Application Note
J.
Cleavage Energy of TiN
The purpose of this study is the computation of the cleavage energy of a material, i.e. the energy
required to split a material into two parts. This could be a bulk material, a grain boundary, or
an interface. To this end, one needs to compute the total energy of the bulk solid and the
material with a free surface.
Outline of Approach
Within a supercell approach, the cleavage process is described as follows:
B
A
B
→
Bulk solid
A
B
+
B
Slab A
Slab B
This process cleaves two A|B interfaces and leads to the formation of free surfaces of A and
B.
1. Optimize Structure of Bulk TiN
The experimental crystal structure of TiN is retrieved with INFO MATICA. Using the search
“Formula is TiN” gives several structures. We select ICSD.105128. Next, a cell optimization
with V ASP 4.6 is performed using “Accurate” precision. All other computational parameters
are left at their defaults, i.e. GGA-PBE-PAW potentials with Ti_pv (the Ti-3p levels are
treated as valence), a geometry Convergence of 0.02 eV/Å, a k-spacing of 0.5 Å-1, and
Methfessel-Paxton smearing with Smearing width σ =0.2 eV. The computed equilibrium
lattice parameter is 4.2332 Å, which is 0.3 % smaller than the original experimental value
of 4.2442 Å.
2. Construction of Slabs A and B
Using the optimized unit cell of TiN, the surface builder is used to construct two slab
models with one slab being terminated on both sides with Ti atoms and the other slab
model terminated with N atoms.
Select Edit≫Build surfaces…,
enter “1 1 1” as Miller indices, then
“Search” for possible cells. A cell is
found with a thickness of 7.33216
Å.
Choose Repeats:3 with a
Gap (Ång) of 0 and hit “Create”.
Copyright © Materials Design, Inc. 2002-2008
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Materials Design Application Note
Cleavage Energy of TiN
This gives a full supercell as shown to the left Make two copies of this full cell using the
command Edit≫Copy.
Full cell
N-terminated
Ti-terminated
Activate the window with the first
copy of the full structure and use
icon in the upper left corner
the
of the MEDEA tool bar and hold left
mouse button and drag to select
seven layers in the middle. Rightclick and hit
Delete selected atoms or use the
DEL key.
This will generate a N-terminated
slab as shown in the figure in the
middle panel.
You can also use the Spreadsheet
button
, sort atoms by zposition and delete the atoms in
this manner.
Starting with the second copy of
the full structure, delete the
complementary atoms above and
below.
This leads to a 7-layer slab model
of a Ti-terminated surface as
shown in the right-most panel.
3. Relax Slabs A and B and Compute Total Energy
Using VASP 4.6, relax the atom positions in the N-terminated and Ti-terminated slabs. To
this end, select “Normal” precision, “Projection Real space” and leave the other
parameters at their defaults. The k-mesh of 0.5 Å-1 leads to a 5x5x1 mesh, which is quite
reasonable.
4. Calculate Reference Energy of Bulk TiN
The highest degree of error cancellation is achieved, when the energy for bulk TiN is
computed with the full supercell using the same computational parameters as in the slab
calculations rather than with the primitive cell of TiN. This is done by performing a single
point energy calculation of the full cell.
Copyright © Materials Design, Inc. 2002-2008
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Materials Design Application Note
Cleavage Energy of TiN
5. Evaluate Cleavage Energy
The energy of cleaving a TiN crystal along a TiN(111) plane is obtained from the formula
A|B
𝐸𝐸cleavage =
1
B
AB �
�𝐸𝐸 A + 𝐸𝐸slab
− 𝐸𝐸bulk
2𝐴𝐴 slab
(1-1)
where A is the surface area in the supercell.
(1)
Analyze Results
The results of the calculation of the cleavage energy are shown in the table below.
Cleavage of TiN in (111) Plane
Value
Total energy (eV) of full supercell, 𝐸𝐸bulk (𝑇𝑇𝑖𝑖9𝑁𝑁9 )
A
Total energy of relaxed N-terminated slab, 𝐸𝐸slab
(Ti5 N6 )
B
Total energy of relaxed Ti-terminated slab, 𝐸𝐸slab
(Ti4 N3 )
1
A
B
Cleavage energy 𝐸𝐸cleavage = �𝐸𝐸slab
+ 𝐸𝐸slab
− 𝐸𝐸bulk �
2
1
Unit cell area (with a=2.993 Å): 𝐴𝐴 = √3 𝑎𝑎2
2
Cleavage energy
Copyright © Materials Design, Inc. 2002-2008
Units
-176.925 eV
-105.085 eV
-65.286 eV
3.28 eV
5.250×10-19 J
7.758×10-20 m2
6.77 J m-2
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Materials Design Application Note
Cleavage Energy of TiN
6. Using a Stoichiometric Slab Model
Both slabs used above are symmetric in terms
of their surface termination, but they are nonstoichiometric. Another option would be to
choose stoichiometric slabs, but accept the
fact that each slab has a dipole moment,
because one side is terminated with Ti and
the other side with N. To explore this
alternative, load the optimized unit cell of TiN
from the bulk calculation. Then invoke the
surface builder, choose (1 1 1) as the Miller
indices and hit Search. Leave the Repeats at
2 and change the Gap (Ång) to 0. Then build
the structure by Create. This model, shown
below on the left, consists of six layers of Ti
and six layers of N. Use Create Centered P1
to generate a model with the all atoms.
Moving the upper and lower planes as shown
in the right panel in the figure, a slab is
highlighted with three layers of each atom
type. Now use Create Centered P1 again to
generate a slab model.
Perform a relaxation of the atoms on the slab model and a single point energy calculation
with the full model using normal precision, real space projection, and leave all other
parameters at their defaults.
Slab A and Slab B are identical, namely Ti3N3 slabs with one side Ti-terminated and the
other side N-terminated. Hence the cleavage energy is
𝐸𝐸cleavage =
1
1
A
A
�𝐸𝐸slab
+ 𝐸𝐸slab
− 𝐸𝐸bulk � =
(2𝐸𝐸slab − 𝐸𝐸bulk )
2𝐴𝐴
2𝐴𝐴
Using this alternative approach, we find E slab = -55.665 eV and Ebulk = -117.969 eV.
(1-2)
This gives Ecleavage = (1/A) 3.32 eV = (1/7.758)×1020 × 1.60219×10-19 × 3.32 = 6.86 J/m2.
The previous approach yielded a value of 6.77 J/m-2. The values are close, but not identical,
which is due to finite thickness effects of the slabs. However, the values are sufficiently
similar to give confidence that the structural models are reasonable.
A similar study on the non-polar TiN(001) surface gives a cleavage energy of 2.60 J/m2.
This demonstrates that the non-polar surface is thermodynamically far more stable than
the polar surface, at least as long as the surfaces are not covered, for example, by a polar
liquid after cleavage.
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