content_image/427213.jpg content_image/427214.jpg content_image/427216.jpg content_image/427217.jpg content_image/427215.jpg content_image/427212.jpg
As a rule this integral cannot be calculated analytically. An exception from this rule is the uniform distribution. If we define the width of the ’rectangle’ as 2Δ E$_{T}$ we get 

and 

We also use the normal distribution 

with the standard deviation as Δ E$_{T}$ and the Lorentz distribution (natural line broadening) 

with the half full width at half maximum (FWHM) as Δ E$_{T}$ . 

Fig. 10 shows the numerical results for the three differ- ent broadening distributions and different width Δ E$_{T}$ = 0 . 02 , 0 . 1 and 0 . 5 meV . Again we took the conduction band edge at the interface E$_{Cs}$ as reference energy, as- sumed that traps exist in the oxide only within 4 nm from the OS interface (constant trap density within this region), and used Eq. (2) instead of (1). We chose ε$_{TsCs}$ ≃ 95 . 7 meV in order to get n$_{sc}$ = 10 $^{11}$cm − 2 and for the 3D trap density N$_{T}$ = n$_{T}$ /Z$_{max}$ = 10 $^{18}$cm − $^{3}$. 

As one would expect, the transition is less abrupt for higher Δ E$_{T}$ and does not vanish even for Δ E$_{T}$ = 0 . 5 meV . If the broadening is indeed caused by potential fluctuations due to disorder or trap-trap interaction, the value of Δ E$_{T}$ could even be much larger than k$_{B}$T and the transition might be smeared out even stronger. 

In the metallic regime the mean trap energy is below the chemical potential. As the normal and the Lorentz distribution have tails, there always remain some charged traps from the upper tail when otherwise all traps would be filled with electrons and therefore would be neutral. On a logarithmic resistivity scale the ρ ( T ) -behavior is changed drastically by the few additional charged traps. 

In the insulating regime the mean trap energy is above the chemical potential. Here a large part of the traps is charged and by contrast the few uncharged traps due to the lower tail of the energy distribution hardly play any role. 

By means of analytical considerations we got the fol- lowing estimations for the temperature T$_{b}$ below which the resistivity becomes almost constant. In general the resistivity ρ varies by some orders of magnitude, so as a criterion for being almost constant we took that re- gion where ρ changes finally by only a factor two down to zero temperature. The markers in Fig. 10 represent these temperatures T$_{b}$ . 

<img src='content_image/427213.jpg'>

<img src='content_image/427214.jpg'>

<img src='content_image/427216.jpg'>

<img src='content_image/427217.jpg'>

According to this definition, for the insulating behavior n$_{s}$ < n$_{sc}$ for all three distributions we get 

and for the metallic behavior n$_{s}$ > n$_{sc}$ 

<img src='content_image/427215.jpg'>

The uniform distribution has no tails so in the metallic regime there is no temperature range where ρ is almost constant. 

In this work we have performed numerical calcula- tions within the dipole trap model for Si-MOS structures. Originally this model was proposed by Altshuler and Maslov with several approximations, in order to be able to get analytical solutions. Due to our numerical treat- ment we could eliminate several approximations. We de- scribe the potential inside the insulator by its detailed spatial dependence instead of the saddle-point approxi- mation with the quadratic dependence around its max- imum, we fix the trap state energy relative to the con- duction band edge instead of relative to the electronic ground state inside the triangular potential well and we have taken into account the detailed change of the chem- ical potential in the two-dimensional electron layer with respect to the bulk material, which seems to be more realistic than the two cases in the original treatment. 

According to our calculations, the metallic regime at high electron densities n$_{s}$ , where the resistivity is de- creasing towards lower temperature, is strongly devel- oped . Also a critical density n$_{sc}$ can be identified with a characteristic temperature dependence different from the metallic and the ’insulating’ region. For electron densi- ties n$_{s}$ < n$_{sc}$ , the resistivity curves satturate towards lower temperature and remain constant when the tem- perature approaches zero. They do not show an insult- ing behavior in the sense that ρ increases towards zero temperature. Such an increase can in principle be caused e. g. by a further decrease of the chemical potential µ with temperature, as in the work of Altshuler and Maslov for case A, where it was assumed that the temperature de- pendence of µ ist the same for the two-dimensional elec- tron layer as it is in the Si-bulk material. In the dipol trap model a constant efficient screening is assumed in order to clarify the effects which are caused when the traps change their charge state. A realistic treatment of the temperatue dependence of the electronic screening could also cause an increase in ρ towards lower temper- ature in that it favors the formation of dipol trap states. 

<img src='content_image/427212.jpg'>

## IX. CONCLUSIONS 

