10 (a) (b) 10 −1 10 0 10 1 10 2 10 3 10 −2 10 −1 10 0 10 1 10 2 T (K)  (h/e 2 ) (c) (d) 10 −1 10 0 10 1 10 2 10 3 23 24 25 26 27 28 T (K) 0  (meV) 10 −1 10 0 10 1 10 2 10 3 1.4 1.5 1.6 1.7 1.8 T (K) n depl  (10 11 cm −2 ) Figure 8. Resistivity ρ ( T, n s ) for the potential across th e oxide layer a) V ins ∝ n s with N T = 10 $^{18}$cm $_{−}$3 a n d ε TsCs ≃ 68 . 4 meV and b) V ins ∝ n s + n$_{depl}$ with N T = 2 . 98 × 10 $^{18}$cm $_{−}$3 and ε TsCs ≃ 95 . 7 meV . The parameters are chose n to give a critical density of n sc = 10 11 cm $_{−}$2 for both cases. At the individ ual curves n s = 0 . 75 , . . . , 1 . 25 × n sc in steps of 0 . 0 5 × n sc from top to bottom, full lines represent ε TsCs = const . (realistic cas e), dashed lines ε $^{Ts}$0 = const . (for comparison). In c) the ground state energy of the inversion layer ε 0 ( T, n s ) is shown for same n s values as before, but now assigned bottom up. I n d) the depletion density n$_{depl}$ ( T, n s ) is shown, but the n s dependence va nishes within the line width. Figure 9. Resistivity ρ ( T, n s ) for rectangular spatial trap profile with Z max = 4 nm with curves of constant n s . The critical density is n sc = 10 $^{11}$cm $_{−}$2 which means that ε TsCs has to be 95 . 7 meV . Densities for full lines are n s = 0 . 75 , . . . , 1 . 25 × n sc in steps of 0 . 05 × n sc , and for d a shed lines n s = 0 . 96 , . . . , 1 . 04 × n sc in steps of 0 . 01 × n sc (a lways top down), with N T = n T /Z max = 2 . 98 × 10 $^{18}$cm $^{$_{−}$3}$. 10 −1 10 0 10 1 10 2 10 3 10 −3 10 −2 10 −1 10 0 10 1 T (K)  (h/e 2 ) α = Δ E$_{T}$ k$_{B}$T , (52) β = E$_{Ts}$ − eV$_{ins}$ Z D − ε$_{D}$ D Z − µ k$_{B}$T (53) η = ˜ E$_{Ts}$ − E$_{Ts}$ Δ E$_{T}$ , (54) g ( ˜ E$_{Ts}$, E$_{Ts}$, Δ E$_{T}$ ) = 1 Δ E$_{T}$ h ( ˜ E$_{T}$$_{s}$ − E$_{Ts}$ Δ E$_{T}$ ) , (55) g ˜ E$_{Ts}$, E$_{Ts}$, Δ E$_{T}$ = 1 Δ E$_{T}$ h ( η ) , (56) P$_{+}$ ( α, β ) = ∫ ∞ −∞ h ( η ) 1 2 exp ( − αη − β ) + 1 dη. (57) the probability P$_{+}$ can be written as and a dimensionless distribution funct ion h ( η ) defined by ˜ E$_{Ts}$ . By introducing the dimensionless parameters