14 and with the chain rule respectively Faá di Bruno’s for- mula to get U$_{d}$ (0)+ U$_{s}$ (0) = 0 which means that the resulting ground state energy is measured against the conduction band edge at the interface E$_{Cs}$ as requested. The Hamiltonian is given by ˆ H = ˆ T + U with the oper- ator for the kinetic energy ˆ T = − ℏ 2 2 m$_{z}$ ∂ 2 ∂z 2 , where m$_{z}$ is the z -component of the effective mass of the semiconductor in the bulk. The ground state energy is the expectation value of the Hamiltonian, calculated with the value of b which makes the total en- ergy per electron where the sum runs over all integer numbers k$_{1}$, . . . , k$_{n}$ ≥ 0 which fulfill minimal. The factor 1 / 2 in the third term prevents from double counting the electron-electron interaction. With the density n ∗ = n$_{depl}$ + 11 $_{32}$n$_{s}$ introduced by AFS 28 one gets We do not have to find this numbers, we only need to know that for n > 1 at least one k$_{j}$ > 0 , therefore each term in (A18) contains a factor d $^{j}$µ$_{E}$$_{0}$ dT j , so for n$_{s}$ = const . , T → 0 the sum vanishes and from Eq. (A16) we get Appendix B: Ground state energy of the inversion layer As mentioned before we calculate the ground state en- ergy of the inversion layer with help of the Ritz vari- ational principle using the Fang-Howard test envelope wave function 34 and as an approximation for the potential N$_{A}$ and N$_{D}$ are the densities of the acceptors and donors respectively. The coefficients have been chosen in order to get α, β, γ, δ > 0 and to get a most simple equation As AFS used e $^{2}$n$_{depl}$ ϵ$_{sc}$ϵ$_{0}$ z ( 1 − z 2 z$_{d}$ ) for 0 < z < ∞ instead of Eq. (B3) they did not get the term proportional to exp ( − bz$_{d}$ ) , under normal circumstances it is very small but formally it is necessary to see that b → 0 , ˜ ε → −∞ is not the global minimum. Furthermore they neglected β and δ ( b ) b − 3 and got The term U$_{d}$ comes from the charged acceptors within the depletion layer with thickness z$_{d}$ , U$_{s}$ describes the inter- action with all other electrons in the inversion layer, and U$_{i}$ the interaction with image charges. To write U$_{s}$ ( z ) in this form we have to assume that only the first sub- band is occupied, this is the so called quantum limit. The conduction band edge is built by U$_{d}$ ( z ) + U$_{s}$ ( z ) , therefore the zero point of the energy scale was chosen d ¯ ε dT = d ¯ ε dµ$_{E}$$_{0}$ dµ$_{E}$$_{0}$ dT , (A17) d $^{n}$¯ ε dT n = ∑ n ! k$_{1}$ ! . . . k$_{n}$ ! d $^{k}$¯ ε dµ k E$_{0}$ × ( 1 1! dµ$_{E}$$_{0}$ dT ) k$_{1}$ . . . ( 1 n ! d $^{n}$µ$_{E}$$_{0}$ dT n ) k$_{n}$ , (A18) k = k$_{1}$ + · · · + k$_{n}$, ε$_{0}$ = 〈 ˆ T 〉 + 〈 U$_{d}$ 〉 + 〈 U$_{s}$ 〉 + 〈 U$_{i}$ 〉 , (B6) k$_{1}$ + 2 k$_{2}$ + · · · + nk$_{n}$ = n. (A19) ˜ ε = 〈 ˆ T 〉 + 〈 U$_{d}$ 〉 + 1 2 〈 U$_{s}$ 〉 + 〈 U$_{i}$ 〉 (B7) ¯ ε ≃ ε$_{F}$$_{0}$ for k$_{B}$T ≪ ε$_{F}$$_{0}$. (A20) ˜ ε = α 2 b 2 + βb + γb − 1 − δ ( b ) 2 b − $^{2}$, (B8) α = ℏ 2 4 m$_{z}$ , (B9) β = e 2 32 πϵ$_{sc}$ϵ$_{0}$ ϵ$_{sc}$ − ϵ$_{ins}$ ϵ$_{sc}$ + ϵ$_{ins}$ , (B10) γ = 3 e $^{2}$n ∗ ϵ$_{sc}$ϵ$_{0}$ , (B11) δ = 12 e $^{2}$( N$_{A}$ − N$_{D}$ ) ϵ$_{sc}$ϵ$_{0}$ (B12) × { 1 − [ ( bz$_{d}$ ) 2 12 + bz$_{d}$ 2 + 1 ] exp ( − bz$_{d}$ ) } . (B13) ϕ ( z, b ) = {√ b 3 2 z exp ( − bz 2 ) for z ≥ 0 0 for z < 0 (B1) U ( z ) = { U$_{d}$ ( z ) + U$_{s}$ ( z ) + U$_{i}$ ( z ) for z ≥ 0 ∞ for z < 0 , (B2) U$_{d}$ ( z ) ≃   $^{e}$$^{2}$$^{n$_{depl}$}$ ϵ$_{sc}$ϵ$_{0}$ z ( 1 − z 2 z$_{d}$ ) for z < z$_{d}$ e $^{2}$n$_{depl}$ 2 ϵ$_{sc}$ϵ$_{0}$ z$_{d}$ for z > z$_{d}$, (B3) e $^{2}$n$_{s}$ 2 U$_{s}$ ( z ) ≃ e $^{2}$n$_{s}$ 2 bϵ$_{sc}$ϵ$_{0}$ [ 6 − ( ( bz ) 2 + 4 bz + 6 ) exp ( − bz ) ] , (B4) U$_{i}$ ( z ) ≃ be 2 32 πϵ$_{sc}$ϵ$_{0}$ ϵ$_{sc}$ − ϵ$_{ins}$ ϵ$_{sc}$ + ϵ$_{ins}$ 1 4 z . (B5) d ˜ ε db = αb + β − γb − 2 + δ ( b ) b − 3 = 0 . (B14) b = ( γ α ) 1 / 3 = ( 12 e $^{2}$n ∗ m$_{z}$ ϵ$_{sc}$ϵ$_{0}$ ℏ 2 ) 1 / 3 . (B15)