5 where Γ is the Gamma function. A comparison yields Ω$_{j}$ = N$_{T}$ Z j +3 / 2 m √ k$_{B}$T ε$_{D}$D × √$_{π}$ ︸︷︷︸ Γ ( 1 $_{2}$) F − 1 / 2 ( ln 2 + ε$_{Ts}$$_{0}$ + ε$_{m}$ − µ$_{E}$$_{0}$ k$_{B}$T ) . (29) On the right hand side j appears only in the exponent of Z$_{m}$ , so within the saddle-point approximation Eq. (17) simplifies to Figure 4. Band bending and notation on semiconductor side of the OS interface for thermal equilibrium between 2D layer and bulk. The electronic ground state energy of the inversion layer E 0 , the conduction band edge E C , its values at the in- terface E Cs and in the bulk E Cb , the valence band edge E V , and the chemical potential µ are shown schematically, also the ground state energy relative to the conduction band edge ε 0 , the chemical potential relative to conduction band edge in the bulk µ Cb and relative to the ground state energy µ E 0 are noted. ¯ Z 2 / 3 = Ω$_{2}$$_{/}$$_{3}$ Ω$_{0}$ ≃ Z 2 / 3 m . (30) III. CHEMICAL POTENTIAL IV. APPROXIMATION FOR LOW TEMPERATURES 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 The band bending E$_{Cb}$ − E$_{Cs}$ adjusts so that E$_{Cb}$ − E$_{Cs}$ = ε$_{0}$ + µ$_{E}$$_{0}$ − µ$_{Cb}$. (34) µ$_{E}$$_{0}$ = k$_{B}$T ln [ exp ( ε$_{F}$$_{0}$ k$_{B}$T ) − 1 ] . (31) If the band bending is increased, the quantum well gets narrower and the ground state energy increases. Thus ε$_{0}$ itself is a function of the band bending, a self consistent calculation solves this problem (fix point iteration). If only one subband of the inversion layer is occupied (quantum limit), the Fermi energy relative to its ground state energy is given by 28 Another possibility is that the 2D electron system is at low temperatures decoupled from the bulk substrate and no common chemical potential exists. We do not treat this case in this work, but expect a similar behavior. Fig. 5 shows the behavior of µ ( T, n$_{s}$ ) which is crucial for the understanding of the behavior of ρ ( T, n$_{s}$ ) . The chemical potential relative to the ground state energy of the inversion layer µ$_{E}$$_{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. 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 ε$_{F}$$_{0}$ = 2 π ℏ $^{2}$n$_{s}$ g$_{s}$g$_{v2D}$m$_{d2D}$ , (32) n$_{s}$ = g$_{s}$g$_{v2D}$m$_{d2D}$ 2 π ℏ 2 ︸ ︷︷ ︸ 2D density of states ∫ ∞ E$_{0}$ 1 exp ( E − µ k$_{B}$T ) + 1 ︸ ︷︷ ︸ Fermi − Diracdistribution dE. (33) Our assumptions are shown schematically in Fig. 4 (semiconductor side of the OS interface) and are as fol- lows. In thermal equilibrium there is a single chemical potential µ throughout the structure. For a certain tem- perature T , µ is determined in the bulk by the (residual) doping density (giving µ$_{Cb}$ = µ − E$_{Cb}$ , for details see App. B). In the inversion layer the position of µ relative to E$_{0}$ (i. e. µ$_{ε}$$_{0}$ ) follows directly from the 2D density n$_{s}$ . 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