Bayesian inference of high-purity germanium detector impurities based on capacitance measurements and machine-learning accelerated capacitance calculations

The impurity density in high-purity germanium detectors is crucial to understand and simulate such detectors. However, the information about the impurities provided by the manufacturer, based on Hall effect measurements, is typically limited to a few locations and comes with a large uncertainty. As the voltage dependence of the capacitance matrix of a detector strongly depends on the impurity density distribution, capacitance measurements can provide a path to improve the knowledge on the impurities. The novel method presented here uses a machine-learned surrogate model, trained on precise GPU-accelerated capacitance calculations, to perform full Bayesian inference of impurity distribution parameters from capacitance measurements. All steps use open-source Julia software packages. Capacitances are calculated with SolidStateDetectors.jl, machine learning is done with Flux.jl and Bayesian inference performed using BAT.jl. The capacitance matrix of a detector and its dependence on the impurity density is explained and a capacitance bias-voltage scan of an n-type true-coaxial test detector is presented. The study indicates that the impurity density of the test detector also has a radial dependence.


Introduction
Advanced scientific applications of high-purity germanium (HPGe) detectors often require a quantitative understanding on how the detector signal depends on the event topology.This requires a realistic simulation of the detector.Example applications are searches for physics beyond the Standard Model such as neutrinoless double-beta decay [1][2][3][4][5] and dark matter [6][7][8].In such rare-event searches, it is essential to distinguish between signal and background events based on the shape of the detector signal.e-mail: lhauert@mpp.mpg.de(corresponding author) A very important input to the simulation of HPGe detectors is the impurity density distribution, , of the electrically active impurities in the germanium crystal as it strongly influences the electric potential, Φ.The calculation of Φ is the first step in the simulation of an HPGe detector.Software packages like SolidStateDetectors.jl [9] (SSD), MJDSigGen [10] or the AGATA Detector Library [11] are able to perform a full simulation of HPGe detectors from field calculation to signal formation.It is crucial, though, to use the correct  to obtain the correct electric field, which influences the drift of the charge carriers and, thus, the formed pulses.Often, the impurity density is measured via the Hall effect at different heights of a drawn crystal ingot by cutting thin slices out of the ingot.However, there is a rather large uncertainty on these impurity measurements and, in addition, assumptions have to be made how the impurity density changes in between.Usually, only a linear or quadratic change of the impurity density between the bottom and the top of a cylindrically shaped crystal is assumed.If an incorrect model for  is assumed, wrong conclusions can be drawn in studies involving the subsequent parts of the simulation, e.g. in studies involving the mobility tensor.
An HPGe detector is, in principle, a  −  diode operated in reversed bias mode.The extent of the depleted volume for different bias voltages,   , depends on .The undepleted volumes are extensions of the detector contacts.Thus, the capacitance between those contacts depends on   and .Therefore,  can be studied by measuring the capacitance for different   , a C-V curve, and comparing it to simulated C-V curves for different .
Past work by Bruyneel et al. [12,13] has shown that it is indeed possible to determine impurity density parameters based on C-V measurements.It was, however, limited to a best-fit approach without uncertainty estimates and with very few free parameters.Moving beyond this to impurity density models that have many free parameters and fully exploring such parameter spaces is very challenging due to the high numerical cost: Simulating a C-V curve requires repeated and numerically expensive field calculations.Even with the GPU-accelerated implementation in SSD and using multiple GPUs in parallel, it takes a few minutes to calculate one C-V curve.It is, therefore, prohibitive to perform these calculations directly during parameter inference as a proper exploration of the parameter space would take a very long time.The novel method presented here circumvents this problem by replacing the exact capacitance calculations with a machine-learned approximation function.It comprises the following steps: 1. Definition of a model for  for a detector and including the allowed parameter space for its parameters, .The generation of  via field calculations can take a few days.Afterwards, the trained DN N can predict capacitances very quickly and with sufficient accuracy.This makes it possible to perform a Bayesian exploration of the full parameter space.Note that this method can be used for detector optimisation in general: not only to fit  but also to find optimal values for other design parameters of a detector.

