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

16 2. Operation and Radiation Damage of Silicon Detectors2.1.3 The Fermi LevelThe occupation probability for electrons is given by theFermi-Dirac distribution function ;1f(E) = 1+e E;E Fk B T(2.14)For E C ;E F k B T (non-degenerate semiconductors) it converts to Boltzmann statistics,givingn = N C e ; E C ;E Fk B T(2.15)for the concentration of electrons in the conductive band [8]. Similarly, for the concentrationof holes in the valence band holdsp = N V e ; E F ;E Vk B T: (2.16)The Fermi level for the intrinsic case (n = p = n i ) can be determined from the chargeneutrality conditionE F (intrinsic) =E i = E C + E V2+ k BT2 ln NVN C(2.17)and the intrinsic carrier density is as given in eq. 2.1. It can be seen that in an intrinsicsemiconductor the Fermi level lies very close to the middle of the bandgap.In case of a doped semiconductor donor or acceptor impurities introduce newenergy levels in the bandgap. To preserve charge neutrality in thermal equilibrium theFermi level must adjust. Charge neutrality demandsn + N ; A = p + N + D(2.18)where N + D and N ; Aare numbers of ionised donors and acceptors respectively. In a caseof standard shallow level dopants (P ,B) most dopants are ionised at room temperature,thus N + D N D, N ; A N A. The concentration of electrons and holes in an n type sample(N D > N A ) follows from eq. 2.18 and 2.1 and for N D N A and N D ; N A n i theFermi level equalsE F = E C ; k B T lnNC(2.19)N Dwith the equivalent equationE F = E V + k B T lnNVN A(2.20)