Characterization of silicon drift detectors with electrons for the TRISTAN project

Several 7-pixel SDD arrays have been manufactured at the semiconductor laboratory of the Max Planck Society (HLL) [27]. The chips are produced from monolithic silicon wafers with a thickness of 450 μm. They feature a thin entrance window, terminated by a 10 nm thick SiO₂ layer. The hexagonal pixel shape enables an arrangement without dead area. For this work SDD chips with 2 mm pixel diameter, shown in figure 1, were used. Each of these pixels features twelve drift rings and a small anode capacitance of approximately 100 fF.

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Each anode is wire bonded to a charge-sensitive preamplifier application-specific integrated circuit (ASIC), situated in close vicinity to the chip. The 'CUBE' ASIC has been developed by Politecnico di Milano and XGLab for applications in high-count rate and low-noise spectroscopy [6, 27]. Its field effect transistor is based on complementary metal oxide semiconductor technology and operates in pulsed-reset mode.

The 'DANTE' digital pulse processor is used as a back-end electronics. DANTE is provided by XGLab and features a waveform-digitizing analog-to-digital converter with a sampling frequency of 125 MHz and 16 bit resolution. Two trapezoidal filters are applied for event triggering and energy reconstruction, optimizing the system for high-count-rate applications [24]. Characterization measurements of the TRISTAN SDD with a ⁵⁵Fe x-ray source demonstrated an energy resolution of 139 eV (FWHM) at 5.9 keV with trapezoidal filter peaking times of about 1 μs [32] and a detector temperature of −30 °C.

⁸³Rb/^83mKr source. The krypton calibration source consists of a mono-layer of ⁸³Rb, evaporated onto a highly oriented pyrolytic graphite carrier substrate [38]. ⁸³Rb decays with a half-life of 86.2 days via electron capture to ^83mKr. In this decay, predominantly Kr–K_α and Kr–K_β x-rays are emitted. The occurrence of both photon and electron lines in one spectrum enables an in situ calibration and characterization at the same time. The isomeric state ^83mKr is the second excited state of krypton and has an energy of about 41.6 keV [29]. It deexcites via a cascade of two γ-decays with 32.2 keV and 9.4 keV, respectively. In both transitions, conversion electrons occur with energies

$$\begin{equation}{E}_{\mathrm{c}\mathrm{e}}={E}_{{\gamma}}+{E}_{{\gamma}}\left(\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{i}\mathrm{l}\right)-{E}_{\mathrm{c}\mathrm{e}}\left(\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{d}\right)-{E}_{\mathrm{c}\mathrm{e}}\left(\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{i}\mathrm{l}\right),\end{equation} \tag{ 1 }$$

defined by the energy of the γ-transition E_γ , the binding energy of an electronic shell of the krypton atom E_ce(bind), and the recoil energies of the atom after the emission of the γ-ray E_γ (recoil) and the conversion electron E_ce(recoil). The emission of a conversion electron from the K-shell (E_ce(bind) = 14.3 keV) is energetically forbidden in the second deexcitation. Photon and electron peak energies of specific interest for this work are listed in table 1. All other lines are situated in a low-energy continuum and thus unsuited for the investigation.

During the measurement, the source was positioned about 1 cm underneath the detector entrance window, such that the entire pixel array was illuminated homogeneously. A typical spectrum of ⁸³Rb/^83mKr measured with the TRISTAN SDD detector is shown in figure 2. While the photon peaks are symmetric, peaks from conversion electrons show a pronounced low-energy tail. Furthermore, the electron peak position ${\bar{E}}_{i}$ of a peak i is shifted towards lower energies compared to the respective theoretical value ${\bar{E}}_{i}^{\mathrm{t}\mathrm{h}}$:

$$\begin{equation}{\bar{E}}_{i}={\bar{E}}_{i}^{\mathrm{t}\mathrm{h}}-{{\Delta}}_{i}^{\mathrm{e}\mathrm{w}}-{\delta }_{\mathrm{s}\mathrm{c}}-{\delta }_{{\Phi}}.\end{equation} \tag{ 2 }$$

This shift is due to energy losses in the entrance window ${{\Delta}}_{i}^{\mathrm{e}\mathrm{w}}$ of the detector, potential energy losses in the source δ_sc, and the potential difference between source and entrance window δ_Φ [26].

