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with C = 8 (AM) or C = 16 (our assumption). (We use in this work the notation of capital E for absolute energies and Greek ε for energy differences.) Finally the trap energy can be written as 

where the subscript e in ε$_{e}$ stands for ’electrostatic’. The shape of E$_{T}$ ( Z ) is shown in Fig. 2 together with the relevant energy notation. 

Without the presence of a magnetic field the probabil- ity of a trap to be charged is given by 

This formula is similar to the Fermi distribution function, differences are the minus sign in the exponent and the prefactor 1 / 2 . A trap is charged when it is not occupied by an electron, thus the minus sign. If the trap is charged there are two possibilities for the spin orientation but only one if it is not charged, leading to the factor 1 / 2 . As mentioned before p$_{+}$ is determined by E$_{T}$ − µ (the vertical lines in Fig. 2) and the temperature. 

AM assume that a positive charged trap is screened by the electrons in the 2DEG and the trap forms together with that image charge a dipole. For the transport scat- 

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tering cross section σ$_{t}$ of this dipole they found classically 

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where Z is the distance between the trap and the oxide semiconductor interface, ε is the energy of the scattered electron relative to the ground state energy of the inver- sion layer E$_{0}$ : 

and ϵ ∗ is an effective dielectric constant 

where ϵ$_{sc}$ is the dielectric constant of the semiconductor. We have recalculated Eq. (8) and got the same result as AM. 

The Drude formula together with the Boltzmann equa- tion in relaxation time approximation yields the resis- tivity dρ ( Z ) caused by the charged traps within the layer [ Z, Z + dZ ] . By integrating this contributions over the whole oxide [0 , D ] one gets 

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where m$_{c}$ is the conductivity mass of the free electrons within the inversion layer and ¯ ε is an effective elec- tron energy as used by AM. In the corresponding AM equation the argument of the cosh function contains the Fermi energy, but should be replaced by the chemical potential. 31 Furthermore ε$_{F}$$_{0}$ = E$_{F}$ − E$_{0}$ is the Fermi energy, µ$_{E}$$_{0}$ = µ − E$_{0}$ is the chemical potential, each rel- ative to the ground state energy of the inversion layer, and N + T is the 3D density of charged traps. With the 3D trap density N$_{T}$ ( Z ) it is given by 

Now we like to present Eq. (12) in a form similar to the Drude formula 

and further find a term for the scattering rate 1 / 〈 τ 〉 in the usual form that scattering rate is equal to scattering cross section times density of scattering centers times ve- locity of scattered particles, which leads to one of the basic equations used throughout this work 

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