The electromagnetic properties of a two-dimensional electron gas (2DEG) are usually characterized by a surface conductivity dependent on surface charge density $N_s$ and scattering time $\tau$ as:
\begin{equation} \sigma_s(\omega) = \frac{N_s e^2 \tau}{m^{\ast}}\frac{1}{1 - i \omega\tau} \end{equation}
where $e$ and $m_{\ast}$ are electron charge and effective electron mass respectively. During analysis, it is common practice to consider 2DEG as an infinitesimally thin sheet of charge. Some commercial EM codes as well as some past papers [1](eq. 3.54) associate a finite thickness $d$ to the 2DEG with a subsequent dielectric function as:
\begin{equation} \varepsilon(\omega) = 1 + i \frac{\sigma_s(\omega)}{\varepsilon_{b} \omega d} \end{equation}
where $\varepsilon_b$ is the permittivity of background. How can one justify such procedure in order to determine dielectric properties of a 2DEG?
[1]. Ando, T., Fowler, A. B., & Stern, F. (1982). Electronic properties of two-dimensional systems. Reviews of Modern Physics, 54(2), 437.
