content_image/422499.jpg content_image/422500.jpg content_image/422511.jpg content_image/422512.jpg content_image/422513.jpg content_image/422514.jpg content_image/422515.jpg content_image/422502.jpg content_image/422504.jpg content_image/422505.jpg content_image/422506.jpg content_image/422507.jpg content_image/422508.jpg content_image/422509.jpg content_image/422501.jpg content_image/422503.jpg content_image/422510.jpg content_image/422516.jpg content_image/422517.jpg
and with the chain rule respectively Faá di Bruno’s for- mula 

<img src='content_image/422499.jpg'>

<img src='content_image/422500.jpg'>

where the sum runs over all integer numbers k$_{1}$, . . . , k$_{n}$ ≥ 0 which fulfill 

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 

<img src='content_image/422511.jpg'>

and as an approximation for the potential 

<img src='content_image/422512.jpg'>

<img src='content_image/422513.jpg'>

<img src='content_image/422514.jpg'>

<img src='content_image/422515.jpg'>

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 

<img src='content_image/422502.jpg'>

<img src='content_image/422504.jpg'>

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 

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 

<img src='content_image/422505.jpg'>

<img src='content_image/422506.jpg'>

<img src='content_image/422507.jpg'>

<img src='content_image/422508.jpg'>

<img src='content_image/422509.jpg'>

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 

<img src='content_image/422501.jpg'>

<img src='content_image/422503.jpg'>

<img src='content_image/422510.jpg'>

<img src='content_image/422516.jpg'>

<img src='content_image/422517.jpg'>

