Fast calculation of capacitances in silicon sensors with 3D and 2D numerical solutions of the Laplace's equation and comparison with experimental data and TCAD simulations
aa r X i v : . [ phy s i c s . i n s - d e t ] O c t Fast calculation of capacitances in silicon sensors with 3Dand 2D numerical solutions of the Laplace’s equation andcomparison with experimental data and TCAD simulations
P. Assiouras, 𝑎, P. Asenov, 𝑎 A. Kyriakis 𝑎 and D. Loukas 𝑎 𝑎 Institute of Nuclear and Particle Physics (INPP), NCSR DemokritosAgia Paraskevi, Greece
E-mail: [email protected]
Abstract: We have developed a software for fast calculation of capacitances in planar silicon pixeland strip sensors, based on 3D and 2D numerical solutions of the Laplace’s equation. The validity ofthe 2D calculations was checked with capacitances measurements on Multi-Geometry Silicon StripDetectors (MSSD). The 3D calculations were tested by comparison with pixel sensors capacitancemeasurements from literature. In both cases the Laplace equation results were compared withsimulations obtained from the TCAD Sentaurus suite. The developed software is a useful tool forfast estimation of interstrip, interpixel and backplane capacitances, saving computation time as afirst approximation before using a more sophisticated platform for more accurate results if needed.
Keywords:
Particle tracking detectors (Solid-state detectors); Simulation methods and programs;Si microstrip and pad detectors; Detector modelling and simulations II (electric fields, chargetransport, multiplication and induction, pulseformation, electron emission, etc) Corresponding author. ontents
Silicon sensors are extensively used in High Energy Physics experiments as tracking detectorsof charged particles. The most commonly used planar silicon detectors in High Energy Physicsexperiments are devices segmented into strips (micro-strip detectors) or pixels (micro-pixel detec-tors). Important parameters in the operation and the design of silicon detectors are the capacitancesbetween adjacent strips or pixels and between the strips or pixels and the backplane. These ca-pacitances are related to signal to noise ratio as well as crosstalk phenomena between neighboringpixels or strips.A numerical algorithm for solving the three dimensional Laplace’s equation and calculatingthe capacitances of a pixel sensor, was presented in [1]. A reduced form of the algorithm hasbeen implemented for calculating the capacitances of micro-strip sensors by solving the Laplace’sequation in two dimensions. Through these algorithms, numerical calculations of the capacitancesbetween adjacent strips or pixels as well as the capacitances between the strips or pixels and thebackplane can be made. These algorithms have been implemented within a software that can beused as a simulation tool for a fast estimation to lower order of the above mentioned capacitances.– 1 –alculations for the pixel sensors obtained with our method are compared with published ex-perimental data and TCAD simulations on various configurations of pixel geometries. Calculationsfor strip sensors, are compared with experimental results and TCAD simulations on multi-geometrystrip sensors (MSSD). The MSSD sensors were kindly provided by the Outer Tracker Sensorworking group for the Phase-2 upgrade of the CMS/LHC collaboration [2].
In the current section we give an outline of the method, for a detailed presentation see [1]. Akey characteristic of this method is that the axes that are parallel to the pixel or strip plane arediscretized in finite elements, while the perpendicular axis is kept continuous. Then by using aFourier transform, the three dimensional problem for the pixel sensors is reduced to two dimensionsand the two dimensional problem for the strip sensors is reduced to one dimension. The problem isthen solved in Fourier space by using a numerical method.
Figure 1 shows the capacitive network of a pixel detector. The capacitances that are calculated withthis method are those formed between the central pixel and the adjacent pixels in the directions thatare parallel to 𝑥 - and 𝑦 - axis respectively ( 𝐶 and 𝐶 ), the capacitances that are formed betweenthe central pixel and the adjacent pixels in the diagonal direction ( 𝐶 ) and the capacitances thatare formed between each pixel and the backplane ( 𝐶 ). These capacitances are strongly relatedto the geometry features of the sensor such as the dimensions of each pixel and the separation gapbetween them and the thickness. Figure 1 : Schematic of the capacitive network of a pixel sensor with 9 pixels. 𝐶 and 𝐶 arethe capacitances that are formed between the central and the adjacent pixels in the directions thatare parallel to the 𝑥 - and 𝑦 - axis respectively. 𝐶 are the capacitances that are formed betweenthe central and the adjacent pixels in the diagonal direction and 𝐶 are the capacitances that areformed between each pixel and the backplane.To calculate the strip sensor capacitances the Poisson’s equation is solved with normalizedboundary conditions by setting 𝑉 ( 𝑥, 𝑦, 𝑧 ) = 𝑉 ( 𝑥, 𝑦, 𝑧 ) = 𝜀 𝑆𝑖 𝐸 𝑆𝑖 ( 𝑥, 𝑦, 𝑧 ) − – 2 – 𝑎 𝐸 𝑎 ( 𝑥, 𝑦, 𝑧 ) =
