13 dM dx = ln [ 1 − exp ( − 1 x )] − 1 [ exp ( 1 x ) − 1 ] x , (A7) $^{2}$M 1 1 d $^{2}$M dx 2 = − 1 [ exp ( 1 x ) − 1 ] x 3 − 1 [ exp ( 1 x ) − 1 ] 2 x $_{3}$. (A8) g$_{a,b}$ = 1 [ exp ( 1 x ) − 1 ] $^{a}$x $_{b}$. (A9) yields dg$_{a,b}$ dx = ag$_{a,b}$$_{+2}$ + ag$_{a}$$_{+1}$$_{,b}$$_{+2}$ − bg$_{a,b}$$_{+1}$, (A10) The second derivative contains only terms of the form Differentiating g$_{a,b}$ yields The first two derivatives are lim x → $_{0}$g$_{j,k}$ ( x ) = lim x → 0 exp ( − j x ) [ 1 − exp 1 x )] j x k = lim x → 0 exp ( − j x ) x k = lim y →∞ y k j $^{k}$exp ( y ) , d $^{n}$µ$_{E}$$_{0}$ ( T ) dT n ∣ ε$_{F}$$_{0}$ = const. , T → 0 = 0 for n ≥ 1 (A1) lim x → $_{0}$g$_{j,k}$ ( x ) = lim y →∞ k ! j $^{k}$exp ( y ) = 0 . (A11) lim x → 0 dM dx = ln 1 − lim x → $_{0}$g$_{1}$$_{,}$$_{1}$ = 0 , (A12) x = k$_{B}$T ε$_{F}$$_{0}$ (A2) µ$_{E}$$_{0}$ ≃ E$_{F}$$_{0}$ for k$_{B}$T ≪ ε$_{F}$$_{0}$. (A13) d $^{n}$¯ ε ( T ) dT n ∣ ε$_{F}$$_{0}$ = const. , T → 0 = 0 for n ≥ 1 (A14) M ( x ) = µ$_{E}$$_{0}$ ( T ( x )) − ε$_{F}$$_{0}$ ε$_{F}$$_{0}$ . (A3) d $^{n}$M ( x ) dx n = ε n − 1 F 0 k n B d $^{n}$µ$_{E}$$_{0}$ dT n for n ≥ 1 . (A4) F j ( η ) ≃ η j +1 Γ ( j + 2) . (A15) d $^{n}$M ( x ) dx n ∣ x → 0 = 0 for n ≥ 1 . (A5) M ( x ) = x ln [ 1 − exp ( − 1 x )] . (A6) ¯ ε ≃ ε$_{F}$$_{0}$ ( ε$_{F}$$_{0}$ µ$_{E}$$_{0}$ ) 5 for k$_{B}$T ≪ ε$_{F}$$_{0}$, (A16) scattering. At very low electron densities this screening becomes weaker and the scattering will finally increase so that the resistivity should be higher in this regime. These effects have not been taken into account in the frame of the current work, as we like to present the basic effects due to charging of trap states. Althogether, our detailed numerical calculations within the dipol trap model show that a pronounced metallic state can be caused by trap states at an appro- priate energy level inside the oxide of Si-MOS structures. For the realistic assumptions of energetic broadening and narrow spatial distribution near the oxide-semiconductor interface, the behavior is in close agreement with exper- imental observations. therefore all higher derivatives d $^{n}$M dx n also contain only terms g$_{j,k}$ with j ∈ { 1 , 2 , . . . , n } and k ∈ { n + 1 , n + 2 , . . . , 2 n − 1 } . Now we can multiply the nu- merator and the denominator of g$_{j,k}$ with exp ( − j/x ) and write ACKNOWLEDGMENTS Work was supported by the Austrian Science Founda- tion (FWF) project no. P16160. We thank our former colleague A. Prinz to perform first calculations within the Altshuler-Maslov Trap model. Appendix A: Temperature behavior of the chemical potential and of the effective electron energy for low temperatures with y = j/x . Applying the rule of L’Hospital k times we get First we show that holds for µ$_{E}$$_{0}$ ( T ) from Eq. (31). In our calculations we use the temperature T and the electron density in the inversion layer n$_{s}$ as independent variables. So it is al- lowed to set n$_{s}$ = const. and therefore also ε$_{F}$$_{0}$ = const. while varying T, see Eq. (32). The limit for the first derivative also vanishes so Eq. (A5) holds and With help of Eq. (A1) we show now that For simplicity we introduce the auxiliary variable and the function With the chain rule we find So Eq. (A1) is equivalent to From Eq. (31) we get holds for Eq. (38). For not too small η the Fermi-Dirac integral F j ( η ) can be approximated by 32 For low temperatures µ$_{E}$$_{0}$ approaches ε$_{F}$$_{0}$ and therefore it is positive, so from Eq. (38) we get