Study of radiation damage in silicon detectors for high ... - F9

3. Irradiation Facility 49width of depleted region w and consequently to p V , i.e. I = A p V . The magnitude ofthe parameter A can be estimated from the conditions at full depletion A = p I FDVFD. Thenthe resistance of the depleted bulk at given depletion voltage is given byR d = V I =VpVA p V = A= V FDI FDpV0(3.13)Using reverse current of I FD 25 A at full depletion voltage 25 of V FD 250 V oneobtainsR d =10M p V 0 : (3.14)One can see that in the described case R d =Z d 20 and Z u =R u 15. Thus thedepleted region can be approximated by a capacitor and the undepleted by a resistor.The signicance of the resistance of the undepleted region is however decreasing withincreasing depletion depth. Thus at half of full depletion depth (V = V FD =4), one alreadyobtains Z d =R u > 15. Thus for large depletion depths the bulk can be described well bya single capacitor and both serial and parallel model should give the same result. Theparallel model however has the advantage that it can take into account changes in thesurface current, presented by the surface resistor. They are hard to predict but can getvery important close to the breakdown where the dierential resistance is small. Since fordetermination of full depletion voltage from the C/V characteristic, we are interested inits shape close to full depletion, the parallel model suits better and was used throughoutthis work.The situation however changes dramatically when the measuring frequency is increased.This is reected in a decrease of impedance of both capacitors and at 1 MHz,Z u has the leading role also in the undepleted region. Then the equivalent circuit can beapproximated by two serial capacitors with a constant sum of capacitances. Thus at 1MHz the measured capacitance is almost independent of applied voltage, as seen in gure3.12. Similar eect is obtained by increasing the temperature. Since resistance of both depletedand undepleted region is approximately proportional 26 to e ;Eg=2k BT , increasing thetemperature by a few 10 C has the same relative eect as the above mentioned increasein frequency.Static resistivities, used in the above estimation, do not describe the system veryaccurately. And although dynamic resistivities are somewhat dierent 27 , the qualitativeconclusions remain the same.25Values of reverse current and full depletion voltage at the minimum after benecial annealing wereused.26For depleted region see eq. 2.38 and for undepleted regions equations 2.11 and 2.1.27A dierence of up to about factor of two was estimated.