0) is applied for keeping the electric field in the interface continuous, where 𝜀 𝑆𝑖 , 𝜀 𝑎 are the dielectric constants and 𝐸 𝑆𝑖 ( 𝑥, 𝑦, 𝑧 ) , 𝐸 𝑎 ( 𝑥, 𝑦, 𝑧 ) are the vertical components of the electricfield in silicon and ambient space, respectively. The ambient space in this work is considered to beair. The detector is considered to be in a fully depleted state and it is free from thermally generatedfree charge carriers (pairs of electrons and holes). This is the state in which a realistic silicondetector works, when a reverse bias voltage is applied to the silicon sensor and the depleted regionis formed. The thermally generated free charged pairs of electrons and holes are swept from theelectric field creating an ionization chamber. The pixels are assumed to be infinitesimally small indepth compared to the fully depleted region of the detector. Also, for simplicity it is considered thatthe volume and the surface of the detector are free from static charges. Under real circumstances,charges exist inside the detector volume. These are stripped ions in the depleted regions and defectsfrom contamination inside the material. However, they produce an electric field component whichis independent of the biasing voltage. The charges add a voltage-independent term in the expressionof the strip charge which does not influence the calculation of the capacitance.Under the aforementioned assumptions Poisson’s equation is reduced to the Laplace’s equation2.1. Subsequently, by using Fourier transform, while keeping the perpendicular axis continuous,the three dimensional problem is reduced to two dimensions (equation 2.2). For details see ref. [1]. ∇ 𝑉 ( 𝑥, 𝑦, 𝑧 ) = F −→ (2.1) 𝜕 𝑉 ( 𝑘 𝑥 , 𝑘 𝑦 , 𝑧 ) 𝜕𝑧 = (cid:16) 𝑘 𝑥 + 𝑘 𝑦 (cid:17) 𝑉 (cid:0) 𝑘 𝑥 , 𝑘 𝑦 , 𝑧 (cid:1) (2.2)where 𝑉 (cid:0) 𝑘 𝑥 , 𝑘 𝑦 , 𝑧 (cid:1) is the potential in Fourier space and 𝑘 𝑥 , 𝑘 𝑦 the corresponding coordinatesin Fourier space. By solving the differential equation with the appropriate boundary conditions,equation 2.3 is derived which gives the electric field in Fourier space ( 𝐸 𝑧 (cid:0) 𝑘 𝑥 , 𝑘 𝑦 , (cid:1) ) as a function ofthe potential in Fourier space ( 𝑉 (cid:0) 𝑘 𝑥 , 𝑘 𝑦 , (cid:1) ), which for 𝑧 = 𝐹 (cid:0) 𝑘 𝑥 , 𝑘 𝑦 (cid:1) is a function of the Fourier coefficients. 𝐸 𝑧 (cid:0) 𝑘 𝑥 , 𝑘 𝑦 , (cid:1) = 𝐹 (cid:0) 𝑘 𝑥 , 𝑘 𝑦 (cid:1) + 𝑒 − 𝐹 ( 𝑘 𝑥 ,𝑘 𝑦 ) 𝑤 − 𝑒 − 𝐹 ( 𝑘 𝑥 ,𝑘 𝑦 ) 𝑤 𝑉 (cid:0) 𝑘 𝑥 , 𝑘 𝑦 , (cid:1) (2.3)The Laplace’s equation is then solved in the Fourier space by using a self-consistent numericalmethod. First an initial guess of the potential is made 𝑉 𝑖𝑛𝑖𝑡 ( 𝑥, 𝑦, ) where for ( 𝑧 = ) it correspondsto the pixel plane. By using Fourier transform, with respect to 𝑥 - and 𝑦 - axis the initial guessis transformed to the potential in Fourier space 𝑉 𝑖𝑛𝑖𝑡 ( 𝑘 𝑥 , 𝑘 𝑦 , ) . Then the vertical componentof the electric field inside the sensor 𝐸 𝑖𝑛𝑖𝑡𝑆𝑖 ( 𝑥, 𝑦, ) is calculated from 2.3 by performing inverseFourier transform. The vertical component of the electric field in the ambient space 𝐸 𝑖𝑛𝑖𝑡𝑎 ( 𝑥, 𝑦, ) is calculated by equation 2.3 setting ( 𝑤 → ∞ ) and using inverse Fourier transform. Next thepotential is redefined by boundary conditions. The values of the vertical component of the electricfield in the ambient space 𝐸 𝑖𝑛𝑖𝑡𝑎 ( 𝑥, 𝑦, ) are used for calculating the new values of the electric fieldinside the sensor 𝐸 𝑖𝑛𝑖𝑡𝑆𝑖 ( 𝑥, 𝑦, ) , by using the boundary condition in the space that is not covered bypixels. This gives a new estimation of the electric field inside the sensor 𝐸 𝑛𝑒𝑤 ( 𝑥, 𝑦, ) and a new– 3 –stimation of the potential 𝑉 𝑛𝑒𝑤 ( 𝑥, 𝑦, ) . The actual solution of the problem is assumed to be alinear combination of the new and the initial potential functions. Then a check for convergence ismade and if it has not been achieved the initial function is set equal to the new potential functions,that has been derived from the linear combination. All the above steps are repeated through severalcycles until convergence is reached.Finally, once convergence has been achieved, the charges stored in each pixel are calculated byintegrating the charge density in the whole pixel surface ( 𝑄 = ∯ ( 𝜀 𝑆𝑖 𝐸 𝑆𝑖 − 𝜀 𝑎 𝐸 𝑎 ) 𝑑𝑆 ). Then thecalculation of pixel capacitances is made. This algorithm can be used for calculating the capacitances in micro-strip detectors with planargeometry. Figure 2 shows the capacitive network of a strip sensor with 7 strips.
