15 eφ$_{0}$ = E$_{Cb}$ − E$_{Cs}$, eφ$_{0}$ = E$_{Cb}$ µ + µ − E$_{0}$ + E$_{0}$ − E$_{Cs}$, eφ$_{0}$ = − µ$_{Cb}$ + µ$_{E}$$_{0}$ + ε$_{0}$ (B23) b = F ( K + 1 K − 1 ) , (B16) F = β 3 α = e $^{2}$m$_{z}$ 24 πϵ$_{sc}$ϵ$_{0}$ ℏ $_{2}$, (B17) K = [ 1 2 ( 3 B − 2 + √ 3 B (3 B − 4) ) ] 1 / 3 , (B18) ∗ B = 9 α $^{2}$γ β 3 = γ βF 2 = 96 πn ∗ F 2 ϵ$_{sc}$ + ϵ$_{ins}$ ϵ$_{sc}$ − ϵ$_{ins}$ . (B19) N$_{C}$ F 1 / 2 ( µ$_{Cb}$ k$_{B}$T ) + N$_{A}$ g$_{A}$ exp ( ε$_{A}$ − ε$_{g}$ − µ$_{Cb}$ k$_{B}$T ) + 1 = = N$_{V}$ F 1 / 2 ( − ε$_{g}$ − µ$_{Cb}$ k$_{B}$T ) (B24) en$_{depl}$ = e ( N$_{A}$ − N$_{D}$ ) z$_{d}$ = √ 2( N$_{A}$ − N$_{D}$ ) ϵ$_{sc}$ϵ$_{0}$eφ$_{0d}$, (B20) eφ$_{0}$$_{d}$ = eφ$_{0}$ − eφ$_{0}$$_{s}$ − k$_{B}$T. (B21) N$_{C}$ = g$_{s}$g$_{v3D}$ ( m$_{de3D}$k$_{B}$T 2 π ℏ 2 ) 3 / 2 , (B25) N$_{V}$ = g$_{s}$ ( m$_{dh3D}$k$_{B}$T 2 π ℏ 2 ) 3 / 2 , (B26) ε$_{g}$ = ε$_{g0Si}$ − α$_{Si}$T 2 T + β$_{Si}$ . (B27) φ$_{0}$$_{s}$ = 3 en$_{s}$ bϵ$_{sc}$ϵ$_{0}$ . (B22) We neglected only δ ( b ) b − $^{3}$, this leads to As can be seen from Fig. 4 for the total band bending holds. The chemical potential relative to the conduction band edge in the bulk µ$_{Cb}$ is determined by the charge neutrality and can be calculated by solving The density n$_{depl}$ can be calculated from the total band bending eφ$_{0}$ = E$_{Cb}$ − E$_{Cs}$ (b=bulk, s=surface), numerically. N$_{C}$ and N$_{V}$ are the effective densities of state in the conduction band and in the valence band, ε$_{A}$ is the acceptor ionization energy, g$_{A}$ the acceptor degen- eracy factor, and ε$_{g}$ the gap energy. According to Sze 27 this quantities are given by Here eφ$_{0d}$ is the band bending caused by the charges in the depletion layer (acceptors and donors) and eφ$_{0}$$_{s}$ that caused by the free electrons in the inversion layer. To get this equations one has to assume that in the deple- tion layer all acceptors are charged and that in the bulk there is charge neutrality. The boundary between the de- pletion layer and the bulk is not sharp, this is described by the term − k$_{B}$T in eφ$_{0}$$_{d}$ . 36 To calculate φ$_{0}$$_{s}$ one has to solve the Poisson equation with the charge density which corresponds with the test wave function ϕ , the result is The values we used can be found in Table I. The ground state energy ε$_{0}$ itself is a (small) part of the the total band bending, this problem is solved by a fix point iteration using ε$_{0}$ = 0 as start value. ∗ gerhard.brunthaler@jku.at 1 S. V. Kravchenko, G. V. Kravchenko, J. E. Furneaux, V. M. Pudalov, and M. DIorio, Phys. Rev. B 50 , 8039 (1994). 2 S. V. Kravchenko, W. E. Mason, G. E. Bowker, J. E. Furneaux, V. M. Pudalov, and M. DIorio, Phys. Rev. B 51 , 7038 (1995). 9 F. Stern, Phys. Rev. Lett. 44 , 1469 (1980). 10 A. Gold and V. T. Dolgopolov, Phys. Rev. B 33 , 1076 (1986). 11 S. Das Sarma, Phys. Rev. B 33 , 5401 (1986). 12 A. Gold, J. Phys.: Cond. Matt. 15 , 217 (2003). 13 S. Das Sarma, M. P. Lilly, E. H. Hwang, L. N. Pfeiffer, K. W. West, and J. L. Reno, Phys. Rev. Lett. 94 , 136401 (2005). 14 S. Das Sarma and E. Hwang, Solid State Comm. 135 , 579 (2005). 3 E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42 , 673 (1979). 4 E. Abrahams, S. V. Kravchenko, and M. P. Sarachik, Rev. Mod. Phys. 73 , 251 (2001). 15 A. M. Finkel’stein, Z. Phys. B Condes. Matter 56 , 189 (1984). 5 M. Y. Simmons, A. R. Hamilton, M. Pepper, E. H. Linfield, P. D. Rose, D. A. Ritchie, A. K. Savchenko, and T. G. Griffiths, Phys. Rev. Lett. 80 , 1292 (1998). 16 C. Castellani, C. Di Castro, P. A. Lee, and M. Ma, Phys. Rev. B 30 , 527 (1984). 17 A. Punnoose and A. M. Finkel’stein, Phys. Rev. Lett. 88 , 016802 (2001). 6 M. P. Lilly, J. L. Reno, J. A. Simmons, I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, K. W. West, E. H. Hwang, and S. Das Sarma, Phys. Rev. Lett. 90 , 056806 (2003). 18 A. Punnoose and A. M. Finkel’stein, Science 310 , 289 (2005). 19 G. Zala, B. N. Narozhny, and I. L. Aleiner, Phys. Rev. B 64 , 214204 (2001). 20 I. V. Gornyi and A. D. Mirlin, Phys. Rev. B 69 , 045313 (2004). 8 S. J. Papadakis and M. Shayegan, Phys. Rev. B 57 , R15068 (1998). 7 V. Senz, T. Ihn, T. Heinzel, K. Ensslin, G. Dehlinger, D. Grützmacher, U. Gennser, E. H. Hwang, and S. D. Sarma, Physica E 13 , 723 (2002).