## 2 Design of Detector Parameters
### Resistive Layer  
The resistive layer and the readout anode can be viewed as a distributed two-dimensional RC network in the finite element approximation, as depicted in Figure 2, where \(R_{s}\) is the sheet resistance of resistive layer and \(C_{total}\) is the total equivalent capacitance between the resistive layer and the anode. \(\rho\) (\(x\),\(y\)) indicates the point where the electron cloud lands on the resistive layer. \(C_{1}\) to \(C_{N}\) represent the capacitances between the germanium film and each anode strip, respectively. The total capacitance \(C_{total}\) is dependent on capacitances \(C_{1}\) to \(C_{N}\) and the inter-electrode capacitances \(C_{\bar{y}}\) (\(i=1\) to \(N\), \(j=1\) to \(N\), \(I\
eq j\)) between anode strips. Additionally, the readout circuits are presented by a charge-sensitive amplifier (CSA) in the red dotted box.  
The induced charge on the anode is affected by the sheet resistance of the resistive layer. If the sheet resistance is too low, the transmission to the electrode decreases, whereas a too-high sheet resistance results in the decrease in resolution and count rate because of the electron cloud overlapping. When the electron cloud diffuses on the resistive layer, the charge density distribution can be expressed as [14]:  
\[\rho(r,t)=\frac{Ne}{2\pi(2t/\tau+\sigma^{2})}\exp\left(-\frac{r^{2}}{2(2t/\tau +\sigma^{2})}\right) \tag{1}\]  
where, in this case, \(\tau=R_{s}C_{total}\). N is the number of incident electrons. Clearly, the variation of the density distribution function of the charge cloud is related to the broadening width \(\sigma\) and time constant \(\tau\). The charge cloud broadening \(\sigma\) can be adjusted by changing the gap distance and gap voltage, which is generally about 0.5 mm. Given that \(T_{rc}\) is the duration of the electron cloud on the resistive layer, which satisfies \(\rho(0,T_{rc})=0.2\times\rho(0,0)\). To avoid overlapping, \(T_{rc}\) needs to satisfy \(t<1/(5\times N_{cuts})\); \(N_{cuts}\) is the counting rate of detector. As much energy as possible needs to be transmitted to the anode to achieve high energy utilization. A sheet resistance of 50 M\(\Omega\) is finally chosen to satisfy the above requirements.  
The width of the induced charge cloud footprint at the plane of the anode obviously must be optimized to match the XS anode. Indeed, a too-narrow footprint results in an under-sampled charge distribution. At the same time, a too-wide charge footprint leads to charge division between a large number of anode electrodes and, consequently, to the reduction in the signal-to-noise ratio (especially at the edge of the charge distribution). The charge density on the resistive layer as a function of radius for different substrate thickness is shown in Figure 3. It is seen that the thicker the substrate thickness, the more dispersed the induced charge footprint, which prevents accurate centroid decoding. The substrate thickness of 0.5 mm is found to be optimal as five to seven strips must be covered.  
Figure 2: A lumped-parameter presentation of the readout mode for charge induction. \(\rho\)(\(x\),\(y\)) is the electron cloud on the resistive layer. R\({}_{s}\) is the sheet resistance of the resistive layer. \(C_{1}\), \(C_{2}\), \(C_{3}\), to \(C_{N}\) are the capacitance between the resistive layer and each anode strip, respectively. \(C_{12}\), \(C_{23}\), \(C_{13}\) to \(C_{1N}\), \(C_{2N}\), \(C_{3N}\) are the inter-electrode capacitance for each anode strip. \(C_{total}\) is the capacitance between the resistive layer and the virtual ground.