9 Ω$_{j}$ ≡ ∫ D 0 N + T ( Z ) Z $^{j}$dZ = ∫ D 0 N$_{T}$ ( Z ) p$_{+}$ ( Z ) Z $^{j}$dZ. (49) N$_{T}$ ( Z ) = { n$_{T}$ Z$_{max}$ for 0 ≤ Z ≤ Z$_{max}$ 0 for Z > Z$_{max}$, (50) P$_{+}$ ( Z ) = ∫ ∞ −∞ g ( ˜ E$_{Ts}$, E$_{Ts}$, Δ E$_{T}$ ) 1 2 exp ( − ˜ E$_{Ts}$ − eV$_{ins}$ Z D − ε$_{D}$ D Z − µ k$_{B}$T ) + 1 d ˜ E$_{Ts}$. (51) VII. SPATIAL TRAP PROFILE VIII. BROADENING OF THE TRAP ENERGY If now the trap density N$_{T}$ is a function of Z it has to remain inside the integral Ω$_{j}$ (compare Eq. 18) For simplicity we use here an rectangular spatial trap profile from the OS interface to an arbitrary depth Z$_{max}$ , When n$_{s}$ is decreased not only µ$_{E}$$_{0}$ but also ε$_{0}$ decreases, thus µ drops off faster against E$_{Ts}$ and the transition is more abrupt. This can be seen in Fig. 8a) where re- sults for ε$_{TsCs}$ = const . (full lines) and ε$_{Ts}$$_{0}$ = const . (dashed lines) are compared. In both cases Ω$_{j}$ was cal- culated numerically. The critical curves n$_{s}$ = n$_{sc}$ coin- cide at T → 0 because the values ε$_{Ts}$$_{0}$ ≃ 42 . 0 meV and ε$_{TsCs}$ ≃ 68 . 4 meV were chosen in order to get the same n$_{sc}$ and for higher temperatures because ε$_{0}$ ( T ) is nearly constant over a wide temperature range (Fig. 8c)). For the 3D trap density we assumed again N$_{T}$ = 10 $^{18}$cm − $^{3}$. where n$_{T}$ is the 2D trap density. Fig. 9 shows ρ ( T, n$_{s}$ ) for Z$_{max}$ = 4 nm , the conduction band edge E$_{Cs}$ was used as reference energy, ε$_{TsCs}$ ≃ 95 . 7 meV was chosen in order to get n$_{sc}$ = 10 $^{11}$cm − $^{2}$. Where the 3D trap density N$_{T}$ does not vanish its value is assumed to be 2 . 98 × 10 $^{18}$cm − 3 as before (see Fig. 8b)), resulting in n$_{T}$ = N$_{T}$ · Z$_{max}$ = 1 . 19 × 10 $^{12}$cm − 3 of which again only a part is charged. It has to be emphasized that we still use Eq. (1). But the 2D density of positive charges in the depletion layer n$_{depl}$ is a byproduct of the calculation of ε$_{0}$ . So it is no longer a problem to use Eq. (2) V$_{ins}$ ∝ n$_{s}$ + n$_{depl}$ instead of (1) V$_{ins}$ ∝ n$_{s}$ . We will do so henceforward. As a result the slope of the line E$_{Ts}$ − eV$_{ins}$Z/D in Fig. 2 is increased, the maximum of E$_{T}$ ( Z ) falls off against µ , the p$_{+}$ ( Z ) - peak gets smaller and narrower. But it is still possible to let coincide the critical curves ρ ( n$_{s}$ = n$_{sc}$ ) by increasing the trap density N$_{T}$ in order to compensate the narrower p$_{+}$ ( Z ) -peak and by increasing ε$_{TsCs}$ in order to get the same critical value n$_{sc}$ . Here it is also important that the 2D charge carrier density of the depletion layer n$_{depl}$ ( T ) has almost no n$_{s}$ dependence for low temperatures as can be seen in Fig. 8d). The resistivity ρ ( T, n$_{s}$ ) , with N$_{T}$ = 2 . 98 × 10 $^{18}$cm − 3 and ε$_{TsCs}$ ≃ 95 . 7 meV , for which we get the same n$_{sc}$ as before, is shown in Fig. 8b). The above used depletion density n$_{depl}$ was calculated under the assumption of a background doping density of N$_{A}$ = 2 × 10 $^{15}$cm − $^{3}$, which is a typical value for high mobility Si-MOS samples. 35 Here the values for N$_{T}$ and ε$_{TsCs}$ were chosen in order to get the requested n$_{sc}$ for the given N$_{A}$ . In reality the value ε$_{TsCs}$ is determined by the chemical nature of the defect and thus the critical density may be different from sample to sample if the background doping is different. As can be seen in Fig. 9, the behavior for low tem- peratures has hardly changed, but for high temperatures ρ ( T, n$_{s}$ ) now saturates as a broadening of the p$_{+}$ ( Z ) -peak beyond Z$_{max}$ does not lead to a further increase in the number of charged scattering centers. This saturation of ρ ( T, n$_{s}$ ) is in fairly good agreement with experiments, where ρ for high T is limited as well. As the trap states will not all be identical and in ad- dition the stochastic position distribution will influence them mutually, their energetic position has to be broad- ened. If now the slope of the energy E$_{Ts}$ − eV$_{ins}$Z/D is higher due to the inclusion of n$_{depl}$ , the variable max ( E$_{T}$ ( Z ) − µ ) which is crucial for the resistivity is less sensitive on V$_{ins}$ and n$_{s}$ , therefore the transition is less abrupt. The n$_{s}$ dependence of n$_{depl}$ does not play a role because it hardly exists. We describe the broadening Δ E$_{T}$ with the help of a normalized distribution function g ( ˜ E$_{Ts}$, E$_{Ts}$, Δ E$_{T}$ ) for the trap energy E$_{Ts}$ which characterizes the trap, see Fig. 2. Now E$_{Ts}$ has the meaning of a mean value. (Mean value should not be understood in a strict mathematical sense, e. g. for the Lorentz distribution the mean value does not exist, but in this case it is obvious to take the energy E$_{Ts}$ where the distribution reaches its maximum.) Furthermore ˜ E$_{Ts}$ is the value for a particular trap. The probability of ˜ E$_{Ts}$ to lie within the interval [ E, E + dE ] is given by g ( E, E$_{Ts}$, Δ E$_{T}$ ) dE . Therefore we replace the probability p$_{+}$ of a trap to be charged by At higher temperatures the large value of k$_{B}$T leads to charged trap states in regions where the chemical poten- tial is even somewhat below the chemical potential (i. e. the p$_{+}$ ( Z ) -peak broadens) and therefore the resistivity is increased to unrealistic high values, see curves at higher temperature in Fig. 8. But an appreciable density of traps should exist only within the strained region of the oxide, 27 and thus the broadening of the peak beyond the width of this region leads to an unrealistic description. We can resolve this problem by introducing a spatial trap density profile N$_{T}$ ( Z ) . The denominator is that of p$_{+}$ , only E$_{Ts}$ is replaced by