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In work Gelmont, B. L., Shur, M., & Stroscio, M. (1995). Polar optical-phonon scattering in three- and two-dimensional electron gases. Journal of Applied Physics, 77(2), 657–660 when author moving from equation 10b to 10c said "using the standard relationship between the Fermi level, $E_f$, and the surface electron carrier concentration, $N_s$, we further obtain...". What did the author mean saying the "...standard relationship..." enter image description here

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From above two equations we can conclude that this "standard relationship" is associated with the expressions in paranthesis e.g.

$$\frac{1}{[1+\exp(-\eta)]\ln[1 + \exp(\eta)]} = \frac{1 - \exp(-\gamma)}{\gamma} $$

I think that searched expression could be realted with a phenomenon "Fermi-level pinning".

After Inmaurer sugesstion e.g. $ \frac{n_s \pi \hbar^2}{mk_BT}= \frac{n_s}{N_2D} = F_0 = \ln[1 + \exp(\eta)] = \gamma $ i find the solution for my own question which i put here:

$$\frac{1}{[1+e^{-\eta}]\gamma} = \frac{1}{\frac{1 + e^{\eta}}{e^{\eta}}} \times \frac{1}{\gamma} = \frac{\textbf{1} + e^{\eta} - \textbf{1}}{1 + e^{\eta}} \times \frac{1}{\gamma} = $$

$$ \left[ 1 - (1 + e^\eta)^{-1} \right] \times \frac{1}{\gamma} = \left[ 1 - e^{ln(1 + e^\eta)^{-1}} \right] \times \frac{1}{\gamma} = $$

$$ = \frac{1 - e^{-\gamma}}{\gamma} $$

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    $\begingroup$ Maybe the following is useful: If you asked me this question without giving me any equation, I would have guessed that you were looking for the relationship given by Eq. (13) in arxiv.org/ftp/arxiv/papers/0811/0811.0116.pdf . I think that you can rewrite your equation as $\frac{1}{\left(1+e^{-\eta}\right)\left(1-e^{-\gamma}\right)} = $ Eq. (13). However, I'm not sure how to interpert the left side of that equation. $\endgroup$ – lnmaurer Oct 12 '19 at 14:12

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