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

2. Operation and Radiation Damage of Silicon Detectors 21In a steady state, trap occupancy is constant in time ( dn T= 0) therefore emissions ofdtelectrons and holes have to be balanced re h = re. e From that and equations (2.29) the trapoccupancy f T can be determinedf T =e h N Ve e N C + e h N V(2.34)and by inserting this value back into emission rates (2.29) the creation rate of electron-holepairs can be determinedr e e = r h e = r pair =e hN V e e N Ce e N C + e h N VN T : (2.35)Inserting (2.31) into (2.35) and assuming N C N V , eT h T and v e v h one obtains 1r pair = N T ehv T eh N CV e ;Eg=2kBT cosh ;1 E T ; 1 k B T 2 (E V + E C ) : (2.36)Equation 2.36 shows that the creation rate of electron-hole pairs is maximal when E T =1(E 2 C + E V ). This means that the most ecient electron-hole pair generation centresare those located close to the middle of the bandgap. The situation is similar even ifN C N V , e T h T and v e v h are not assumed. In that case the trap energy formaximal r pair is shifted toE T (max) = 1 2E C + E V + k B T ln Te v e N Ch T v hN Vand the function deviates somewhat from the symmetrical form 10 .(2.37)The total current density due to generation in the depleted region is given by anintegral of the generation rate over the space charge regionj SCR = e 0 r pair w : (2.38)Since w(V ) / p V also the bulk generation current is proportional to p V as long as thediode is not fully depleted. To determine the temperature dependence, we use E T E i11eq. 2.36 and take into account v eh / p T (eq. 2.30). Thus we obtainEgj SCR / T 2 e ; 2k B T: (2.39)The bulk generated reverse current in an unirradiated diode is of the order ofnA/cm 2 , while for a diode irradiated to 10 14 n/cm 2 it is of the order of A/cm 2 . The10One of the exponential terms included in cosh is multiplied by a factor T e veNC T h v hN V.11Only traps close to mid-gap are considered, since they give the largest contribution to the leakagecurrent.