Study of a two-dimensional electron gas in a magnetic field. I. Weak field

Abstract

A many-body theory of a two-dimensional electron gas in a magnetic field is presented. The field dependence of the Fermi momentum shows a two-dimensional peculiarity. It consists of the spin-dependent paramagnetic part and the orbital diamagnetic part. The former increases more strongly with ${r}_{s}$ than the latter. The paramagnetic and diamagnetic susceptibilities are found as functions of ${r}_{s}$. The field-dependent ground-state energy is evaluated. Its paramagnetic part decreases steadily with ${r}_{s}$, while the diamagnetic part shows a maximum. No such maximum has been found for the three-dimensional case in the same approximation which is valid for high density and low magnetic fields. From the susceptibilities, an effective $g$ factor and an effective mass are derived. They are given approximately by $\frac{{g}^{*}}{g}=1+0.87\ifmmode\times\else\texttimes\fi{}{10}^{6}{n}^{\ensuremath{-}\frac{1}{2}}$ or more accurately by $\frac{{g}^{*2}}{{g}^{2}}=1+1.74\ifmmode\times\else\texttimes\fi{}{10}^{6}{n}^{\ensuremath{-}\frac{1}{2}}\ensuremath{-}0.708\ifmmode\times\else\texttimes\fi{}{10}^{12}{n}^{\ensuremath{-}1}$ and by $\frac{{m}^{*}}{m}=1+3.21\ifmmode\times\else\texttimes\fi{}{10}^{4}{n}^{\ensuremath{-}\frac{1}{2}}$, where $n$ is the number density.