Application Note
N.
Surface Energy of Molybdenum
The surface energy of a material is defined as the energy required to create a surface (h k l)
from the bulk material. Surface energies are usually given in units of J m-2. Surface energies
can be determined with VASP by performing two calculations, namely one for a repeated slab
representing the particular surface (h k l) and one for the bulk system as a reference.
For calculations using a slab model, the surface energy is given by
𝐸𝐸surf =
1
[𝐸𝐸 (𝑛𝑛) − 𝑛𝑛𝐸𝐸bulk ]
2𝐴𝐴 slab
Where 𝐸𝐸surf is the surface energy, A the surface area, 𝐸𝐸slab(𝑛𝑛) is the energy of a slab
containing n chemical formula units, and 𝐸𝐸bulk is the energy per formula unit in the bulk
material.
While the calculation of surface energies is conceptually straightforward, one needs to be
careful in controlling the errors originating from the geometric models, e.g. layer thickness
and spacing between layers, and various computational parameters, e.g. k-meshes. The
following examples demonstrate these points.
1. Select bulk structure
In MEDEA, activate INFOMATICA under Tools and search the structure of Mo in the
CRYSTMET database by using the command Require that Formula is Mo and Require that
Structural Completeness to be Complete. Select, for example, ICSD.53799 and view the
structure by right-clicking on the selected row.
2. Optimize lattice with VASP
The lattice parameter of bulk Mo is now optimized using VASP. To this end, the following
computational parameters are chosen: Structure Optimization:  Relax atom positions,
 Allow cell volume to change,  Allow cell shape to change.
GGA-PBE-PAW with the default Mo_pv potential, Accurate precision,  Increase
planewave cutoff, Projection: Reciprocal space
k-mesh spacing: 0.5/Å (6x6x6 mesh), SCF convergence of 10-6 in the SCF, and MethfesselPaxton smearing. The optimized structure is taken as the starting point for the creation of
surface models. To ensure consistency with the subsequent surface calculations, which are
performed with medium precision, the total energy of the bulk unit cell is recalculated also
with medium precision by a single point total energy.
Note 1: An increased-precision is chosen for the lattice optimization, because in this relies on the
stress tensor, which requires a high-quality basis. In the subsequent surface calculations, the cell
parameters are no longer optimized and thus a Normal level of accuracy is sufficient.
Note 2: A k-mesh of 6x6x6 is reasonable, but still rather coarse for a small unit cell of a metal such
as Mo. For a better converged calculation, a k-mesh of 10x10x10 or finer is recommended.
Copyright © Materials Design, Inc. 2002-2008
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Materials Design Application Note
Surface Energy of Molybdenum
3. Creation of a (001) Surface Model
The optimized structure of bulk Mo is loaded into the MEDEA window, e.g. by opening the
results of previous calculations. Under the Edit pull-down menu, select Build surfaces. Enter
0 0 1 as the desired Miller indices, use Search and to choose4 Repeats and a Gap (Ång)of 6
Å. The result is a unit cell shown at the left. Select P1 symmetry and confirm with Apply.
Now you can select one atom and delete it (DEL)to create the 7 layer cell. If you want, you
can raise the symmetry with Edit≫Edit Structure≫Symmetry or right-clicking into the cell
background and use Edit Symmetry… to Raise Symmetry to P4/mmm. The resulting
surface is centered in the cell, as shown on the right.
SurfaceBuilder
P4/nmm
P1
P1
7 layer
5 layer
3 layer
P4/mmm
P4/mmm
P4/mmm
Copyright © Materials Design, Inc. 2002-2008
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Materials Design Application Note
Surface Energy of Molybdenum
4. Relaxation of the slab model
The atomic positions of the 7 layer model are now optimized with the following parameters
for VASP changed: Uncheck  Increase planewave cutoff,  Allow cell volume to change,
 Allow cell shape to change, but keep  Relax atom positions. We use now a 4×4×1
mesh (as the 0.5/Å spacing is the same and the cell shape changed). Note the value kz=1 due
to the elongated shape of the unit cell in the z-direction. In this step, all lattice parameters are
kept constant, atoms will relax mostly in z direction with slight changes to the vacuum
between slabs.
5. Evaluation of the surface energy
The evaluation of the surface energy is summarized in the spreadsheet below. Due to the
choice of a primitive cell for the bulk bcc Mo calculation, a k-mesh of 6×6×6 corresponds
most closely to the 4×4×1 k-mesh used in the slab calculations. To check the consistency, also
the single-point total energy of a Mo 1×1×6 supercell is calculated, that's approximately the
size of the cell for the 7 layer calculation.
Three different slab thicknesses are tested, namely 3-, 5-, and 7 layers. It can be seen that the
5-layer model is already close to the 7-layer model, thus indicating convergence in the
thickness of the slab.
