Chemistry Reference
In-Depth Information
TABLE 4.2 Surface Energies Calculated for Cu(100) and Cu(111)
from DFT as Function of Slab Thickness, in eV
/
˚
2
(J
/
m
2
)
a
Slab Model
s
, Cu(100)
s
, Cu(111)
5 layers
0.094 (1.50)
0.087 (1.40)
6 layers
0.097 (1.55)
0.089 (1.43)
7 layers
0.098 (1.57)
0.089 (1.43)
8 layers
0.096 (1.53)
0.091 (1.46)
0.114 (1.83)
b
Expt.
a
In each calculation, the bottom two layers were constrained in their bulk positions and
all other layers were allowed to relax.
b
From Ref. 3.
where
E
slab
is the total energy of the slab model for the surface,
E
bulk
is the
energy of one atom or formula unit of the material in the bulk,
n
is the
number of atoms or formula units in the slab model, and
A
is the total area
of the surfaces (top and bottom) in the slab model.
k
Macroscopically, surface
energy is typically expressed in units of joules
m
2
).
/
meters squared (J
/
In DFT calculations it
is more natural
to define surface energies in
˚
2
), so it is convenient to note that
electron volts
/
angstroms squared (eV
/
˚
2
.
The surface energy defined in Eq. (4.2) is the difference of two quantities
that are calculated in somewhat different fashions. In the case of the surface,
one would typically be using a comparatively large supercell, including a
vacuum space, and using comparatively few
k
points. In the case of the
bulk, the opposite would be true. How, then, can we be sure that the difference
in the theoretical treatments does not influence our answer? Unfortunately,
there is no single solution to this problem, although the problem can be mini-
mized by making every effort to ensure that each of the two energies are well
converged with respect to number of layers in the slab model,
k
points, energy
cutoff, supercell size, and the like.
As an example, the DFT-calculated surface energies of copper surfaces
are shown in Table 4.2 for the same set of slab calculations that was described
in Table 4.1. The surface energy of Cu(111) is lower than for Cu(100), mean-
ing that Cu(111) is more stable (or more “bulklike”) than Cu(100). This is con-
sistent with the comment we made in Section 4.4 that the most stable surfaces
of simple materials are typically those with the highest density of surface
atoms. We can compare our calculated surface energy with an experimental
m
2
1J
/
16.02 eV
/
k
This definition neglects entropic contributions to the surface energy.
