1000T FIGURE 22. 4 The inverse temperature dependence of electron concentration in Si; 1 N,=10 7cm-3N=0; 2: N,=10 6 cm3,Nn=1014cm-3 the holes behind. Assuming that the Fermi level is removed from both E and E, by at least 3kBT(a nondegenerate semiconductor), the concentrations of electrons and holes are given by N expI(F-edIkBTI p=N expI(E-F)/kgT (22.5) where N =2(2mtrkgn)2// and N,= 2(2mttkgn)// are the effective densities of states in the conduction and valence bands, respectively, h is Plank constant, and the effective masses m* and m* depend on the detai of the band structure [ Pierret, 19871 ( E /kg)2 n is independent of the doping neutrality condition can be used to show that in an n-type(n>p)semiconductor at or below room temperature n(n+Na)(Na-Na-n)-1=(N/2)exp(-△W∥k) (22.6) where Na and Na denote the concentrations of donors and acceptors, respectively Corresponding temperature dependence is shown for silicon in Fig. 22.4. Around room temperature n AW/2 for n >N, and AW, for n< N The reduction of n compared with the net impurity concentration.2 Na-Na, while at low temperatures n is an exponential function of temperature with the activation energ Na is known as a freeze-out effect. This effect does not take place in the heavily doped semiconductor For temperatures T> T:=(E/2kB)/n[VNN,/(Na-Na)] the electron concentration n=n>>No-N,is no longer dependent on the doping level (Fig. 22.4). In this so-called intrinsic regime electrons come directly from the valence band. A loss of technological control over n and p makes this regime unattractive for electronic e 2000 by CRC Press LLC© 2000 by CRC Press LLC the holes behind.Assuming that the Fermi level is removed from both Ec and Ev by at least 3kBT (a nondegenerate semiconductor), the concentrations of electrons and holes are given by n = Nc exp[(F – Ec)/kBT] (22.4) and p = Nv exp[(Ev – F)/kBT] (22.5) where Nc = 2 (2m* npkBT)3/2/h3 and Nv = 2(2m* ppkBT)3/2/h3 are the effective densities of states in the conduction and valence bands, respectively, h is Plank constant, and the effective masses m* n and m* p depend on the details of the band structure [Pierret, 1987]. In a nondegenerate semiconductor, np = NcNv exp(–Eg /kBT) D = n2 i is independent of the doping level. The neutrality condition can be used to show that in an n-type (n > p) semiconductor at or below room temperature n(n + Na)(Nd – Na – n)–1 = (Nc/2) exp(–DWd /kBT) (22.6) where Nd and Na denote the concentrations of donors and acceptors, respectively. Corresponding temperature dependence is shown for silicon in Fig. 22.4. Around room temperature n = Nd – Na, while at low temperatures n is an exponential function of temperature with the activation energy DWd /2 for n > Na and DWd for n < Na. The reduction of n compared with the net impurity concentration Nd – Na is known as a freeze-out effect. This effect does not take place in the heavily doped semiconductors. For temperatures T > Ti = (Eg /2kB)/ln[ /(Nd – Na)] the electron concentration n ª ni >> Nd – Na is no longer dependent on the doping level (Fig. 22.4). In this so-called intrinsic regime electrons come directly from the valence band. A loss of technological control over n and p makes this regime unattractive for electronic FIGURE 22.4 The inverse temperature dependence of electron concentration in Si; 1: Nd = 1017 cm–3, Na = 0; 2: Nd = 1016 cm–3, Na = 1014 cm–3. N Nc v