My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 5.
THE JELLIUM SLAB
Here ∆ɛ slab is the error in the calculated energy per electron of the slab and ∆σ is
the resultant error in the jellium surface energy. Equation (5.1) has been used to
relate the number of electrons per unit area N/A to the density parameter r s . For a
slab width of 20.0, setting ∆σ = 50 erg cm −2 gives ∆ɛ slab ≈ 0.1 mHa (or 3 meV) per
electron. This gives an idea of the accuracy needed in the Monte Carlo simulations.
5.3 Preliminary investigations
The surface energy of jellium is defined in the limit of an infinitely-wide slab with
infinite extent in the xy-direction. In DFT, it is possible to achieve one of these
limits. Because the external (background) potential does not depend on x or y, the
system is homogeneous in the xy-plane; the non-interacting Kohn-Sham orbitals
have the simple form
φ nk‖ (r) = u n (z)e ik ‖·r ‖
. (5.5)
The density of states in k ‖ -space is therefore constant, and extrapolation to a system
with infinite xy-extent is trivial. This is how the infinite-system density profiles
displayed in figures 5.1 and 5.2 were obtained.
Unfortunately, this simple extrapolation is not possible in QMC simulations,
which must use a finite number of electrons; for this reason, it is useful to study
finite cells in DFT.
The DFT simulations are carried out on a grid in the z-direction which extends
for some distance outside the slab, using a code supplied by Pablo Garcia-Gonzalez
[26]. It is important to ensure that the results are converged with respect to both
the number of grid points used and the spacing of these points (or equivalently, the
effective cell size in the z-direction, which will be denoted as w).
Figure 5.3 shows the results of convergence testing, for both the finite and infinite
horizontal cells. Cells of two different lengths in the z-direction have been compared:
even the smaller of these is around five times larger than the slab width, and the
figure demonstrates that in this regime the cell size is unimportant. What matters is
the sampling of the slab region: this is why the results for a cell size of 105 with 512
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