Another parameter to consider is the spacing between the slabs. The effect of different
separation (thickness of vacuum) is tested here by calculating the total energy of a 5-layer
slab model with a c-parameter of 14 and 18 Å, respectively. The difference is only 12 meV,
which is insignificant in the present case. It should be noted that the dependence on vacuum
thickness might be more critical for systems with a dipole moment or non-metallic systems.
For the (001) the area perpendicular to the elongated axis is 𝐴𝐴 = 𝑎𝑎2 , but as we have two
surfaces in the cell, we need to add the prefactor of 2. In Job.out we find besides the Vasp
energy also the heat of formation for the formula unit and the cell. The heat of formation is
with respect to the reference state of Mo and allows comparing the different layers at a
glance. The surface energy is computed from either energies (𝐸𝐸VASP)or heat of formations
(𝐸𝐸form) by subtracting n-times the bulk reference for Mo (primitive cell), and dividing through
the surface area. (1 eV=96.458 kJ/mol, NA =6.022142⋅1023)
System
Mo bcc (optimization)
Mo bcc (P1)
Mo primitive (P1)
Mo 1x1x6 cell
Mo(001) 3 layers
Mo(001) 5 layers
Mo(001) 5 layers
Mo(001) 7 layers
Mo
Mo2
Mo
Mo6
Mo3
Mo5
Mo5
Mo7
k-mesh EVASP (eV)
6×6×6
-10.79857
4×4×4
-21.62101
6×6×6
-10.80283
4×4×2
-129.56473
4×4×1
-28.05031
4×4×1
-49.97279
4×4×1
-49.96033
-71.58431
4×4×1
Copyright © Materials Design, Inc. 2002-2008
Comment Eform [kJ/mol] Esurf [J/m2]
a=3.168 Å
4.69
7.08
4.28
c=19.00 Å
29.04
c=18.67 Å
433.35
3.48
c=14.0 0Å
411.35
3.23
c=18.67 Å
412.55
3.24
c=18.67 Å
419.30
3.22
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Materials Design Application Note
Surface Energy of Molybdenum
A point more subtle is the usage of energies from cell volume optimizations, there is a small
difference of 0.4 kJ/mol. The rather coarse k-spacing also shows up when comparing the
different Mo bulk cells; for surface energies, we used the primitive Mo cell.
Surface Reconstruction
The Mo(001) surface is known to undergo a reconstruction, forming zigzag chains of surface
atoms with a c(2×2) periodicity. This reconstruction is not taken into account in the present
calculations. The energy gained by this reconstruction, about 0.05 J/m2, is small compared
with the surface energy.
Cleavage Energy
In the case of a simple metal such as Mo, the surface energy is uniquely defined by the Miller
plane such as (001), allowing the creation of a slab model with two equivalent surfaces.
However, in the general case, it is not always possible to create a slab with two identical
surfaces. In this case, one needs to be careful in the definition of a surface energy. In fact, in
such a situation the concept of a “cleavage energy” may be more appropriate. In the present
case of the Mo(001) surface, the cleavage energy would be exactly twice the surface energy.
6. Compute Energy of Mo (110) Surface
For comparison, we compute the (110) surface energy, as this is the most densely packed
surface for a bcc structure. The same unit optimized unit cell is taken as the starting point for
the surface builder to create a (100) surface model with 3 Repeats and a Gap (Ång) of 6 Å.
We keep the settings from the surface optimizations and allow only for  Relax atom
positions.
The resulting heat of formation is higher than for the (100) surface, but with the larger
surface of the slab, 𝑎𝑎 = 4.4803, 𝑏𝑏 = 3.1681, the resulting surface energy is 25% lower. The
actual number of k-points varies depending on the symmetry: 8×8×1 with Cmma, and 3×4×1
with P1 symmetry, both are based on a spacing of 0.5/Å, but the effective spacing for Cmma
is 0.304×0.496×0.323, where for the P1 cell it is 0.467×0.496×0.323
System
k-mesh EVASP (eV)
Comment Eform [kJ/mol] Esurf [J/m2]
Mo primitive
Mo
6×6×6
-10.80283
4.28
Mo(110) 6 layer Mo12 8×8×1
-125.22086 c=19.44 Å
477.164
2.49
Mo(110) 6 layer Mo12 3×4×1
Results
-124.28553 c=19.44 Å
567.409
3.02
It takes less energy, 2.5 to 3 J/m2, to create a (110) surface compared with 3.22 J/m2 to
create a (001) surface. So the Mo (110) surface is computed to be more stable than the (001)
surface. This is consistent with experimental evidence.
The variation of surface energies with respect to the used k-mesh indicate that a denser kmesh is necessary to reach convergence.
Copyright © Materials Design, Inc. 2002-2008
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