Detector capacitance matrix
An HPGe detector with  ≥ 2 contacts can be seen as a system of  conductors which are capacitively coupled.A schematic of the capacitances for a system of two conductors is shown in Fig. 1.The mutual capacitance,    , between two conductors  and  is defined as where   is the change in charge on conductor  for a change in potential,   , of conductor .The mutual capacitance is symmetric:    =   .The self capacitance of conductor ,   , can be understood as a mutual capacitance where the other conductor is a grounded closed surface surrounding conductor .In absence of any surroundings, this surface can be imagined as a grounded sphere with an infinite radius.
In case of a typical HPGe detector in a grounded cryostat, the surface is defined through the grounded walls of the cryostat and the grounded parts of the holding structure of the detector.The different    are the elements of the so-called mutual capacitance matrix C.However, when working with a system of conductors, it is usually not very practical to work with C as the elements cannot be studied individually.Therefore, the so-called Maxwell capacitance matrix [14], is introduced which is generally more practical as it connects the potentials of all conductors, ì  = ( 1 , ...,   ), with the charges on all conductors, ì  = ( 1 , ...,   ) via The C * notation can be distinguished from the C notation through its negative off-diagonal elements with The elements of C * can be calculated [15] through the weighting potentials of the contacts, W  : where  0 is the vacuum permittivity and   the relative permittivity of the medium at position ì .The integral is over the closed system volume  W .
Since the  *   depend on W  and   (ì ), they depend on the geometry of the detector and its environment.If the detector is not fully depleted, the contacts which are touching these The weighting potential W  of contact  in an HPGe detector relates the position of a charge to the induced signal in the contact .undepleted regions become enlarged, which is an effective change of geometry.The enlargement of the contacts depends on  and   .Thus, for a detector in a fixed environment, the capacitances become dependent on two variables: The absolute values of  *   decrease with increasing   .The   at which the detector becomes fully depleted is called the full-depletion voltage,  fd  ().For   >  fd  (), the values of  *   basically do not change anymore.Thus, there are lower limits on the absolute values for the elements of C * for a given : It should be noted here that a capacitance is an electrostatic quantity and is not frequency dependent.The reactance is the quantity which introduces a frequency dependence.

The experimental setup K1 and the detector Super-Siegfried
The test stand K1 is a small vacuum chamber with a cooling finger submerged directly in a liquid-nitrogen dewar.The type true-coaxial HPGe detector Super-Siegfried [16] was mounted on a special base plate which fitted on the cooling finger inside K1.The detector, together with the necessary holding structure, is depicted in Fig. 2. Closely around the holding structure a so-called hat is placed on top of the base plate, see in Fig. 3.The hat has an inner radius of 55 mm, an inner height of 105 mm and serves as an infrared shield.The base plate, the holding structure and the hat are all grounded.
The detector has a length of  D = 70 mm and a radius of 37.5 mm.The borehole has a radius of 5 mm and at the top and the bottom; it widens to a radius of 10 mm within about 3 mm.The inner borehole of Super-Siegfried is the only  + contact and the mantle is divided into 19  + segments.For the measurements presented in this paper, the segments were not read-out separately, but were connected together into one single  + contact.The  + contact is lithium drifted and the  + segments are established through boron implantation.The manufacturer provided two values for the impurity level at the top and at the bottom of the detector: The operation voltage of the detector suggested by the manufacturer is 3000 V.
As the  + segments are connected to form only one contact, the capacitance matrix of the detector is a 2×2 matrix as shown in Fig. 1, where the  + and the  + contacts are the two conductors 1 and 2. The base plate, holding structure and hat

