A calculation of the width dependence of the singularities in the polarisability function of a model two-dimensional electron gas on a narrow strip

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Abstract
The electronic polarisability function is calculated for a simple model of a noninteracting 2D electron gas on an infinitely long strip of width w. Periodic boundary conditions are applied both along and across the strip. An explicit formula for the dynamic polarisability function is given at temperature T which takes account of collisional broadening gamma . Detailed calculations of the static polarisability for T=0 and finite gamma are made as functions of wavenumber q and width w. As w is increased so that more and more sub-bands have minima below the Fermi level h(cross)2kF/2m*, the singular behaviour at q=2kF changes from the broadened logarithmic form characteristic of a 1D electron gas through an intermediate region exhibiting similar peaks associated with each occupied sub-band to reach a 2D limit in which the peaks coalesce to give the broadened cusp singularity which is characteristic of a 2D electron gas.