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Scanning electron microscope. SEM are generally used to visualize small structures, which cannot be resolved optically. In this work the SEM was used to generate mono-energetic electrons as calibration source for the TRISTAN SDD detector.

An image area is repeatedly scanned with an electron beam of about 10 nm diameter ¹⁵ . The electrons are accelerated to kinetic energies of up to 30 keV and are focused by a fast-changing magnetic field onto the sample inside a vacuum chamber. The beam intensity is determined by the temperature of a heated tungsten spiral, from which the electrons are emitted. The TRISTAN detector was positioned on the sample holder and electrically connected to the DAQ system via a feedthrough in a flange of the chamber. First investigations with a single channel detector have shown a good applicability of this method [20]. During the measurement of the 7-pixel detector, the beam rapidly scanned over the entire array, averaging the response over all positions of beam incidence. A typically recorded electron energy spectrum is displayed in figure 3.

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Main peak. The majority of electrons deposit almost their entire initial energy in the sensitive volume of the detector, leading to a main peak in the energy spectrum. Its general shape is well approximated by a Gaussian function

$$\begin{equation}{I}_{\mathrm{G}}\left(E\right)={A}_{\mathrm{G}}\cdot \mathrm{exp}\left(-\frac{{\left(E-\mu \right)}^{2}}{2{\sigma }^{2}}\right),\end{equation} \tag{ 3 }$$

where A_G is the amplitude, μ is the mean and σ is the standard deviation.

Low-energy tail. The entrance-window surface is covered with a 10 nm thick silicon oxide (SiO₂) layer. Charge deposited in this layer cannot be detected. Moreover, the electric fields in the silicon volume close to this layer are too weak to efficiently transport the charge carriers to the read-out contact. These undetected energy depositions lead to an asymmetry of the main peak at its low-energy shoulder, which is modelled with the following function:

$$\begin{equation}{I}_{\mathrm{D}}\left(E\right)={A}_{\mathrm{D}}\cdot \mathrm{exp}\left(\frac{E-\mu }{\beta }\right)\left(1-\text{erf}\left[\frac{E-\mu }{\sqrt{2{\sigma }^{2}}}+\frac{\sigma }{\sqrt{2}\beta }\right]\right).\end{equation} \tag{ 4 }$$

It is composed of a 'washed-out' step function expressed by the error function and an exponential tail towards lower energies. Amplitude and slope of the function are given by A_D and β, respectively.

Silicon escape peak. Incident radiation leads to the ionization of silicon atoms in the detector material, most often on the K-shell. In the subsequent K_α de-excitation, an x-ray with ΔE_esc = 1.74 keV is emitted. If this photon leaves the detector, the energy ΔE_esc remains undetected. Hence, the silicon escape peak is modeled as a scaled projection of the main peak with amplitude A_esc, shifted towards lower energies by ΔE_esc:

$$\begin{equation}{I}_{\mathrm{e}\mathrm{s}\mathrm{c}}\left(E\right)={A}_{\mathrm{e}\mathrm{s}\mathrm{c}}\cdot \mathrm{exp}\left(-\frac{{\left(E-\left[\mu -{\Delta}{E}_{\mathrm{e}\mathrm{s}\mathrm{c}}\right]\right)}^{2}}{2{\sigma }^{2}}\right).\end{equation} \tag{ 5 }$$

Backscattering tail. A fraction of primary and secondary electrons scatter back from the detector surface or escape after incomplete energy deposition. The backscattering probability for electrons with energies of around 20 keV is about 20% at perpendicular electron incidence and increases with the incident angle [17]. This effect leads to a backscattering tail, dominating the spectral shape between silicon escape peak and detection threshold. The probability for an electron to be backscattered is the largest at the detector surface and decreases with increasing penetration into the detector material. Consequently, the backscattering tail rises towards lower energies. We describe this tail with a multiplication of two power functions given by

$$\begin{equation}{I}_{\mathrm{B}}\left(E\right)={A}_{\mathrm{B}}\cdot {\left(\frac{E}{\mu -a}\right)}^{b}\cdot {\left(1-\frac{E}{\mu }\right)}^{c}.\end{equation} \tag{ 6 }$$

The first term with exponent b describes the low-energy region just above the threshold energy a, whereas the second term with exponent c describes the higher end of the backscattering tail. Parameter A_B is the overall amplitude of the function.