Figure 2 : Schematic of the capacitive network of a micro-strip detector with 7 strips. 𝐶 are thecapacitances between the central and the first adjacent strip, 𝐶 between the central and the secondadjacent strip, 𝐶 between the central strip and the third adjacent strip, and 𝐶 between each stripand the backplane, where w is the strip width, s is the interstrip space and p is the strip pitch and dis the detector thicknessThe capacitances that are calculated with this algorithm are those between a central strip andthe backplane ( 𝐶 ), the capacitances between the central and the first adjacent strips ( 𝐶 ) andbetween the central and the second adjacent strips ( 𝐶 ). While the software calculates capacitancesup to the third ( 𝐶 ), these are negligible and omitted from our study. The same method as in thethree dimensional case is followed, with the difference that in the strip sensor case the problem isreduced to two dimensions and by using Fourier transform the Laplace’s equation is solved in onedimension. In this case the axis that is parallel to the strip plane ( 𝑥 - axis) is discretized in finiteelements, while the perpendicular ( 𝑦 - axis) is kept continuous. Technology computer-aided design (TCAD) is used in the semiconductor industry in order todevelop and optimize semiconductor processing technologies and devices. It can be used in orderto simulate the fabrication procedure, the operation and the reliability of the semiconductor devices.The TCAD suite that was used for this work is the commercial software package TCAD Sentaurusfrom Synopsys [3]. – 4 –CAD follows a finite element analysis scheme. Firstly, the device is designed in two or threedimensions and the properties of each region of the device, such as the doping concentration, thematerials or shape are defined. Another way to create a device is by simulating the actual fabricationprocedure, but this approach is beyond the scope of the present work. Afterwards the device issubdivided into finite elements by following a Delaunay triangulation algorithm [4] which createsa mesh of the device.The next step is to activate the desired physical models and parameters before initiating thedevice simulation program. Some of the physical models that were used in this work are the Augerrecombination, Shockley-Read-Hall recombination, avalanche electron-hole generation, trap-to-trap recombination, band-to-band tunneling, doping dependence mobility, high field saturation andcarrier-carrier scattering [5].The fundamental partial differential equations for semiconductors (Poisson’s, continuity equa-tions for electrons and holes) are solved at each of the generated mesh point and the desired physicalquantities are calculated. In order to calculate the capacitances a small signal AC analysis is per-formed at 1 kHz for the backplane capacitance and at 1 MHz for the interstrip capacitance. Thesefrequencies correspond to the frequencies that the experimental measurements of this sensors wereperformed.
Figure 3 shows the simulated structure of the MSSDs. The design resembles a perpendicular cross-section of the sensor to the strip plane. The final results are scaled to the actual sensor strip length.The structure has 5 strips instead of 32 of the actual MSSD sensors with implant and aluminumwidths as denoted in table 1. The different layers of each strips are depicted in the top-right partof figure 3 as well as the generated mesh consisting of triangular segments. The metal depictedin gray is extended a few µ m in the interstrip space. This technique is called metal-overhang andis used in order to overcome the junction curvature effect which limits the breakdown voltage ofplanar junctions [6], [7]. In the space between two strips two additional structures (depicted incyan) have been designed with high dose of p-implant resembling the actual p-stop structures of thesensor. The bulk doping concentration is assumed to be equal to 3 . × cm − (p-type), while thestrip doping concentration is assumed to be equal to 1 . × cm − (n-type). The deep diffusiontechnique on the backplane is simulated by using an error function doping profile. More details onthe parameters that were used in order to produce these simulations can be seen in Appendix A.Most of the parameters that are used in the simulation have been chosen by following the workspresented in [8] and [9].The interstrip capacitance between two strips i and j for an AC coupled sensor are calculated,according to [10], with the following formula 3.1. 𝐶 𝑖𝑛𝑡 = 𝐶 𝑀 𝑖 − 𝑀 𝑗 + 𝐶 𝐼 𝑖 − 𝐼 𝑗 + 𝐶 𝑀 𝑖 − 𝐼 𝑗 + 𝐶 𝐼 𝑖 − 𝑀 𝑗 (3.1)where 𝐶 𝑀 𝑖 − 𝑀 𝑗 is the capacitance between the metal of the i th strip and the metal of the j th strip, 𝐶 𝐼 𝑖 − 𝐼 𝑗 is the capacitance between the implant of i th strip and the implant of j th strip, 𝐶 𝑀 𝑖 − 𝐼 𝑗 isthe capacitance between the metal of i th strip and the implant of j th strip, 𝐶 𝐼 𝑖 − 𝑀 𝑗 is the capacitancebetween the implant of i th strip and the metal of j th strip.– 5 – igure 3 : The structure that was used for simulating the MSSD sensors. In the bottom middlethe whole structure that was used with 5 strips is shown. The color variation depicts the dopingconcentration. The top-left figure shows a close up view between two strips. Two p-stop structuresare designed, depicted in cyan, in order to achieve strip isolation. The top-right figure shows aclose-up view of the simulated structure near to one strip edge. The n-type implant is displayed inred which is simulated with a Gaussian profile, the aluminum contacts are displayed with gray, the 𝑆𝑖𝑂 is displayed with brown and the p-type silicon bulk is displayed with green. For pixel sensors a 3D simulation approach is employed. The simulated structures consist of 9orthogonal pixels with two different pixel geometries one with pixel area of 50 × µ m , (figure4a), and one with pixel area of 100 × µ m , (figure 4b). Both structures are DC-coupled with an + p configuration and an active thickness of 150 µ m. A guard ring structure surrounds the device,providing a homogeneous electric field inside the sensitive area and minimizing the edge effects.The simulations were made for different pixel layouts with varying separation gap between 5 µ m to50 µ m with a 5 µ m step. The capacitances were calculated by performing a small signal AC analysiswith frequencies at 1 kHz and 1 MHz for the backplane and interpixel capacitances, respectively.These are the configurations under development for the Phase-2 upgrade of the CMS/LHC [2] andAtlas/LHC [11] silicon trackers at CERN. Simulation parameters are shown in Appendix A.– 6 – a) (b) Figure 4 : Simulated structure for pixel sensors with 50 × µ m , (figure 4a) and with 100 × µ m , (figure 4b) pixel area. Figure 5 shows an actual picture of a Multi Geometry Silicon Strip Detector (MSSD). The MSSDscontain 12 individual regions and they all have their own bias and guard rings. Each region contains32 AC coupled strips on n + p configuration with pitches varying from 70 to 240 µ m resulting towidth-to-pitch ratios (w/p) varying from 0 .
