4 Z O z S E$_{C}$ E$_{Cs}$ E$_{0}$ E$_{Ts}$ ϕ E$_{T}$(Z) p$_{+}$(Z) Figure 3. Oxide-semiconductor interface. Trap energy E T ( Z ) (full line), its Taylor approximation (dashed line), and the resulting probabilities p + ( Z ) in arbitrary units with µ as zero point. 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 ¯ Z 2 / 3 = ∫ D 0 N + T ( Z ) Z 2 / $^{3}$dZ ∫ D 0 N + T ( Z ) dZ = ∫ D 0 N + T ( Z ) Z 2 / $^{3}$dZ n + T (17) − for this factor in σ$_{t}$ ( ¯ ε, ¯ Z ) = c$_{σ}$ ¯ ε − 1 / 3 ¯ Z 2 / $^{3}$. In order to calculate the resistivity ρ + + In order to calculate the resistivity ρ the knowledge of n + T is not necessary, n + T 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 see Fig. 2 and 3. Now (18) can be written as Ω$_{j}$ ≃ N$_{T}$ ∫ D 0 Z $^{j}$dZ 1 2 exp ( µ$_{E}$$_{0}$ − ε$_{Ts}$$_{0}$ − ε$_{m}$ + ε$_{D}$ D Z 3 m ( Z − Z$_{m}$ ) 2 k$_{B}$T ) + 1 , (25) ε$_{Ts}$$_{0}$ = E$_{Ts}$ − E$_{0}$ = const. (26) Ω$_{j}$ ≡ ∫ D 0 N + T ( Z ) Z $^{j}$dZ = = N$_{T}$ ∫ D 0 Z $^{j}$dZ 1 2 exp ( − E$_{T}$ ( Z ) − µ k$_{B}$T ) + 1 . (18) AM assume that the energy E$_{Ts}$ relative to the ground state energy E$_{0}$ is constant, but we believe that rather the conduction band edge at the interface E$_{Cs}$ has to be used as reference energy, i. e. ε$_{TsCs}$ = E$_{Ts}$ − E$_{Cs}$ = const. This issue will be further treated in section VI. In the last step we followed AM and assumed that the trap density is constant within the oxide, respectively in the region where p$_{+}$ ( Z ) does not vanish. Now we can write The integrand is a peak around Z$_{m}$ which drops off ex- ponentially on both sides. In order to come to the same result as AM, we further apply the following approxima- tions: (i) Because of the exponential decrease one can integrate from −∞ to ∞ . (ii) The integrand is domi- nated by the denominator, so Z j ≃ Z j m can be set in the numerator. Now the integrand is symmetric around Z$_{m}$ and with help of the substitution Z = ε$_{D}$D Z 3 $_{m}$k$_{B}$ T ( Z − Z$_{m}$ ) 2 one gets σ$_{t}$ ( ¯ ε, ¯ Z ) = c$_{σ}$ ¯ ε − 1 / 3 ¯ Z 2 / 3 = c$_{σ}$ ¯ ε − 1 / $_{3}$Ω$_{2}$$_{/}$$_{3}$ Ω$_{0}$ , (19) + n T = Ω$_{0}$ , (20) ρ = √$_{2}$$_{m}$$_{c}$$_{c}$$_{σ}$$_{¯}$ ε 1 / $^{6}$Ω$_{2}$$_{/}$$_{3}$ n$_{s}$e 2 . (21) 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. Ω$_{j}$ = N$_{T}$ Z j +3 / 2 m √ k$_{B}$T ε$_{D}$D × ∫ ∞ 0 Z − 1 / $^{2}$d Z exp ( µ$_{E}$$_{0}$ − ε$_{Ts}$$_{0}$ − ε$_{m}$ k$_{B}$T − ln 2 + Z ) + 1 . (27) This corresponds to the integral in equation (9c) in $^{Ref. 26 (}$Z corresponds to x $^{2}$). We brought it into the form above as it corresponds now to a Fermi-Dirac integral 32 Z$_{m}$ = D √ ε$_{D}$ eV$_{ins}$ , (22) ε$_{m}$ = − 2 eV$_{ins}$ε$_{D}$, (23) ε$_{e}$ ( Z ) ≃ ε$_{m}$ − ε$_{D}$ D Z 3 m ( Z − Z$_{m}$ ) $^{2}$, (24) F k ( η ) = 1 Γ ( k + 1) ∫ ∞ 0 Z $^{k}$d Z exp ( Z − η ) + 1 , (28)