cm
Fig. 2 Super-Siegfried within its grounded holding structure as mounted on the grounded base plate.form the grounded shell around the two contacts.This is also shown in Fig. 4, which shows the schematic for the measurements of the mutual capacitance between the two contacts,  12 .This is similar to what has been described in [13].  is applied to the  + contact and the  + contact is held at ground over a termination resistor,  2 .A pulse generator is connected to the  + contact over its internal resistor,  PG , and an additional resistor  1 which serves in combination with  2 as a voltage divider.To measure  12 , rectangular pulses with an amplitude of  PG were generated and injected into the  + contact.This corresponds to a change in the potential  2 which translates into a change of charge on the  + contact,  1 , via Eq.( 3): Thus, given the electronic circuit, the measured capacitance,  m 12 , can be calculated for different   from the measured  1 (  ) as where the values for the different components are  PG = 50 Ω,  1 = 6190 Ω,  2 = 51.4Ω and  PG = 126 mV. 1 (  ) was extracted from the induced pulses in the  + contact as follows: The  + contact was connected to a chargesensitive preamplifier circuit typically used to read-out germanium detectors.The amplified signals were recorded with a sampling rate of 250 MHz and a pulse length of 20 µs by a Struck SIS3316 [17] analog-to-digital converter unit (ADC).The recorded pulses were inverted (⇒ − 1 becomes + 1 in Eq. ( 9)), corrected for the decay of the charge in the amplification circuit and were calibrated [18,19].The parameters of the decay correction and calibration of the read-out circuit (preamplifier together with the ADC) were determined from pulses of background gamma events of known energy of a measurement at   = 3000 V.An assumption made here is that the parameters of the decay correction and calibration are independent of   or, respectively,  12 .After the decay correction and calibration, recorded pulses are often given in units of energy.For this study, the values were converted into charge pulses, (), in units of charge, pC, based on the ionisation energy of germanium.
For   ≥  fd  , the recorded pulses are rectangular pulses and their amplitude corresponds to  1 .This is, however, not the case for   <  fd  .This is shown for three different   in Fig. 5. Partially depleted detectors have to be modeled differently [20] within an electric circuit as shown in Fig. 6.
The undepleted volume can be described as an additional RC component introducing also a frequency dependence to the signal and leading to longer pulses.This can be calculated for one dimensional systems [20].In reality, that is usually very complicated.However, here, it is not necessary as we are only interested in the total charge,   , flowing through the circuit which can be determined by fitting the tail of () with where   ,   and   are fit parameters.This is also illustrated in Fig. 5.

Measurement of the C-V curve
The detector was operated at 60 different bias voltages, Each measurement lasted 600 s.For lower  , , most of the observed pulses were induced by the pulse generator.
For increasing  , , more pulses induced by gammas from natural radioactivity were recorded as the depleted volume and, thus, the active volume of the detector increased.However, the peak created by the pulse generator was always clearly identifiable.The spectra of   of all pulses from all 60 measurements are shown in Fig. 7.For all measurements, the events induced by the pulse generator form a peak.The mean values of these peaks,  μ,m t, , were determined by fits of scaled normal distributions as shown in Fig. 7. Fig. 7 Histograms of  t for different   .The most dominant peak in each spectrum comes from pulser events.The   from pulser events decreases for increasing   .For higher bias voltages, also events from background events, mainly from environmental gammas, become visible in the range [0, 0.14] pC.The peak corresponding to events from 2614 keV gammas from the Tl 208 decay is labelled.Inset: The fit to the peak of pulser events at   = 25 V.The fitted parameter  μ,m t,=1 is shown as a vertical line.
Using Eq. ( 9), the determined  As mentioned earlier, in theory,  12 should not change anymore for   >  fd  .This does not take into account that the contacts are regions of the detector, which are doped more than three orders of magnitude higher than the bulk.With increasing   , also very small volumes of the contacts become depleted.However, for reasonable values of   below the break-through voltage, the contacts never become completely depleted and − m 12 /  never becomes entirely zero.This is demonstrated in Fig. 8b.Therefore, it is not trivial in general to define  fd  .Here, we define it as −Δ m 12 /Δ  != 10 −3 pF/V:  fd  ≈ 2600 V.For   > 2600 V, simulations are expected to result in a fully depleted detector for the correct .