133 to 0 . Figure 5 : A Multi-Geometry Silicon Strip Sensor. The sensor has 12 regions with different pitchesand width-to-pitch ratio (w/p). Three of these sensors with physical thickness of 320 µ m and activethickness of 120, 200 and 320 µ m used for the measurements Table 1 : Geometrical characteristics of the MSSD sensors for each region.
Label 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Region 1 2 3 4 5 6 7 8 9 10 11 12Pitch 120 240 80 70 120 240 80 70 120 240 80 70Width 16 34 10 8 . . . . . . .
133 0 .
142 0 .
125 0 .
121 0 .
233 0 .
242 0 .
225 0 .
221 0 .
333 0 .
342 0 .
325 0 . ( . ) ( . ) ( . ) ( . ) ( . ) ( . ) ( . ) ( . ) ( . ) ( . ) ( . ) ( . ) – 7 –he three MSSD sensors that we measured in this work originate from 3 different wafers offloat zone silicon (FZ). They have the same physical thickness of 320 µ m, but three different activethicknesses of 120 µ m (FZ120P), 200 µ m (FZ200P), 320 µ m (FZ320P). The manufacturer achievesthis by a deep diffusion technique. In order to achieve isolation of the strip implants an additionalhigh dose of p+ implantation is made between the strips surrounding each n+ strip. These structuresare called p-stop.The measurements were performed with a semi-automated probe station (Carl Susse PA 150).The whole setup was electrically shielded inside a light-tight metal box. The capacitances aremeasured with an HP4192A LCR meter which supplies a small AC signal superimposed uponthe DC bias voltage on the HIGH terminal. The amplitude and phase are measured on the LOWterminal. Backplane capacitances were measured by using the bias ring which connects togetherall the 32 strips via the bias resistors. The measurements were performed at a frequency of 1 kHzwith an amplitude of 250 mV. The bias voltage (detector HI) was applied to the backplane whilethe bias ring was grounded. The bias voltage was ramped up from 0 V to 400 V. The interstripcapacitances were measured by performing an automatic strip scan in each strip, of each region, ofthe sensor. In this measurement,two neighboring strips were contacted with the probes connectedone with the HI terminal along with the backplane and the strip under test to the LOW terminal. Theinterstrip capacitance measurements were performed at 1 MHz with an amplitude of 250 mV. Thefrequencies and the amplitudes that have been chosen correspond to those that yield the optimumC-V characteristics of these sensors with the particular experimental setup for the backplane andinterstrip capacitances respectively. Histograms 6a , 6b and 6c show the comparison between the backplane capacitances for the 3 MSSDsensors. The regions 3 and 12 in the FZ200P sensor and 2 and 9 in the FZ320P sensor were damaged.In the majority of the measured samples with different w/p ratio the measured backplane capacitanceis larger than the simulated one, probably due to parallel parasitic capacitances introduced duringthe measurement. The TCAD simulated capacitance seems to be closer to the measured one inthe majority of our samples. However, our simpler but much faster Laplace solver gives quitecomparable results for the backplane capacitance. width to pitch ratio C apa c i t an c e [ p F / c m ] Comparison of the Backplane Capacitance for the MSSD_FZ120P
ExperimentalTCAD simulationsLaplace solver (a) width to pitch ratio C apa c i t an c e [ p F / c m ] Comparison of the Backplane Capacitance for the MSSD_FZ200P
ExperimentalTCAD simulationsLaplace solver (b) width to pitch ratio C apa c i t an c e [ p F / c m ] Comparison of the Backplane Capacitance for the MSSD_FZ320P
ExperimentalTCAD simulationsLaplace solver (c)
Figure 6 : Comparison of the experimental results (red), the results of the related TCAD simulations(green) and the Laplace solver (blue) for the backplane capacitance in FZ120P 6a, FZ200P 6b andFZ320P 6c sensors. – 8 –igures 7a and 7b show the relative errors of the backplane capacitance defined as ( 𝐶 𝑒𝑥 𝑝 − 𝐶 𝑠𝑖𝑚 )/ 𝐶 𝑒𝑥 𝑝 %, of the backplane capacitance where 𝐶 𝑠𝑖𝑚 are the simulated results from our Laplacesolver and the TCAD simulations respectively and 𝐶 𝑒𝑥 𝑝 are the experimental values. A Gaussianfit is also shown for comparison. The numerical calculations made with the Laplace solver have amean value of 14% with a standard deviation of 7% while the calculations made with TCAD havea mean value of 12% and a standard deviation of 6%. (a) (b) Figure 7 : Histograms of the relative error of the simulated results to the experimental values forthe backplane capacitance for the Laplace solver 7a and for the TCAD simulations 7b from all 3MSSD sensors
Histograms 8a , 8b and 8c show measured and simulated interstrip capacitances for the 3 MSSDsensors. As in the backplane case, in the majority of the measured samples with different w/pratio the measured interstrip capacitance is larger than the simulated one, probably due to parallelparasitic capacitances introduced during the measurement. width to pitch ratio C apa c i t an c e [ p F / c m ] Comparison of the Interstrip Capacitance for the MSSD_FZ120P
ExperimentalTCAD simulationsLaplace solver (a) width to pitch ratio C apa c i t an c e [ p F / c m ] Comparison of the Interstrip Capacitance for the MSSD_FZ200P
ExperimentalTCAD simulationsLaplace solver (b) width to pitch ratio C apa c i t an c e [ p F / c m ] Comparison of the Interstrip Capacitance for the MSSD_FZ320P
ExperimentalTCAD simulationsLaplace solver (c)
