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Here σ$_{t}$ ( ¯ ε, ¯ Z ) is the transport scattering cross section from Eq. (8) for the effective electron energy ¯ ε and an effective distance ¯ Z between the traps and the OS inter- face, n + T = ∫ D 0 N + T ( Z ) dZ is the 2D density of charged traps, and v (¯ ε ) is the electron velocity which corresponds with ¯ ε = m$_{c}$v $^{2}$/ 2 . Comparing Eq. (12) and (16) we find 

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for this factor in ( ) In order to calculate the resistivity ρ of n + T is not necessary, n + T 

the knowledge cancels out with the denom- inator of ¯ Z 2 / 3 within σ$_{t}$ ( ¯ ε, ¯ Z ) , AM do not use it. As mentioned in the introduction we are interested in the metal-insulator transition depending on the electron den- sity n$_{s}$ , i. e. we want to know the temperature behavior of ρ as a function of n$_{s}$ . In this context n + T is very useful in order to understand on the basis of Eq. (16) that it contributes the main variations to the resistivity ρ ( n$_{s}$, T ) whereas σ$_{t}$ ( ¯ ε, ¯ Z ) and v (¯ ε ) show only weak dependence on n$_{s}$ and T . The benefit of Eq. (16) against (12) is, that the physical meaning of the terms is immediately clear. 

The integral in the numerator and that in the denom- inator of ¯ Z 2 / 3 can be treated in quite the same way, so we define 

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In the last step we followed AM and assumed that the trap density is constant within the oxide, respectively in the region where p$_{+}$ ( Z ) does not vanish. Now we can write 

To be able to calculate the integral which corresponds with Ω$_{2}$$_{/}$$_{3}$ AM expanded the electrostatic energy ε$_{T}$ ( Z ) into a Taylor series about the point Z$_{m}$ where it reaches its maximum ε$_{m}$ . This procedure is called saddle-point approximation. 

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see Fig. 2 and 3. Now (18) can be written as 

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AM assume that the energy E$_{Ts}$ relative to the ground state energy E$_{0}$ is constant, but we believe that rather the conduction band edge at the interface E$_{Cs}$ has to be used as reference energy, i. e. ε$_{TsCs}$ = E$_{Ts}$ − E$_{Cs}$ = const. This issue will be further treated in section VI. 

The integrand is a peak around Z$_{m}$ which drops off ex- ponentially on both sides. In order to come to the same result as AM, we further apply the following approxima- tions: (i) Because of the exponential decrease one can integrate from −∞ to ∞ . (ii) The integrand is domi- nated by the denominator, so Z j ≃ Z j m can be set in the numerator. Now the integrand is symmetric around Z$_{m}$ and with help of the substitution Z = ε$_{D}$D Z 3 $_{m}$k$_{B}$ T ( Z − Z$_{m}$ ) 2 one gets 

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$^{Ref. 26 (}$Z corresponds to x $^{2}$). We brought it into the form above as it corresponds now to a Fermi-Dirac integral 32 

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