Simulation of the C-V curve
The C-V curve for a given  was simulated with SolidStat-eDetectors.jl (SSD).Since version v0.7, SSD can be used to calculate C * of a detector while taking the influence of the environment into account.Since v0.8, it can also perform the required 3d field calculations for Φ and W  efficiently on GPUs.
In SSD, Φ and W  are calculated by solving Gauss's law on adaptive 3d (cylindrical or Cartesian) grids via the iterative successive over-relaxation (SOR) algorithm [19].As the potentials are calculated on grids, the integral in Eq. ( 5) is turned into a sum over the grid of W  .The gradient of W  is determined through interpolation onto the grid of W  as the two final grids are usually not identical due to the adaptive grid refinement.
In order to calculate the elements of the matrix via Eq.( 5), Φ and all W  have to be calculated first.In contrast to W  , Φ does not occur in Eq. (5).It is, however, required to determine the depleted volume which depends on  and   [19].This information is then passed to the calculation of the weighting potentials.In the calculation of the weighting potentials, the relative permittivity,   , inside undepleted volumes was scaled by 10 5 as an approximation of infinity.This makes these areas quasi-conductive and results in equal potentials over these volumes.Thus, undepleted volumes in touch with a contact become extensions of this contact and will be on the same potential as applied to the contact.
The detector Super-Siegfried and the grounded holding structure and base plate as implemented in SSD are shown in Fig. 9.For the simulations, a cylindrical grid was chosen and the grid was limited to the dimensions of the hat.Fixed boundary conditions of 0 V were set at the outer edge in  and at both edges in  to mimic the grounded closed shell of the hat and base plate.In , the grid was limited from 0 • to 120 • and periodic boundary conditions were set.