Figure 8 : Comparison of the experimental results (red) the results of the related TCAD simulations(green) and the Laplace solver (blue) for the interstrip capacitance in FZ120P 8a, FZ200P 8b andFZ320P 8c sensors.Figures 9a and 9b show the relative errors of the simulated results ( 𝐶 𝑠𝑖𝑚 ) from our Laplacesolver and the TCAD simulations respectively, to the experimental values ( 𝐶 𝑒𝑥 𝑝 ) of the interstripcapacitance as it has been calculated with our Laplace solver and with the TCAD simulations,– 9 –espectively. The numerical calculations made with the Laplace solver have a mean value of 27%with a standard deviation of 4% while the calculations made with TCAD have mean value of 4%and a standard deviation of 8%. (a) (b) Figure 9 : Histograms of the relative error of the simulated results to the experimental values forthe interstrip capacitance for the Laplace solver (figure 9a) and for the TCAD simulations (figure9b). It must be noted that the numerical calculations from the Laplace solver are tailored to theideal case where the sensor is free from static charges. In addition, no oxide and no aluminumcontacts above the strip plane are taken into account. Thus, it calculates only the capacitancebetween adjacent implants 𝐶 𝐼 𝑖 − 𝐼 𝑗 . On the other hand, TCAD makes a more detailed simulation withmore accurate physical models. Moreover, the interstrip capacitances are calculated by followingequation 3.1 where the capacitances between adjacent aluminum 𝐶 𝑀 𝑖 − 𝑀 𝑗 and between adjacentaluminum and implants 𝐶 𝑀 𝑖 − 𝐼 𝑗 , 𝐶 𝐼 𝑖 − 𝑀 𝑗 are taken into account. In ref. [10] it is noted that when themetal overhang is absent the implant-implant capacitance is the dominant component of the totalinterstrip capacitance. Otherwise, when the overhang is present and begins to increase, the otherthree components of the interstrip capacitance start to increase with a simultaneous decrease of theimplant-implant capacitance.The numerical calculations as described in section 2.2 should calculate the interstrip capaci-tances in the case where the metal overhang is absent and thus the implant-implant component ofthe interstrip capacitance is dominant. In order to check the validity of our software in calculatingthe implant-implant component of the capacitance we have simulated also the case where the metaloverhang is absent with TCAD. In this case the aluminum width is taken to be equal to implantwidth (figure 10). The other properties of the simulated structure were kept the same as describedin section 3.1.Histograms, 11a, 11b, 11c show the simulated results of the implant-implant component ofthe capacitances for two cases with metal-overhang (depicted in green) and without metal-overhang(depicted in yellow), compared with results from Laplace solver(depicted in blue).The calculated results of our software agree with the simulated results for the implant-implantcomponents of the interstrip capacitance in the case were metal overhang is absent with an averageaccuracy of ≈ igure 10 : The simulated structure without metal-overhang close to one strip edge. The n-typeimplant is displayed in red which is simulated with a Gaussian profile, the aluminum contacts aredisplayed with gray, the SiO is displayed with brown and the p-type silicon bulk is displayed withgreen. width to pitch ratio C apa c i t an c e [ p F / c m ] Comparison of the implant-implant capacitance for the MSSD_FZ120P
Laplace solverTCAD without overhangTCAD with overhang (a) width to pitch ratio C apa c i t an c e [ p F / c m ] Comparison of the implant-implant capacitance for the MSSD_FZ200P
Laplace solverTCAD without overhangTCAD with overhang (b) width to pitch ratio C apa c i t an c e [ p F / c m ] Comparison of the implant-implant capacitance for the MSSD_FZ320P
Laplace solverTCAD without overhangTCAD with overhang (c)
Figure 11 : Comparison of the calculted results (blue) with TCAD simulations in the case weremetal-overhang is present (green) and without metal-overhang (yellow) for the FZ120P 11a, FZ200P11b and FZ320P 11c sensors.dielectric constant of the ambient space to that of
𝑆𝑖𝑂 doesn’t improve the results. However, itsightly decreases the interstrip capacitance while the backplane remains the same. In ref. [13], pixel capacitances have been measured on pixel sensors from 6 different n-type waferswith fixed pitch=100 µ m and with varying separation gap between the pixel implants from 5 µ mto 30 µ m with a 5 µ m gap step, resulting in an implant width that ranges from 95 µ m to 70 µ m.For comparison we have calculated the capacitances for those sensors with our 3D Laplace solver.Figure 12 shows the calculated total interpixel capacitances 𝐶 𝑡𝑖 𝑝 for the six different structures incomparison with the literature measurements taken from ref. [13]. In this case, due to the squaregeometry of the pixels and with reference to figure 1, 𝐶 = 𝐶 , while with reference to figure 2of [13], 𝐶 ≡ 𝐶 𝑖 𝑝 , which is the orthogonal interpixel capacitance and 𝐶 ≡ 𝐶 𝑑𝑖𝑎𝑔 which is thediagonal interpixel capacitance. The data are compared with the total interpixel capacitance, as itis sensed by a virtually grounded preamplifier, given by equation 4.1. The orthogonal ( 𝐶 , 𝐶 )and diagonal 𝐶 components are drawn as well for clarity. 𝐶 𝑡𝑖 𝑝 = 𝐶 + 𝐶 + 𝐶 (4.1)– 11 –he calculated results approximate the lower error limit of the experimental data of the totalinterpixel capacitance. The diagonal capacitance in ref. [13] was approximated to 2 . . . µ m gap and 1 . µ m gap. The orthogonalinterpixel capacitance in [13] range, between 1 − µ m gap) to 11 −
12 fF (5 µ m gap) whilefrom our calculations the orthogonal capacitance range between 3 .
