Many-body effects in the Si metal-oxide—semiconductor inversion layer: Subband structure

For the system of electrons in the Si metal-oxide—semiconductor inversion layer, the corrections to Stern's Hartree approximation due to the exchange and the electron-electron residual interactions are worked out for the ranges of temperature $200\lesssim T\lesssim 300$ K and the surface electron density ${10}^{11}\lesssim N_{\mathrm{inv}}\lesssim {10}^{12}$ ${\mathrm{cm}}^{-2\mathrm{}}$. It is found that the electrons in a subband $\lambda$ and with a momentum $\overrightarrow{\mathrm{p}}$ parallel to the Si surface has a broadened distribution in energy as measured from the chemical potential, such that their state density is given by ${\rho }_{\lambda }(\epsilon , \overrightarrow{\mathrm{p}})={\left(\pi {\Gamma }_{\lambda }\right)}^{-1\mathrm{}}\times {\{1-\frac{{[\epsilon -{\hat{\epsilon }}_{\lambda }(\overrightarrow{\mathrm{p}}\left)\right]}^2}{{\left(2\mathrm{}{\Gamma }_{\lambda }\right)}^2}\}}^{\frac{1}{2}}$; the peaks of the electron number distribution $\frac{{\rho }_{\lambda }(\epsilon , \overrightarrow{\mathrm{p}})}{(e^{\beta \epsilon }+1)}$ are shifted to the lower-energy side as compared with the corresponding peaks of the state density and, remarkably, their spacings are almost identical with those of the Hartree subband bottoms. The widths ${\Gamma }_{\lambda }$ are as large as $k_{B}T$ and the spacings of the peaks of the electron number distribution. The perturbation calculation requires respecting the charge neutrality condition, which can only be satisfied by taking into account the charges on the gate electrode that has been out of consideration so far.