mm 70 mm
Fig. 9 The geometry of the HPGe detector Super-Siegfried (dark grey), together with its holding structure and base plate (light grey) as implemented in SSD.The blue lines are segment boundaries of the  + contact and the red cylinder is the  + contact.
It is very important to define the geometry of the detector, especially the geometry of the contacts, as realistically possible as it also influences the capacitance.Particularly,   12 depends on the exact geometry.For all simulations presented in this paper, the  + contacts of the detector were fixed to a thickness of 0.5 µm as measured previously [19].Lithium drifted contacts typically have thicknesses of O (mm).This thickness was not measured previously since irradiation of the inner borehole is not simple to achieve.Since the  + contact thickness,  Li , is on the mm scale, it impacts   12 on a measurable scale.The  + contact geometry is implemented as a tube with the inner radius being fixed at the borehole of the detector.The thickness of the tube,  Li , is a free parameter in the fit presented in this paper.The contact does not cover the widening of the borehole.At the bottom and the top of the contact, the outer edge of the tube is rounded off.
In the field calculation, the potential values of grid points inside the defined volumes of the contacts are fixed to the potential applied to the corresponding contact.In principle, the  + contact could instead be modeled through .The current implementation of a fixed potential inside the volume corresponds to a jump from the bulk impurity density to infinity at the surface of the contact volume as shown in Fig. 10.In future, it is envisioned to smooth this hard edge transition by adding some continuous function to the impurity at the  + contact .This smooth edge transition is also shown in Fig. 10.However, this is not yet part of the studies presented in this paper.In SSD, custom signed impurity-densities can be defined where the sign of the given density determines the sign of the fixed space charges of the minority charge carriers at the specific location.Thus, the sign is used to specify the The  + contact thickness is on a much smaller scale, O (µm), and, thus, it would not be feasible to resolve a smooth edge transition on that scale.type ( or ) of the semiconductor at a specific location, ì  = (, , ): (ì ) < 0 cm −3 ⇔ −type region .
Typically, in simulations of HPGe detectors, a simple linear or quadratic change of  is assumed along the crystal pulling axis , based on certain levels of impurity provided by the manufacturer for some  values.A radial component is usually not assumed.Thus, for the detector Super-Siegfried, the signed impurity-density model based on manufacturer values becomes which comprises a linear profile in  and no modulation in .
The depleted volumes for  M and  Li = 1 mm were calculated with SSD for all  , .The undepleted volumes are shown in Fig. 11 for selected  , .It shows how the detector depletes from the mantle of the detector towards the borehole.It also shows that even for a bias voltage of 2975 V the detector does not become fully depleted.Thus, the overall impurity level of  M seems to be too high since  fd  was determined to be ≈ 2600 V.The simulated C-V curve,  s,   12 , in comparison to the measured C-V curve, is shown in Fig. 12.An uncertainty of is assigned to  s, 12 .The absolute uncertainty of 1 pF is motivated by a possible imperfect implementation of the geometry in comparison to reality.The 1% relative uncertainty is motivated by studies on the simulated capacitance for different levels of the fineness of the final grids of the calculated fields.A relative uncertainty is chosen for this source of uncertainty because the depleted volume is smaller for lower bias voltages (larger capacitances) requiring a finer refinement.In SSD, the refinement of the grids in the field calculations can be tuned  Figure 12 shows that the simulation predicts that the detector is not yet fully depleted for any  , as the simulated C-V curve still decreases and does not reach its lower limit at  ,60 .However, at  ,60 , the simulated capacitance is lower than the measured capacitance limit  , 12 =  m,60 12 .This indicates that  Li might be larger or that the implemented geometry does not perfectly describe reality.For low bias voltages,  s, 12 is larger than  m, 12 .This means that the detector depletes faster in reality than in the simulation, indicating that the  M is too large at larger radii.
Figure 13 also shows that a logarithmic change in  c between   and  u results in a change of  s 12 on a linear scale.Therefore, the (, )-dependent signed impurity-density,  RZ (, |), is defined as where  is set to 1000 for this study and (, ) is a function to and above −  .model  RZ on a logarithmic scale.The sign of (, ) is used to determine the type of the semiconductor at (, ), see Eq. ( 12) and Eq. ( 13).
According to the simulation, the measurement is not sensitive to impurity densities below   and above −  .A parameter transformation, T , is defined to model  RZ between The same sensitivity is assumed for -type densities as for the inverted system, inverted charge distribution and contact potentials, the same values for the capacitances will be calculated.
[− u ,  u ] and significantly reduce the influence of the parameter interval [−  ,   ]:  18), on the left axis and the parameter transformation  = T ( ), see Eq. ( 20), on the right axis as a function of .
The spatial dependence of the model  RZ (, ) is implemented as a spatial dependence of : where the  dependence is modeled by  bot () at  = 0 mm and  top () at  =  D .Both,  bot () and  top (), are modeled as two cubic splines defined for four specific radial positions  b,1 = 20 mm,  b,2 = 28 mm,  b,3 = 33 mm and  b,4 = 37.5 mm.The gradient of the splines at their left boundary,  b,1 , is set to zero, the gradient at the right boundary,  = 37.5 mm, is not fixed.Thus, the model is defined by a set of 8 parameters,   , either defined in impurity levels or in values of , as they can be transformed into each other via Eq.( 18) to Eq. ( 20) and their inverse functions.An example impurity distribution of  RZ based on the values provided by the manufacturer is shown in Fig. 15 with The model allows to modulate the bulk impurity density of a detector including a possible boundary between -type and -type volumes as demonstrated with the example density, see Fig. 15.There, the main bulk of the detector is -type but is -type close to the mantle.For this detector, such a -type volume close to the mantle is motivated by two aspects.First, when pulling the crystal via the Czochralski method, there could be some radial modulation of the impurities due to the process.Especially, since natural germanium is -type and -type dopants have to be added to the molten germanium.Secondly, the  + contacts are heavily over-doped layers, O (10 12 cm −3 ) or even higher.The thickness of the undepleted boron layers is very small, 0.5 µm, but there could be diffusion.This could lead to impurities reaching the magnitude of the bulk densities , O (10 8−10 cm −3 ), penetrating deeper, O (mm), into the -type germanium leading to compensation and type conversion.