37 fF (30 µ m gap) to 9 .
70 fF (5 µ m gap). m] m pixel seperation [ C apa c i t an c e [f F ] ExperimentalTotal Calculated Orthogonal Calculated Diagonal Calculated
Interpixel Capacitances
Figure 12 : Total interpixel capacitance for sensors with fixed pitch=100 µ m and varying separationgap in comparison with experimental data extracted from [13]. Also the calculated results for theorthogonal and the diagonal capacitances are shown.In ref. [14] pixel capacitances have been measured for irradiated and non-irradiated sensorsfrom LBNL and Atlas/LHC test structures along with simulations with HSPICE [15] and IESCoulomb [16] in two and three dimensions. Each structure includes six 3 × µ m pitch in their short direction and 536 µ m pitch in their long direction. Theimplant width and the separation gap vary in each test structure. We have calculated the capacitancesfor the LBNL n-type test structures with the 3D Laplace solver. The experimental and simulated datafor the total interpixel capacitance (equation 4.1) reported in [14] in comparison with calculationsmade with the 3D Laplace solver are presented in figure 13. The Laplace solver results agree withthe measurements with a relative error which is less than 32%.Figure 14 shows the experimental results for the backplane capacitance along with simulationresults from IES Coulomb in two and three dimensions and with calculations with our software byusing both the two and three dimensional methods. The calculated results from our software areinside the error assessment of ± m] m pixel width [ C apa c i t an c e [f F ] Total interpixel capacitances
ExperimentalHSPICEIES2DIES3DLaplace3D
Figure 13 : Experimental and simulated data with HSPICE ,IES Coulomb in two and three di-mensions of LBNL n-type unirradiated sensors extracted from table 2 and table 8 of ref. [14] incomparison with results from Laplace solver in three dimensions for the total interpixel capacitance.
10 15 20 25 30 35 40 m] m pixel width [ C apa c i t an c e [f F ] Backplane capacitances
ExperimentalIES2DIES3DLaplace2DLaplace3D
Figure 14 : Experimental and simulated data with IES Coulomb in two and three dimensions ofLBNL n-type unirradiated sensors extracted from table 6 and of ref. [14] in comparison with resultsfrom Laplace solver in two and three dimensions for the backplane capacitance.
