My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 5.
THE JELLIUM SLAB
-7.5
Energy per electron of slab system (mHa)
-8
-8.5
-9
-9.5
-10
Infinite cell
Finite cell
10 15 20 25
Slab width
Figure 5.4: The dependence of the LDA energy per electron on the slab width; for this system,
r s = 2.07, the number of electrons is 360 and the cell size in the z-direction is around 105.
ɛ slab = ɛ bulk + 8πr3 sσ
3s . (5.6)
When r s = 2.07, the surface energy is negative, which means that ɛ slab should tend
to the bulk value from below as the slab width s is increased; this behaviour is
exhibited by the infinite-system curve in figure 5.4. The figure also shows that the
errors introduced by using a finite simulation cell can be significant (∼ 1 mHa).
The small oscillations in the infinite-system curve are caused by sub-bands: as
the slab gets wider, more sub-bands begin to be filled. They decrease in magnitude
as the slab becomes wider; the true surface energy is defined in the limit of infinite
slab width, when they disappear altogether.
To calculate the surface energy within DFT using equation (4.28), the energy
per electron in the bulk system (the homogeneous electron gas) is required. The
correct approach is to use the same exchange-correlation energy functional in the
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