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where Γ is the Gamma function. A comparison yields 

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On the right hand side Z$_{m}$ , so within the saddle-point approximation Eq. (17) simplifies to 

AM described two scenarios for the temperature be- havior of the chemical potential: (A) The chemical po- tential of the 2DEG and of the Si substrate coincide. (B) The 2DEG is disconnected form the substrate. For the case (A) they assumed that the temperature behavior in the 2DEG is the same as in the bulk. However, they did not take into account that the chemical potential in the 2DEG is measured against the ground state energy E$_{0}$ and in the bulk against the conduction or valence band edge, i. e. they assumed E$_{0}$ and the band bending to be fixed. 

For (B) AM used an equation analogous to 

If only one subband of the inversion layer is occupied (quantum limit), the Fermi energy relative to its ground state energy is given by 28 

where g$_{s}$ , g$_{v2D}$ , and m$_{d2D}$ are the spin degeneracy, the valley degeneracy (for the 2DEG), and the density-of- states mass (2D) respectively. The two equations above can be derived from 

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(semiconductor side of the OS interface) and are as fol- lows. In thermal equilibrium there is a single chemical potential µ throughout the structure. For a certain tem- perature T , µ is determined in the bulk by the (residual) doping density (giving µ$_{Cb}$ = µ − E$_{Cb}$ , for details see App. B). In the inversion layer the position of µ relative to E$_{0}$ (i. e. µ$_{ε}$$_{0}$ ) follows directly from the 2D density n$_{s}$ . 

## III. CHEMICAL POTENTIAL 

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The band bending E$_{Cb}$ − E$_{Cs}$ adjusts so that 

If the band bending is increased, the quantum well gets narrower and the ground state energy increases. Thus ε$_{0}$ itself is a function of the band bending, a self consistent calculation solves this problem (fix point iteration). 

Another possibility is that the 2D electron system is at low temperatures decoupled from the bulk substrate and no common chemical potential exists. We do not treat this case in this work, but expect a similar behavior. 

Fig. 5 shows the behavior of µ ( T, n$_{s}$ ) which is crucial for the understanding of the behavior of ρ ( T, n$_{s}$ ) . The chemical potential relative to the ground state energy of the inversion layer µ$_{E}$$_{0}$ ( T, n$_{s}$ ) increases for decreasing T and for increasing n$_{s}$ . 

The parameters, which were used here and for the fol- lowing calculations are collected in Table I. We assumed a {001} plane for the silicon surface. 

## IV. APPROXIMATION FOR LOW TEMPERATURES 

For given values T and n$_{s}$ we calculate the resistivity ρ with help of the equations (21) and (29). Beside the explicit temperature dependence also the chemical poten- tial µ$_{E}$$_{0}$ and the effective electron energy ¯ ε are functions of T . 

As a first step we replace the integral in (13) by a 

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