My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN
QUASI-2D SYSTEMS
0
-0.02
-0.04
δn ind
-0.06
-0.08
-0.1
bulk (x, z)
surface (x)
surface (z)
0 2 4 6 8 10
x, z
Figure 6.4: The charge disturbance created by an electron-like test charge at the surface (z 0 = 0)
and in the bulk (z 0 = 10). The plots show the value of δn ind (r) at the points (x, 0, z 0 ) (labelled x)
and (0, 0, z 0 + z) (labelled z). In the bulk, the x and z curves are identical.
The induced charge density is then readily obtained from equations (6.30), (6.31)
and (6.34) and is plotted in figure 6.4. It is clear from the figure that the ‘hole’
spreads out in the xy-plane when the charge disturbance is near to the surface. In
the bulk, the parallel and perpendicular directions are equivalent; at the surface,
the hole is deeper and dies away more slowly in the x-direction. The important
issue is the effect of the cell size on the shape of the hole. Figure 6.5 shows that the
change is much more pronounced when the disturbance is at the surface, because
of the greater extent of the hole in the xy-direction. The slow decay of the surface
hole means that the result in the small system is significantly different from that in
the two larger cells. The bulk hole, which decays more rapidly, does not display the
same effect.
Returning to a more realistic slab, where the analytic form of n(z) is not known,
the procedure for calculating the induced charge density will now be outlined.
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