10 15 20 25 30 35 40 m] m pixel width [ C apa c i t an c e [f F ] Ortogonal interpixel capacitances on the short side of LBNL n-type sensors
IES3DLaplace3DLaplace2D (a)
10 15 20 25 30 35 40 m] m pixel width [ C apa c i t an c e [f F ] Second neighbor ortogonal interpixel capacitances on the short side of LBNL n-type sensors
IES3DLaplace2D (b)
Figure 15 : Comparison of simulated data for the orthogonal interpixel capacitance on the shortside with results from Laplace solver in three and two dimensions (figure 15a) and comparisonof simulated data for the second neighbor capacitance on the short side of LBNL n-type sensors(figure 15b). The simulated data were extracted from table 9 of ref. [14].– 13 – .3 Comparison between TCAD and 3D Laplace solver for various pixel geometries
We have simulated the capacitances for pixel sensors with pixel geometries 50 × µ m and 100 × µ m and thicknesses of 150 µ m. These configurations are appropriate to the developmental workin progress for the Phase-2 upgrade of the pixel systems in the CMS/LHC and the ATLAS/LHCexperiment at CERN. The separation gaps vary between 5 µ m to 50 µ m with a 5 µ m step size.Simulations were performed by using both TCAD and the 3D Laplace solver. Figure 16 showsthe simulated results for the backplane capacitances for sensors with 50 × µ m pixels and forsensors with 100 × µ m pixels. The backplane capacitance calculated with the Laplace solveris systematically larger than the one obtained from the TCAD simulations in all the cases for about1 . m] m pixel seperation [ C apa c i t an c e [f F ] m m pixels and wafer thikness 150 m m Backplane capacitance for 50x50
Laplace solverTCAD simulations (a) m] m pixel seperation [ C apa c i t an c e [f F ] m m pixels and wafer thikness 150 m m Backplane capacitance for 100x25
Laplace solverTCAD simulations (b)
Figure 16 : Simulated results by using TCAD (green line) for the backplane capacitance com-pared with simulated results from our program (red line) which implements the numerical methoddescribed in 2.1 for sensors with 50 × µ m
16a and 100 × µ m pixel area respectively 16b.Figures 17 and 18 show the simulated results for the interpixel capacitances. Both simulationsshow a very good agreement in the calculation of the orthogonal and diagonal interpixel capacitancesespecially for larger separation gaps. Figure 19 shows the simulated results for the total capacitances,where the total capacitance is the sum of all the total interpixel capacitance (equation 4.1) includingthe backplane. Again the two simulations agree well in the calculation of the total capacitanceespecially for gaps larger than 15 µ m.Figures 20a and 20b show the relative difference between the calculated results with Laplacesolver and TCAD simulation results, respectively. It can be noted that the difference between thetwo simulations decrease with the increase of the separation gap. The relative difference for thebackplane capacitance ranges between 29 % and 12 % for sensors with 50 × µ m pixels andbetween 38 % and 12 % for sensors with 100 × µ m pixels. For the orthogonal interpixelcapacitances it ranges between 28 % ans 12 % for sensors with 50 × µ m pixels, while forthe diagonal capacitances it is less than 2 % and for sensors with 100 × µ m pixels it rangesbetween 32 % and 16 % for x- axis and between 28 % and 15 % for y- axis, while for the diagonalcapacitances it is less than 3 %. For total capacitance it ranges between 19 % and 0.5 % for sensorswith 50 × µ m pixels and between 19 % and 0.9 % for sensors with 100 × µ m pixels.– 14 – m] m pixel seperation [ C apa c i t an c e [f F ] m m pixels and wafer thikness 150 m m Interpixel capacitance in orthogonal-axes for 50x50
Laplace solver x- and y- axesTCAD simulations x- and y- axes (a) m] m pixel seperation [ C apa c i t an c e [f F ] m m pixels and wafer thikness 150 m m Interpixel capacitance in orthogonal-axes for 100x25
Laplace solver x-axisLaplace solver y-axisTCAD simulations x-axisTCAD simulations y-axis (b)
Figure 17 : Simulated results by using TCAD (green-yellow lines) for the orthogonal interpixelcapacitance in x- and y- axes, compared with simulated results from our program (red-blue lines)for sensors with 50 × µ m
17a and 100 × µ m pixel area respectively 17b . m] m pixel seperation [ C apa c i t an c e [f F ] m m pixels and wafer thikness 150 m m Interpixel capacitance in diagonal-axes for 50x50
Laplace solverTCAD simulations (a) m] m pixel seperation [ C apa c i t an c e [f F ] m m pixels and wafer thikness 150 m m Interpixel capacitance in diagonal-axes for 100x25
Laplace solverTCAD simulations (b)
Figure 18 : Simulated results by using TCAD (green line) for the diagonal interpixel capacitance,compared with simulated results from our program (red line) for sensors compared with simulatedresults from our program (red-blue lines) for sensors with 50 × µ m
18a and 100 × µ m pixelarea respectively 18b . m] m pixel seperation [ C apa c i t an c e [f F ] m m pixels and wafer thikness 150 m m Total capacitance for 50x50 Laplace solverTCAD simulations (a) m] m pixel seperation [ C apa c i t an c e [f F ] m m pixels and wafer thikness 150 m m Total capacitance for 100x25
Laplace solverTCAD simulations (b)
Figure 19 : Simulated results by using TCAD (green line) for the total capacitance, compared withsimulated results from our program (red line) for sensors compared with simulated results from ourprogram (red-blue lines) for sensors with 50 × µ m
19a and 100 × µ m pixel area respectively19b. – 15 – m] m pixel seperation [ A b s . R e l . D i ff e r en c e [ % ] pixels m m Absolute relative difference for sensors with 50x50
Total CapacitancesBackplane CapacitancesOrthogonal Capacitances x- axisOrthogonal Capacitances y- axisDiagonal Capacitances (a) m] m pixel seperation [ A b s . R e l . D i ff e r en c e [ % ] pixels m m Absolute relative difference for sensors with 100x25
Total CapacitancesBackplane CapacitancesOrthogonal Capacitances x- axisOrthogonal Capacitances y- axisDiagonal Capacitances (b)
Figure 20 : Absolute relative difference between the calculated results with Laplace solver and theTCAD simulation results for sensors with 50 × µ m
20a and 100 × µ m pixel area respectively20b. Table 2 shows the time that is needed to calculate the capacitances by using the Laplace solver.The variable resolution is the number of elements that are used for the numerical calculations. Twoexamples are given for comparison with TCAD running on an 8-core processor at 3.70
𝐺 𝐻𝑧 . Inthe case of strip sensors, for a 2D Laplace resolution of 3.5 10 , the run time is 7.3 s, while forthe TCAD simulation with the same resolution the runtime is 30 min. In the pixel case, for a 3DLaplace resolution of 2.0 10 the run time is 34.9 s, while for the TCAD simulations with the sameresolution the runtime is 4 h. Table 2 : Calculation time for different values of discretization for pixel and strip sensors.resolution time s2.6 10 The fast numerical solution of the Laplace’s equation described in this work, gives an accurateapproximation of the experimental results and of the TCAD simulations. For the strip sensorsthe mean value of the relative error of all the regions for the 3 MSSD sensors is 14% for the– 16 –ackplane capacitance and 27% for the interstrip capacitance, while the implant-implant componentfor sensors without overhang is approximated with a mean relative error of 7% , compared withTCAD simulation results. For pixel sensors the relative error of the calculations was found to beless than 32% compared to experimental results found in literature. In addition, compared to theTCAD simulations the calculated results show a very good agreement, especially the calculationsof the interpixel and total capacitances for large inter-pixel gaps.As a general conclusion, the program that implements the three and two dimensional numericalsolution of the Laplace’s equation that is described in this paper can be used in order to provide afast approximation of detector capacitances for planar silicon strip and pixel sensors before a moredetailed simulation with EDA tools is performed. This tool is foreseen to be implemented into aweb-based application.
