2 substrate or (B) as in a 2D electron system with constant electron density. In this work, we precise and develop the AM trap model further in order to better understand the influ- ence of charged traps on the metallic state in Si-MOS structures. We present detailed numerical calculations of the temperature and density dependent resistivity ρ due to electronic scattering in the dipole trap model. Due to the numerical treatment we were able to drop the approx- imations a6), a7), and a8) of the analytic AM model. In addition, we have further extended our calculations for the more realistic case with energetic broadening and spa- tial distribution profile of the defect states, i. e. dropping also approximations a1) and a2). Figure 1. Schematic representation of the trap states together with the 2D electron system in the Si inversion layer. For E T > µ ( T ) the trap state is positively charged and scatters electrons in the 2D layer whereas for E T < µ ( T ) the trap is neutral. denotes µ ( T = 0) . As a result of our calculations, we find good agreement with the calculations of AM. There are mainly deviations in the overall behavior of the resistivity at low electron densities and at high temperatures. In order to under- stand the approximations of AM, we have also recalcu- lated the analytical model and were able to formulate it in a simplified way. Several mathematical terms are rear- ranged so that the scattering efficiency is expressed in the same form as in the usual Drude formulation. In addi- tion, their result contains two integrals, which we could replace by Fermi-Dirac integrals. Thus the known ap- proximations for the Fermi-Dirac integrals lead to simple equations for the resistivity ρ at low temperatures. A potential gradient due to an applied gate voltage V$_{g}$ causes a decrease of the trap energy E$_{T}$ = E$_{Ts}$ − eV$_{ins}$Z/D , with the unscreened trap energy E$_{Ts}$ at the oxide semiconductor (OS) interface, the voltage drop across the oxide (insulator) V$_{ins}$ , the distance from the OS interface Z , and the thickness of the oxide D . In their corresponding equation AM use the symbol V$_{g}$ in- stead of V$_{ins}$ . . But the total gate voltage V$_{g}$ is equal to the voltage drop across the oxide V$_{ins}$ plus the voltage drop across the depletion layer V$_{depl}$ . Later in their paper AM write that the threshold voltage is incorporated into V$_{g}$ , nevertheless they use practically an equation equiva- lent to V$_{ins}$ = en$_{s}$D/ϵ$_{ins}$ϵ$_{0}$, (1) V$_{ins}$ = e ( n$_{s}$ + n$_{depl}$ ) D/ϵ$_{ins}$ϵ$_{0}$ (2) V$_{g}$ = V$_{depl}$ + V$_{ins}$. (3) ε$_{D}$ = e 2 Cπϵ$_{ins}$ϵ$_{0}$D (4) II. TRAPMODEL The paper is organized as follows. In Sec. II the an- alytical formulation of the dipole trap model is given in detail and we show that within the saddle-point approxi- mation the result can be written in terms of Fermi-Dirac integrals. Section III treats the chemical potential and Sec. IV the analytic approximations for low tempera- tures. From the numerical integration, first results are given in Sec. V whereas in Sec. VI the calculations are extended for the case that the conduction band is the ref- erence energy for the trap energy and not the electronic ground state E$_{0}$ . In Sec. VII the model and the calcula- tion are extended for a spatial density distribution of the trap states and in Sec. VIII energetic broadening of trap states is taken into account. Conclusions are drawn in Sec. IX. In two appendices the behavior of the chemical potential and the ground state energy of the inversion layer are described in detail. Please note that we use SI units throughout this work. where ϵ$_{ins}$ is the dielectric constant of the oxide and ϵ$_{0}$ is the electric field constant. But if the threshold volt- age is incorporated into V$_{g}$ , the latter cannot be used to calculate the slope of the electrical potential within the oxide, as a part of V$_{g}$ falls off between the OS interface and the bulk layer. As the charges within the depletion layer (2D charge density − en$_{depl}$ ) also contribute to the gradient of the potential, we use the equation together with According to AM another term has to be added to the trap energy E$_{T}$ which describes the interaction with the two dimensional electron gas (2DEG) because of the im- age force. In Ref. 26 this term is given as − e $^{2}$/ 8 πϵ$_{ins}$ϵ$_{0}$Z (translated to the SI unit system). We think there should be a factor 16 in the denominator, see for instance Ref. 27 or Ref. 28. This factor does not change the re- sults of the trap model qualitatively therefore we write − e $^{2}$/Cπϵ$_{ins}$ ϵ$_{0}$Z . AM define an energy In the AM model it is assumed that a large number of hole trap states exists in the oxide at a certain trap energy E$_{T}$ . If the trap energy lies above the chemical potential µ , the trap is empty (in the electron picture, or has captured a hole in the equivalent description) and is positively charged, whereas if E$_{T}$ lies below µ it is filled with an electron and thus is neutral, see Fig. 1. Please note that we use in this work the terminology µ ( T ) for the chemical potential, the Fermi energy E$_{F}$