Both equations are solved using a finite differences (iterative relaxation) method, which takes some time to converge.

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1 WEIGHTFIELD 2D Silicon Strip Detector Simulation V November 2011 Abstract WEIGHTFIELD is a program that allows simulating a silicon strip detector in two dimensions (crosssection). It determines both drift and weighting fields and calculates the induced current on the readout electrode as a function of the location of a traversing MIP (minimum ionizing particle). Introduction The silicon detector has a thickness of 300µm and a section of 1000µm width is considered. This area is divided into a grid of 1x1µm 2 units (resulting in 300k cells). According to the user s input of strip width and pitch, depletion and bias voltage, bulk and strip implant doping types, the program iteratively calculates the potentials in this grid. This is done by solving Poisson s equation 2 φ d = ρ ε for the drift potential, considering space charges according to the bulk doping concentration (which is derived from the depletion voltage) and the boundary conditions of strip and backplane electrodes. The weighting potential is similarly attacked, but in that case no charges need to be taken into account, so the right side of Poisson s equation becomes zero, leading to Laplace s equation 2 φ w = 0 for the weighting potential. Here, the boundary conditions also apply to all electrodes, but only the readout strip is set to unity potential, while all others are at zero. Both equations are solved using a finite differences (iterative relaxation) method, which takes some time to converge. Once the potentials are available, the program derives the vectors for both drift and weighting fields (negative gradients of the potentials) and displays cuts of potential and field through the sensor thickness. Moreover, a traversing minimum ionizing particle can be simulated in a user selectable position, creating electrons and holes, which are homogeneously distributed along the detector thickness. These charges then move under the influence of the drift field and induce a current in the readout strip according to the weighting field. The currents due to electrons and holes are calculated separately and are also integrated over time to obtain collected charges. How to install? WEIGHTFIELD runs on Windows. It was developed in the LabWindows/CVI 8.1 environment by National Instruments (similar GUI as LabView, but with C code behind). If you have that software installed, you just need the binary file (weightfield.exe) and the GUI panel file (weightfield.uir) and that s all. If you don t have it, use the installer package (run setup.exe), which will install the binary 1

2 file as well as the LabWindows 8.1 run-time library. Alternatively, you can also download that library from and then just run the executable file. If you do not have Windows, it probably requires some porting of the code. Of course the full sources are also provided. The GUI Here is a screenshot of the graphical user interface (GUI). Essentially, all the controls are on the right edge of the window, while the results are displayed in the color map and the two graphs below. The color map in the center shows the cross-section of the detector having a width of 1000µm (x axis) and a thickness of 300µm (y axis, zero is at the backplane at bottom). The readout strip is always centered at the top edge of the detector (x=500µm, y=300µm); the backplane is located on the bottom (y=0µm). The switch on top allows selecting what to display: Freeze: keep the current display forever Weighting Potential: display the weighting potential of the center strip Drift Potential: display the drift potential The present screen can always be saved (in 24 bit PNG format) by hitting the Save Screen button and the Quit button (or the X on the top right corner of the window) ends the program without saving any data. 2

3 Calculating Potentials After starting the program, the memories for the potentials are empty, so we first either have to calculate them or load them from a file. If you want to load a previously saved file, you can skip this section. The controls that are highlighted below are relevant for calculating the potentials. The user can select Strip Pitch (500µm max) Strip Width (must be <= strip pitch) Depletion voltage (Vdepl) Bias voltage (Vbias, must be >= depletion voltage) Bulk material type (n or p type doping) Strip implant type (n or p type doping) Number of Iterations for the calculation of potentials The freedom of choosing bulk and strip types also allows simulating the condition of inversion. Assuming n-type bulk with p-strips (like in the CMS Strip Tracker), just switch to p-type bulk to simulate the situation after type inversion. The calculation can be started by hitting the Start button, and is done for the selected number of iterations (default: 1M). Each iteration is triggered by a timer with a frequency set by the Calc Interval control, which is by default set to zero (but you can decelerate if you really want). The present potential map is brought to the screen every 10 seconds by default; this interval can be adjusted by the Plot Update control. The current iteration is displayed as well as the overall progress 3

4 in percent. The calculation can always be aborted by hitting the Stop button, but cannot be resumed thereafter. The iterative relaxation method is a kind of diffusion process where the electrode potentials set by the initial boundary conditions are averaged over neighboring cells, adding a contribution for the space charge (see for details). Consequently, the progress is very slow it typically takes 100k to 1M events to converge, or a few hours of computing time on a modern CPU. The process could be considerably accelerated by using a multi-grid approach, where calculation starts on a course grid with subsequent refinement. Alternatively, the finite elements method could also provide faster calculation, but is also more complicated. Any contributions are highly welcome! After the potentials are done, they can be saved on disk by hitting the Save button. This allows to skip the time-consuming job next time by just loading previously calculated potentials. The file extension of the binary potentials maps is pmp. Each file needs 4.57MB. The user selected control (pitch and width, voltages, ) values are also saved and thus also loaded. Please note that the color scale for the weighting potential is nonlinear, while the drift field is linear. Displaying Cuts Once the potential maps are finished (either after calculation or by using the Load button), the corresponding electric fields are calculated and displayed on some locations of the grid as black lines with a length according to the absolute value and the proper direction. 4

5 By default, the bottom graphs display cuts through the sensor thickness for potential and field (y components only) of the following locations: Centered on the readout strip Centered between readout and first neighboring strip Centered on the first neighbor strip (weighting potential only) Centered on the second neighbor strip (weighting potential only) The latter two are identical to the readout strip in case of the drift field and thus are not shown if that display is selected. Alternatively, the Cursor field can be selected, which allows to display both potential and field for any x position of the sensor. The position can either be selected by entering the numeric value (in units of µm) in the Cursor X field or by dragging the vertical cursor line in the potential map with the mouse. The x position display is updated accordingly. The granularity of the potential maps is 1µm and the readout strip is at the (top) center at x=500µm. With a pitch of 50µm, as shown in the screenshot below, the first neighbor strips are centered at 450µm and 550µm, respectively. Using the cursor, the electric fields are displayed in the right graph for both x and y components as well as their absolute value which is calculated using E 2 2 d,x + E d,y. 5

6 Calculating the MIP Induced Currents Probably the most interesting feature is to simulate the current that flows through the readout strip following a MIP traversing the detector in perpendicular direction. In order to see those, both Cursor and Current switches need to be turned on. The vertical cursor works in the same way as when displaying cuts, and the induced currents due to electron and hole motions and their sum are now displayed in the right bottom plot showing current over time. Moreover, their integration (the collected charge) is displayed in the numeric indicators on the bottom right side (see screenshot on page 2). Please note that the calculation of currents might take a few seconds, depending on the speed of your CPU. In this calculation, electrons and holes are placed in the silicon detector along the selected path and their motion (due to the drift field) is simulated in steps of 0.1ns. At the same time, the induced currents in the readout electrode are calculated (determined by the weighting field). The simulation stops at 50ns, no matter whether all moving charges are collected or not. It is interesting to see that the electron-hole pairs generated beneath a neighboring strip (not the readout strip) induce both positive and negative currents in the readout strip, which result in a total integrated charge of approximately zero. That charge being not exactly zero (but less than 1% of the MIP deposition) demonstrates the limits of this simulation. Nonetheless this software is accurate enough to show that a typical strip detector primarily collects charges due to either electrons or holes (depending on the strip donor type), even though the Shockley-Ramo theorem states that both carriers are equally important however, this is only true for a parallel-plate configuration, which can be approximated by setting strip pitch and width to the maximum value of 500µm, as shown in the screenshot below. 6