The Model Potential

The electron gas densities are again generated self-consistently from the LDA, using a model potential of the form,

$$v_{\hbox{ ext}} ({\bf r}) = v_0 \, \hbox{cos} \left( \frac{2 \pi x}{l} \right) \, ,$$ (6.10)

where $l$ is the unit cell length. This yields densities with a single maximum along one of the unit cell directions. The inhomogeneity of the system, which will be characterised by the full width at half maximum (FWHM) of the density distribution, in relation to $\lambda _{\hbox {{F}}}^0$, is therefore determined by the size of the amplitude $v_0$. A large value of $v_0$ on the scale of the Fermi energy $\varepsilon_{\hbox{{F}}}^0$, gives rise to a narrow density profile and therefore a small FWHM. The quasi-2D limit is approached by increasing $v_0$ from small values up to some maximum, $v_0^{\hbox{{max}}}$, which results in an electron gas that is extremely confined along one direction. The value of $v_0^{\hbox{{max}}}$ is determined when further (non-negligible) increases yield negligible changes in the self-consistent density profile.