Each mono-energetic electron peak measured with ^83mKr or at the electron microscope is fit with the sum of the terms described above. In the case of the K-32 and M-32 peaks, the terms I_B(E) and I_esc(E) are not considered, as the fit is only performed in a region close to each conversion electron peak because various peaks are overlapping. An example of the fit to the K-32 peak is shown in figure 4(a). The fit to the obtained spectrum at the electron microscope measurement is displayed in figure 4(b).

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A very good agreement of the empirical model with the data is found. The dependence of the model parameters on both energy and angle can be investigated by measuring the response at various energies and incident angles. For the analysis of the continuous tritium β-decay spectrum, it is conceivable to use this empirical description of the detector response to obtain a model of the measured tritium spectrum. To this end, the mono-energetic spectra are combined into a response matrix, which is then multiplied by the theoretical β-decay spectrum. A parameterization of the response is particularly advantageous as it allows one to easily include systematic uncertainties in the data analysis. The feasibility of this approach has been demonstrated in [11].

To eliminate possible influences of the bias voltage δ_Φ and source effects δ_sc on the peak position shift (see equation (2)), the tilted beam method is applied [23]. By tilting the detector relative to the electron source by an angle α, the effective dead-layer thickness for incoming electrons increases. This concept is illustrated in figure 5. By comparing measurements with and without tilt angle, the influence of the entrance window ${{\Delta}}_{i}^{\mathrm{e}\mathrm{w}}$ is isolated as a relative shift of the main energy peaks:

$$\begin{equation}{\Delta}E=E\left(\alpha \right)-E\left({0}^{{\circ}}\right).\end{equation} \tag{ 7 }$$

Measurements were performed with the SEM and the ⁸³Rb/^83mKr source with perpendicular electron incidence (α = 0°) and tilted detector by α = 60°. The measured spectra are fit with the empirical response model described in section 3.1. The resulting peak position differences ΔE are calculated and illustrated in figure 6(a). For a 14 keV electron ΔE = 50 eV, and, as expected, the energy loss decreases to about ΔE = 30 eV at incident energies of about 30 keV.

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In this study we describe the region of incomplete charge collection at the entrance window with a single parameter, the dead-layer thickness d_DL. To relate the observed shift in energy ΔE to a dead-layer thickness, MC simulations of electrons are performed with two incident angles of α = 0° and α = 60° and for several incident energies E_i. For each case the simulation is performed with eleven different dead-layer thicknesses in a range of 40–60 nm in steps of 2 nm. The 10 nm thick SiO₂ layer is included in this dead layer model. The simulations are performed with the KESS software, which was developed by the KATRIN collaboration specifically to describe scattering of low-energy electrons in silicon [13–16, 36].

Figure 6(b) shows a good agreement of the measured and a MC simulated spectrum. A minimization of the squared data-to-simulation residuals for each measured energy yields the best dead-layer thickness and uncertainty of

$$\begin{equation}{d}_{\mathrm{D}\mathrm{L}}=\left(49{\pm}3\right)\enspace \text{nm}.\end{equation} \tag{ 8 }$$

Using this dead-layer thickness, an overlay of the simulated and measured energy shifts is illustrated in figure 6(a).

To reach the targeted sterile-neutrino sensitivity an energy resolution of approximately 300 eV at 20 keV is required [34]. An excellent energy resolution is needed to avoid washing out the characteristic kink-like signature of a sterile neutrino. The ability to detect or rule out this local signature makes the search for sterile neutrinos robust against large classes of systematic uncertainties.

Energy loss in the dead layer is a statistical process and thus different for each incident electron. This variation leads to a smearing of the spectrum and can be interpreted as a worsening of the energy resolution. Figure 7 illustrates the resulting additional contribution to the energy resolution for different dead-layer thicknesses. This contribution is especially large at low energies, where the energy loss in the dead layer is the largest. With the measured effective dead-layer thickness (see equation (8)) the requirement on energy resolution is met.