A Properties used for the TCAD simulations
Table 3 : Geometrical properties and doping concentrations used for the TCAD simulation of stripsensors.
Material FZ 120P FZ 200P FZ 320PBulk doping concentration [ 𝑐𝑚 − ] . 𝑒 Strip doping concentration [ 𝑐𝑚 − ] . 𝑒 Backplane doping concentration [ 𝑐𝑚 − ] . 𝑒 p-stop doping concentration [ 𝑐𝑚 − ] . 𝑒 𝑆𝑖𝑂 thickness between strips [ µ m] 0 . 𝑆𝑖𝑂 thickness between metal-strip [ µ m] 0 . 𝑆𝑖𝑁 thickness [ µ m] 0 . µ m] 0 . µ m] 1 . µ m] 215 125 33 Table 4 : Geometrical properties and doping concentrations used for the TCAD simulation of pixelsensors.
Material n + pBulk doping concentration [ 𝑐𝑚 − ] . 𝑒 Pixel doping concentration [ 𝑐𝑚 − ] . 𝑒 Backplane doping concentration [ 𝑐𝑚 − ] . 𝑒 Guard ring doping concentration [ 𝑐𝑚 − ] . 𝑒 𝑆𝑖𝑂 thickness [ µ m] 1 . µ m] 0 . µ m] 1 . µ m] 20 – 17 – cknowledgments The authors of this paper would like to thank the outer tracker sensor working group of CMS/LHCfor providing the MSSD sensors.This research is co-financed by Greece and the European Union (European Social Fund- ESF)through the Operational Programme Human Resources Development, Education and LifelongLearning 2014 2020 in the context of the project "New generation of sensors and electronics for theupgrade of the CMS/LHC experiment at CERN"- MIS 5047807
References [1] S. Kavadias, K. Misiakos and D. Loukas,
Calculation of pixel detector capacitances through threedimensional numerical solution of the Laplace equation , IEEE Transactions on Nuclear Science (1994) 397.[2] CMS C ollaboration collaboration, The Phase-2 Upgrade of the CMS Tracker , Tech. Rep.CERN-LHCC-2017-009. CMS-TDR-014, CERN, Geneva, Jun, 2017.[3] , 2019.[4] B. Delaunay,
Sur la sphere vide , Bulletin de l’ Académie des Sciences de l’URSS, Classe des SciencesMathématiques et Naturelles (1934) 793.[5] Synopsys, Sentaurus Device User Guide . Synopsys, Inc.[6] S. Sze and G. Gibbons,
Effect of junction curvature on breakdown voltage in semiconductors , Solid-State Electronics (1966) 831 .[7] D. Passeri, P. Ciampolini, A. Scorzoni and G. M. Bilei, Physical modeling of silicon microstripdetectors: influence of the electrode geometry on critical electric fields , IEEE Transactions on Nuclear Science (2000) 1468.[8] T. V. Eichhorn, Development of Silicon Detectors for the High Luminosity LHC , Ph.D. thesis, U.Hamburg, Dept. Phys., Hamburg, 2015. 10.3204/DESY-THESIS-2015-024.[9] R. Eber,
Untersuchung neuartiger Sensorkonzepte und Entwicklung eines effektiven Modells derStrahlenschädigung für die Simulation hochbestrahlter Silizium-Teilchendetektoren , Ph.D. thesis,2013. 10.5445/IR/1000038403.[10] S. Chatterji, A. Bhardwaj, K. Ranjan, Namrata, A. K. Srivastava and R. Shivpuri,
Analysis ofinterstrip capacitance of si microstrip detector using simulation approach , Solid-State Electronics (2003) 1491 .[11] ATLAS Collaboration collaboration,
Technical Design Report for the ATLAS Inner Tracker PixelDetector , Tech. Rep. CERN-LHCC-2017-021. ATLAS-TDR-030, CERN, Geneva, Sep, 2017.[12] K.-H. Hoffmann,
Campaign to identify the future cms tracker baseline , Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment (2011) 30 . – 18 –
13] V. Bonvicini, M. Pindo and N. Redaelli,
Junction and interdiode capacitance of silicon pixel arrays , Nuclear Instruments and Methods in Physics Research A (1995) 88.[14] G. Gorfine, M. Hoeferkamp, G. Santistevan and S. Seidel,
Capacitance of silicon pixels , Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment (2001) 336 .[15] , 20019.[16] , 20019., 20019.