTD thermal sensor with a bare germanium as to increases the fraction of athermal phonons. The hole is circular of radius rcenter. Its default value of 1.5 mm is meant to host a NTD of small standard area(2ײ)mm.

The design FID38 has a more complex ionization channel composed of four electrodes la-beled AthroughD. The denomination of these electrodes in this work follows the convention:

{^{Top Veto,}Top Collect,Bottom Veto,Bottom Collect} ⇔ {^A,B,C,D} ⇔ {^{1, 2, 3, 4}} ^(4.34) As displayed on the scheme4.9, each electrode is an association of aluminium pads and rings.

For real detectors, this association is made by linking the aluminum deposits with aluminium wires bridging over the germanium surface. Inside the electrostatic simulation software, mul-tiple deposits can be assigned to the same electric terminal thus sharing their potential and charges. It is therefore possible to avoid the simulation of the wire-bonding bridges between the different aluminium deposits. These deposits consists in two pads and multiples rings whose shape depends on several parameters. The radius of the circular central pads on the top and bottom surfaces is notedrcenterand is by default set to 0.25 mm. The width of the concentric alu-minium rings is by default set to the minimal value ofwAl =80 µm. The outermost planar rings are attributed a specific independent widthwouter with the same default value. The number of total separated deposits, pads and rings, is notedn_planfor the planar surface andn_latfor the lat-eral surfaces. The default configuration sets these numbers tonplan=7 andnlat=2. The radius

of the outermost ring is limited to the radius of the germanium crystal minus a band of width r_edge = 0.3 mm. This comes from the limits of the aluminium deposition technique employed:

it is not possible to have a thin aluminium electrode at less than 0.3 mm from the edge of the planar surface of the germanium crystal. The concentric aluminium rings are spaced evenly on the planar surfaces with a distanced_plan=2.40 mm between their center. On the lateral surface, aluminium rings respect the mirror symmetry with the equator of the cylindrical germanium crystal. The two most centered lateral electrodes, deemed equatorial rings, are distant from each other bydeq =2 mm. The remaining lateral electrodes are evenly separated by the lateral spacing of default valued^lateral_lat = 1.98 mm between the equatorial rings and the crystals corners. Now that the aluminium deposits are defined, they can be attributed to an electrode. The electrodes respect a top-bottom symmetry withAandBbeing the top electrode withCandDbeing their bottom counterpart. The electrodesBandDcorresponds to the main collecting electrodes, also called collect electrodes. Their function is to record the signal induced by recoils located in the bulk of the crystal. The electrodesAandCcorresponds to auxiliary veto electrodes, also called veto electrodes. Their function is to collect the electric charges produces near the surface of the absorber. This surface tagging ability is the motivation behind the FID-like detector designs. In order to create a usable veto volume on the surface, the equatorial rings, the outermost planar electrodes and the central pads are attributed to the veto electrodes A andC. The remaining rings are attributing in order to obtain a succession of collect and veto electrodes. This attri-bution process imposed constraints on the number of planar and lateral aluminium deposits.

Indeed, the interleaving of collect and veto deposits necessitatesnplanto be an odd integer and n_latto be even.

For the PL38 and FID38 designs, the polarization of the electrodes is controlled by several parameters. First is the voltage bias of the detector, notedVbias, corresponding to the voltage between the two main collecting electrodes. For both design, its default value is 2 V. Then comes the bias symmetry factorSbiasrepresenting the symmetry of the top and bottom electrodes potential with respect to the 0 V ground voltage. At the default value of 0.5, polarizations are considered symmetric with opposite-side electrodes having opposite electric potential. With this two parameters, the polarization of the PL38 design is expressed as:

(VA=VbiasSbias

VB = −^Vbias(1−^Sbias) ⇒^{with S}bias =0.5,

(VA=Vbias/2

VB = −^Vbias/2 (4.35) The default polarization of the PL38 design corresponds to:

(VA,VB) = (+1,−¹)V (4.36) As such, the top electrode Ashould collect the electrons of negative electric charge while the bottom electrodeBcollects the positively charged holes.

With two auxiliary veto electrodes, the FID38 necessitates an additional parameters to fix the polarization of its electrodes. The veto ratio factor notedRveto is used to define the voltage difference between the main collecting and the auxiliary veto electrodes. Its default value for the FID 38 design is−0.25. The polarization of the FID38 design is therefore expressed as:



The default polarization of the FID38 design corresponds to:

(VA,VB,VC,VD) = (−^0.25,+1,+0.25,−¹)V (4.38) The electric equations are solved only in the insulators domains corresponding to the germa-nium crystal and the surrounding vacuum inside the copper chassis. The alumigerma-nium electrodes are set to a fixed potential and thus their interior is excluded from the simulation.