7 We like to explain the shape of ρ ( T, n$_{s}$ ) starting from a value n$_{s}$ > n$_{sc}$ . The maximum of ε$_{T}$ ( Z ) lies below µ , therefore the p$_{+}$ ( Z ) -peak is very small and narrow. When n$_{s}$ is decreased µ drops off (see Fig. 5) and the resistivity ρ rises fast because − ( E$_{T}$ − µ ) /k$_{B}$T is the ex- ponent in the denominator of p$_{+}$ . It rises the faster the smaller the temperature is. When µ reaches the max- imum of the trap energy E$_{T}$ the p$_{+}$ -peak has a height of 2 / 3 , therefore further decreasing of n$_{s}$ and µ cannot increase the height of the peak appreciable (on a loga- rithmic scale, which covers some orders of magnitude), only the width. For increasing temperature T also the chemical poten- tial µ measured against E$_{0}$ decreases which results in increasing resistivity ρ . Additionally the p$_{+}$ ( Z ) -peak is broadened because the Fermi distribution function de- clines over several k$_{B}$T . Applying the appropriate approximations results in and a critical density on the sign of ε$_{Ts}$$_{0}$ + ε$_{m}$ − ε$_{F}$$_{0}$ . (Here µ$_{E}$$_{0}$ is replaced by ε$_{F}$$_{0}$ as by definition µ$_{E}$$_{0}$ → ε$_{F}$$_{0}$ for T → 0 .) This is conform with AM’s definition of the transition point, ( ε$_{Ts}$$_{0}$ + ε$_{m}$ − ε$_{F}$$_{0}$ ) /k$_{B}$T = 0 . Accordingly we define ε$_{mF}$ = ε$_{Ts}$$_{0}$ + ε$_{m}$ − ε$_{F}$$_{0}$, ε$_{mF}$ > 0 → insulating , ε$_{mF}$ < 0 metallic , ε$_{mF}$ = 0 → transition point n$_{sc}$ = n$_{s}$ | ε$_{mF}$ =0 . (45) Ω$_{j}$ ≃     $^{2}$$^{N$_{T}$}$$^{Z}$$^{j}$$^{+3}$$^{/}$$^{2}$ m √ k$_{B}$T ln 2 ε$_{D}$D + ε$_{mF}$ ε$_{D}$D for ε$_{mF}$ > 0 2 N$_{T}$ Z j +3 / 2 m k$_{B}$Tπ ε$_{D}$D exp ( ε$_{mF}$ k$_{B}$T ) for ε$_{mF}$ < 0 F − 1 / $_{2}$(ln 2) ︸ ︷︷ ︸ ≃ 0 . 891 N$_{T}$ Z j +3 / 2 m √ k$_{B}$Tπ ε$_{D}$D for ε$_{mF}$ = 0 . (46) The saddle-point approximation works best for small temperatures T and densities n$_{s}$ > n$_{sc}$ , in this domain the p$_{+}$ ( Z ) -peak is very narrow and therefore the Taylor approximation of the trap energy is accurate within the peak. For large T and/or n$_{s}$ < n$_{sc}$ the deviations of the saddle-point approximation from the numerical calcula- tions is visible in Fig. 6. It should be noted that according to the calculations, the resistivity ρ drops to arbitrarily low values in the metallic regime. This is caused by the fact that electron scattering is taken into account only by the trap states at a single trap energy. If this trap energy is below the Fermi energy, with decreasing T the number of charged scattering centers goes to zero. Only if other scattering effects, like residual impurities, surface roughness, accep- tor states in the depletion layer, etc. are included, the low- T resistivity would be limited. As will be seen later, an energetic broadening of the trap states will have a similar effect. Please note an interesting behavior. When setting ε$_{mF}$ = 0 in the first two equations for T > 0 they converge nei- ther into each other nor into the third one. This apparent discrepancy can be understood, as | ε$_{mF}$ | gets smaller and smaller the maximum temperature where the approxima- tions for the Fermi-Dirac integral are just applicable also gets smaller and smaller and finally vanishes for ε$_{mF}$ = 0 . Indeed for ε$_{mF}$ = 0 and T → 0 the three cases yield the same result, i. e. Ω$_{j}$ = 0 . If the bands in the semiconductor and in the oxide are bent due to an applied gate voltage all energies move up and down with the bands. The energetic position of the trap states E$_{Ts}$ should thus be fixed relative to the conduction band edge and not to the ground state energy E$_{0}$ of the inversion layer as assumed by AM. VI. CONDUCTION BAND AS REFERENCE ENERGY V. COMPARISON OF ANALYTIC AND NUMERICAL RESULTS µ$_{Cs}$ = µ − E$_{0}$ + E$_{0}$ − E$_{Cs}$ = µ$_{E}$$_{0}$ + ε$_{0}$, (47) In order to get rid of the restrictions which came from the saddle-point approximation we have performed nu- merical calculations of the integrals Ω$_{j}$ (Eq. (25)). In Fig. 6 we show the resistivity ρ depending on the tem- perature T and the 2D electron density in the inversion layer n$_{s}$ calculated with the help of the saddle-point ap- proximation (full lines), the approximation for low tem- peratures (dashed lines), and the numerical integration (markers). We chose ε$_{Ts}$$_{0}$ ≃ 42 . 02 meV in order to get n$_{sc}$ = 10 $^{11}$cm − $^{11}$, for the 3D trap density we assumed N$_{T}$ = 10 $^{18}$cm − $^{3}$. The value for N$_{T}$ seems to be quite high, but only in a very narrow layer the traps will in- deed be charged (where the trap states are above the chemical potential) and contribute to scattering. A fur- ther limitation of the available trap states into a narrow region besides the interface will follow later in this work. The critical behavior of ρ ( T ) versus n$_{s}$ for the homo- geneous trap density N$_{T}$ can be seen most clearly on the double logarithmic Figs. 6a) and c). We added 6b) as it is in this form directly comparable with Fig. 1b) in the work of AM. 26 From Eq. (7) we see that E$_{T}$ − µ = E$_{Ts}$ + ε$_{e}$ ( Z ) − µ determines the probability p$_{+}$ of a trap to be charged. So if we measure E$_{Ts}$ against E$_{Cs}$ we also have to know µ$_{Cs}$ = µ − E$_{Cs}$ . We find where ε$_{0}$ = E$_{0}$ − E$_{Cs}$ is the ground state energy relative to the conduction band edge at the OS interface. The chemical potential µ$_{E}$$_{0}$ follows from Eq. (31), but an accurate calculation of ε$_{0}$ is rather complex. For sim- plicity we follow the calculation method of Ando, Fowler