Deep neural network for fast capacitance predictions
For a given set of values for ( Li ,    ), a C-V curve can be calculated with SSD and, in principle, a fit to the measured C-V curve could be done.However, for each  , three 3d field calculations would need to be performed resulting in 180 field calculations for the whole measured C-V curve.Even though each set of three 3d field calculations (with the specified refinement settings) takes less than a minute on a GPU in SSD, it would not be feasible to set up an optimiser or a Bayesian fit.
Therefore, a deep neural network, DN N , was developed which is able to predict the capacitance,  p 12 , for a set of parameters ( Li ,    ) much faster, O(µs).Here,   is added to the input parameters of the model.Even though this adds an extra dimension to the model, it simplifies the output as only one capacitance is predicted instead of a whole C-V curve: The impurity model parameters  D form a 10-dimensional parameter space P D .The following uniform distributions U (, ) were chosen to quasi-randomly draw sets of input parameters for the generation of the training and test capacity datasets: The distribution of  Li is chosen based on typical lithium layer thickness values.  has to cover all values of  , .As Super-Siegfried is an -type germanium detector, the range of the four parameters describing the density towards the borehole, at the top and bottom, are limited to the  region corresponding to an -type density.The four parameters describing the density towards the mantle at the top and bottom, however, are allowed to include -type impurity levels.
Table 1 Prior distribution for all parameters from B Z .See main text for reasoning.

Prior distribution 𝑑
The marginalised posterior distribution and the prior distribution of  Li is shown in Fig. 20.The fit indicates a thickness of about 3 mm, which is quite thick.However, this parameter of the model is mainly sensitive to the end of the C-V curve and probably heavily impacted by possible imperfections of the implementation of the geometry of the detector.In addition, the detector is old and some growth of  Li is expected.
All truncated distributions presented in this paper are cut off at the endpoints of the interval of the respective parameter space., broad truncated normal distributions centred around 0 were chosen based on the conclusions drawn from B Z , that assumed impurities in B Z were too high at larger radii.The electric field strength close to the contacts differs significantly for the three cases.The field strength from  B Z is significantly less radius dependent than that from  * M .This causes the pulse to become faster at the end.The radial decrease of impurities in  B RZ further reduces | ì E | close to  + contact.The field strength close to the  + is further increased.Nevertheless, the pulses for  B Z and  B RZ are very similar.
However, the effect of  on the simulated pulses also depends on the charge drift model describing the mobility tensor and its dependence on the electric field.For the simulated pulses shown in Fig. 29b, the charge drift model from the AGATA Detector Library [11,28] as implemented in SSD [9] was used with its default parameters.The differences in the pulses for the three  models show how important it is to use the correct impurity density distribution when using pulse shapes from measurements to tune the parameters of the drift model.

Summary and Outlook
The capacitance matrix of a germanium detector was explained in detail and it was shown how the capacitances depend on the depletion of the detector and, thus, on the impurity density distribution of the crystal.The setup K1 and the true-coaxial -type germanium detector Super-Siegfried were introduced and it was explained how to measure one of the elements of the capacitance matrix of the detector for different bias voltages.The measured C-V curve was compared to a C-V curve simulated for the impurity density distribution as provided by manufacturer.The comparison suggested a radial dependence of impurity densities.This was confirmed by a Bayesian fit which optimised the impurity density model with only a dependence on the -axis of the detector.A model including a radial dependence of the impurity density was introduced.The Bayesian fit of this model to the measured C-V curve provided a good description of the data.This indicates that the crystal under study really has an  dependent impurity density distribution with a very low level of electrically active impurities close to the detector edge.
A novel method was introduced that uses a deep neural network, trained on GPU-accelerated capacitance calculations, to enable full Bayesian parameter inference on complex impurity density models.
The possibility to determine impurity density distributions from capacitance measurements opens a road to study mobility tensors and drift models by comparing measured and simulated pulses without the uncertainties otherwise introduced by the lack of knowledge on these impurity densities.The knowledge of the impurity densities is also important for pulse-shape analysis used in rare-event searches where the exact understanding of the pulse formation is critical to discriminate between signal and background events.
It should be noted that the method presented here can also be used to optimise general detector properties during the detector design phase.In addition, the method has the potential to determine impurity distributions based on impuritysensitive detector properties other than capacitance.Inferring impurity from voltage-dependent properties like the shape of the depletion volume, determined by Compton scanning, or the total active volume will be the subject of future work.

