Study of radiation damage in silicon detectors for high ... - F9
Study of radiation damage in silicon detectors for high ... - F9
Study of radiation damage in silicon detectors for high ... - F9
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2. Operation and Radiation Damage <strong>of</strong> Silicon Detectors 21In a steady state, trap occupancy is constant <strong>in</strong> time ( dn T= 0) there<strong>for</strong>e emissions <strong>of</strong>dtelectrons and holes have to be balanced re h = re. e From that and equations (2.29) the trapoccupancy f T can be determ<strong>in</strong>edf T =e h N Ve e N C + e h N V(2.34)and by <strong>in</strong>sert<strong>in</strong>g this value back <strong>in</strong>to emission rates (2.29) the creation rate <strong>of</strong> electron-holepairs can be determ<strong>in</strong>edr e e = r h e = r pair =e hN V e e N Ce e N C + e h N VN T : (2.35)Insert<strong>in</strong>g (2.31) <strong>in</strong>to (2.35) and assum<strong>in</strong>g N C N V , eT h T and v e v h one obta<strong>in</strong>s 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 <strong>of</strong> 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 <strong>of</strong> 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 <strong>for</strong>maximal 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 <strong>for</strong>m 10 .(2.37)The total current density due to generation <strong>in</strong> the depleted region is given by an<strong>in</strong>tegral <strong>of</strong> the generation rate over the space charge regionj SCR = e 0 r pair w : (2.38)S<strong>in</strong>ce w(V ) / p V also the bulk generation current is proportional to p V as long as thediode is not fully depleted. To determ<strong>in</strong>e the temperature dependence, we use E T E i11eq. 2.36 and take <strong>in</strong>to account v eh / p T (eq. 2.30). Thus we obta<strong>in</strong>Egj SCR / T 2 e ; 2k B T: (2.39)The bulk generated reverse current <strong>in</strong> an unirradiated diode is <strong>of</strong> the order <strong>of</strong>nA/cm 2 , while <strong>for</strong> a diode irradiated to 10 14 n/cm 2 it is <strong>of</strong> the order <strong>of</strong> A/cm 2 . The10One <strong>of</strong> the exponential terms <strong>in</strong>cluded <strong>in</strong> cosh is multiplied by a factor T e veNC T h v hN V.11Only traps close to mid-gap are considered, s<strong>in</strong>ce they give the largest contribution to the leakagecurrent.