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To reach ppm-level sensitivity to sterile neutrinos, systematic effects which influence the spectral shape need to be described with high precision. As presented in this work, the shape of the response of SDDs to electrons depends on the entrance-window thickness and the backscattering probability. To estimate the impact of an uncertainty of these properties, we performed a sensitivity study based on a semi-analytical detector model [25].

For the study, we assume a differential measurement with total statistics of 10¹⁸ electrons (corresponding to three years' data taking with KATRIN at a 100-fold reduced column density). The detector response function is based on multiple interpolated MC simulations with KESS. The individual MC simulations were performed in a fine grid of various incident energies and incident angles. For this study, the detector response model takes into account the effect of the so-called post-acceleration (PA) electrode. This electrode, boosts the kinetic energy of all electrons by up to 20 keV on their way to the detector [5]. ¹⁶

We investigate the impact of the uncertainty of a parameter of the response function (e.g. the entrance-window thickness), by generating 10³ MC samples of the measured spectrum, each time varying the parameter of interest. From these MC samples the variance of all data points and their covariance is deduced, i.e. the covariance matrix C is constructed.

The sensitivity of the experiment is derived by minimizing the squared data-to-model residuals for 40 × 40 grid points in the (m_s, sin²(Θ))-plane:

$$\begin{equation}{\chi }^{2}\left({m}_{\mathrm{s}},{\mathrm{sin}}^{2}\enspace {\Theta}\right)={ \overrightarrow {r}}^{\mathsf{T}}{C}^{-1} \overrightarrow {r},\end{equation} \tag{ 9 }$$

$$\begin{equation} \overrightarrow {r}\equiv \overrightarrow {r}\left({m}_{\mathrm{s}},{\mathrm{sin}}^{2}\enspace {\Theta}\right)={ \overrightarrow {R}}_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}\left({m}_{\mathrm{s}},{\mathrm{sin}}^{2}\left({\Theta}\right)\right)-{ \overrightarrow {R}}_{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{t}\mathrm{h}}\left(0,0\right),\end{equation} \tag{ 10 }$$

where ${ \overrightarrow {R}}_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}$ depicts the model expectation in case of a sterile neutrino with mass m_s and mixing amplitude sin²(Θ), and ${ \overrightarrow {R}}_{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{t}\mathrm{h}}$ depicts the MC truth, for which no sterile neutrino is assumed.

At each grid point a minimization with respect to the spectrum normalization N, a constant background rate B and the spectrum endpoint E₀ is performed. The sensitivity at 90% C.L. is given by the contour of ${\Delta}{\chi }^{2}={\chi }^{2}-{\chi }_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}^{2}=4.6$, where ${\chi }_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}^{2}$ is the χ² value found for the case (m_s, sin²(Θ)) = (0, 0).

In the first study we investigate the impact of an uncertainty on the entrance-window thickness of 10%. As shown in figure 8(a), a 10% uncertainty on a 50 nm thick dead layer reduces the sensitivity by up to a factor of ten in certain mass ranges. The effect is significantly reduced for a smaller dead-layer thickness of 10 nm. Moreover, the effect can be almost fully eliminated by applying a PA energy of 20 keV, as the increased energy of the β-electrons reduces the relative fraction of energy loss in the dead layer.

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In the second study we test the impact of an uncertainty of the backscattering probability. This uncertainty can for instance arise from an uncertainty of the incidence angle of the electrons, which itself could arise from an uncertainty of the exact orientation of the detector. For this study, we emulate this effect by assuming electrons with an incident angle of (0 ± 5)°. This translates to an uncertainty in the backscattering probability of 0.4% [36] as well as in the experienced dead-layer thickness of 0.38% (see figure 5).

The result, displayed in figure 8(b) shows, that an uncertainty on the backscattering probability of 0.4% significantly reduces the sensitivity of the experiment. Again, the PA of electrons can fully mitigate the effect as high-energy electrons reach further into the detector and are less likely to backscatter.