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SEITE 6 COMSOL ANWENDERKONFERENZ 2006 Charge Carrier Motion in Semiconductors Björn Kreisler, Gisela Anton, Jürgen Durst, Thilo Michel Physikalisches Institut Abt.IV, Erwin-Rommel-Straße 1, D-91058 Erlangen, [email protected]. Abstract The motion of free charge carriers in semiconductors was simulated using the convection and diffusion module in COMSOL. The focus of this work is the sensor layer of the Medipix2 x-ray detector, in our case made of silicon. The charge cloud generated by photon interactions within the sensor material moves through the material due to an ap- plied electric field. The charges are collected by the pixel electrodes attached to the bottom of the sensor layer. The Medipix2 readout ASIC is designed to count single photon interactions and therefore each pixel triggers on the coll- ected charge. The distribution of the charge between the electrodes is the main topic of this work. The interaction point of the x-ray photon in the sensor material, i.e. the sta- ring point of the charge cloud, was varied and the different charge distributions on the electrodes were calculated. Keywords diffusion, charge sharing, semiconductor, x-ray detector, Medipix, photon counting 1. Introduction X-ray imaging is a state of the art technique for medical diagnostics, as insights of the human body are possible wi- thout surgery. For these insights, specially resolved detec- tion of the transmitted x-ray photons is necessary. The de- tection can be done with films, gas detectors, scintillation detectors and direct converting semiconductor detectors. The detection efficiency of film is rather bad and as the linear regime of the intensity projection is quite poor, the future is pointed to the direct converting semiconductor detectors. In those detectors, x-ray photons are converted to free moving charges in a semiconductor sensor layer. These free moving charges drift through the sensor ma- terial due to a externally applied electric field. The time it takes the charge cloud to reach the pixelated electrode at- tached to the bottom of the sensor layer enables the charge cloud to broaden its diameter due to diffusion. The spatial resolution of the detection is therefore strongly dependent on the size of the pixels and on the distribution of the char- ge on the pixelated electrode. 2. Simulation 2.1 Governing Equations The motion of the charge cloud in the semiconductor is governed by drift and diffusion. The applied electric field E across the sensor layer results in a drift motion of the charges with direction to the pixelated electrodes and a speed v = mu*E where mu is the mobility constant of the charges in the semiconductor. For very high electric fields, the mobility constant may not be a constant any more, but this effect can be modelled by fitting a curve to the measu- red mobility. The isotropic diffusion process leads to a broadening of the charge cloud. The driving equation for the conservati- ve diffusion process of a concentration c is where D is the diffusion coefficient of the material. 2.2 Methods and Numerical Model The free charges in the semiconductor were simulated by moving a concentration with properties like charges. The concentration was given a specific mobility and diffusion constant. Special care has to be taken of the initial con- centration distribution. As the diffusion process is heavily dependent on the gradient of the concentration, no sharp edges and discontinuities are allowed. In the work presen- ted here, the initial concentration was modelled by a shif- ted cosine which was cut at the first minimum: where r=x-x 0 and r 0 is the maximal radius of the initial con- centration distribution. The variable A allows to control the total amount of charge to be simulated. The sampling rate needs to be of a very good quality for assuring the continuity. The simulation of the charge motion starts with the static calculation of the electric field in the sensor layer with the electrostatic module of COMSOL. The boundary conditi- ons need to be chosen reasonably. In the work presented here, the geometric setup allows the use of the symmetry of the electric field with respect to the centre of the pixel electrode. The motion of the concentration was coupled to this elec- tric field by linking the components of the velocity to the components of local electric field. As mentioned above, the mobility constant needs to be approximated by a function due to the saturation of the velocity for very high electric fields. The diffusion of the concentration due to concen- tration gradients was taken into account by applying an isotropic diffusion constant. Direct electrostatic repulsion of the charges inside the charge cloud was neglected, as the force between the charges is screened by the sensor material even at very short distances. The boundaries for Excerpt from the Proceedings of the COMSOL Users Conference 2006 Frankfurt