22 Fig. 1
Fig. 1 Schematic of the capacitances for a system of two conductors on potentials   with charges   together with the corresponding mutual, C, and Maxwell, C * capacitance matrices.

Fig. 4
Fig. 4 Schematic of the electronics used for the measurements of  12 of Super-Siegfried in K1.

Fig. 5
Fig.5 Recorded response of the  + contact of Super-Siegfried in K1 to generated rectangular pulses injected into the  + contact for different   .The zoom-in plot shows the fit of  (), see Eq.(10), to the tail of the pulse for   = 25 V.

Fig. 6
Fig. 6 Schematic of the electronics describing a) a fully depleted and b) an only partially depleted detector with only two contacts.The depleted part is modeled with  12 while the undepleted part is modeled as an RC circuit with resistance  u and capacitance  u .

Fig. 8 a
Fig. 8 a) C-V curve of Super-Siegfried as measured in K1.The error bars represent uncertainties conservatively determined by assuming 3% uncertainties on  μ,m t, ,  PG ,  1 ,  2 and  PG and using Gaussian error propagation.The uncertainties are highly correlated.b) Negative differences, −Δ m 12 /Δ  , between the data points in a).

Fig. 10
Fig.10 One-dimensional illustration of the envisioned modelling of the  + layer for future simulations with SSD via a continuous increase of the impurity density from the impurity density of the bulk,  bulk , to the impurity density of the  + contact.The hard transition corresponds to the current implementation of the  + layer in SSD, where the potential values inside the contact volume are fixed to the set contact potential.

Fig. 11
Fig. 11 Cross-section of Super-Siegfried mounted in K1 at  = 36.7 • showing the undepleted volumes for  M and selected  , in steps of 250 V (200 V for the last step to 2975 V) as differently shaded areas.

Fig. 12 C
Fig. 12 C-V curve,  s, 12 , as simulated with SSD for  M together with the measured C-V curve,  m, 12 , already shown in Fig. 8a.

Fig. 15
Fig. 15 Example impurity distribution  RZ for   ,E as a function of a)  and b)  .See text for details.
− u ,  u ] ,   ∼ U (10 V, 3000 V).Note that the distributions of bot/top  are not uniform due to the nonlinear transformation defined between  bot/top  and  bot/top  .

Fig. 24 𝜁
Fig. 24  RZ for the global mode from B Z as a function of a)  and b)  .

Fig. 28
Fig.28 Global mode of the fitted impurity density of B RZ .

Fig. 29
Fig. 29 a) The electric field strength at ( = 30 • , z = 35 mm) over  and b) normalised pulses of the  + contact of an event spawned at (r = 37 mm,  = 30 • , z = 35 mm) as simulated with SSD for the three different impurity density distributions  BZ ,  BRZ and  * M .
2. Quasi-random generation of  parameters sets:  = {  } for  ∈ [1, ]. 3. Calculation of the capacitance,   , via SSD for each element of :  = {  } for  ∈ [1, ]. 4. Training of a deep neural network, DN N , on the generated data set: (| ). 5. Bayesian inference of the model on a measured C-V curve using the trained DN N .

Table 2
Prior distribution for all parameters from B RZ .See main text for reasoning.