This document summarizes a talk given by Yoon Tiem Leong at the School of Physics, Universiti Sains Malaysia on January 24, 2014 about growing graphene epitaxially on a 6H-silicon carbide substrate using a simulated annealing method. The growth process was investigated using two empirical potentials: the Tersoff potential and its refined version called the TEA potential. Single layer graphene formation was achieved using the TEA potential at an annealing temperature of 1200K, agreeing with experimental observations. The formation of graphene was evaluated by calculating changes in the carbon-carbon bond length, binding energy, and distance between the graphene layer and underlying silicon atoms during the simulated annealing process.

1. Epitaxial growth of
graphene on 6H-silicon
carbide substrate by
simulated annealing method
Yoon Tiem Leong
School of Physics, Universiti Sains Malaysia
Talk given at the Theory Lab, School of Physics,
USM
24 Jan 2014

2. Abstract
We grew graphene epitaxially on 6H-SiC(0001) substrate by
the simulated annealing method. The mechanisms that
govern the growth process were investigated by testing two
empirical potentials, namely, the widely used Tersoff
potential and its more refined version published years later
by Erhart and Albe (TEA potential). We evaluated the
reasonableness of our layers of graphene by calculating
carbon-carbon (i) average bond-length, (ii) binding energy.
The annealing temperature at which the graphene structure
just coming into view at approximately 1200 K is
unambiguously predicted by TEA potential and close to the
experimentally observed pit formation at 1298 K.

4. Single layer graphene formation

5. How we construct the unit cell and
supercell for 6H-SiC substrate

We refer

http://cst-www.nrl.navy.mil/lattice/struk/6h.html

to construct our 6H-SiC substrate.

6. The snapshots from the above
webpage
Figure 1: Snapshot from http://cst-www.nrl.navy.mil/lattice/struk/6h.html. The 6HSiC belongs to the hexagonal class. For crystal in such a class, the lattice parameters and
the angles between these lattice parameters are such that a = b ≠ c ; α = β = 90 degree, γ
= 120 degree.

7. Figure 2: Snapshot from http://cstwww.nrl.navy.mil/lattice/struk.xmol/6
h.pos

8. Structure of the unit cell




Each unit cell of the 6H-SiC has a total of 12
basis atoms, 6 of them carbon, and 6 silicon.
Figure 2 displays:
(1) The coordinates of these atoms (listed in the
last 12 rows in Figure 2). We note that only the
Cartesian coordinates are to be used when
preparing the input data for LAMMPS.
(2) Primitive vectors a(1), a(2), a(3) in the {X, Y,
Z} basis (i.e. Cartesian coordinate system).

9. Procedure to construct rhombusshaped 6H-SiC substrate






First, we determine the lattice constants, a , b (= a),
c:
From Figure 2, the primitive vectors, a(1), a(2), a(3)
are given respectively (in unit of nanometer) as
a(1) = (1.54035000, -2.66796446, 0 .00000000)
a(2) = (1.54035000, 2.66796446, 0.00000000)
a(3) = (.00000000, .00000000, 15.11740000).
Squaring a(1) and adding it to a(2) squared, we
could easily obtain the value for the lattice
parameter a, which is also equal to b by definition of
the crystallographic group.

10. Lattice parameters
2
2
1
1
1
 1

a (1) + a (2) =  aX − 31/2 aY ÷ +  aX + 31/2 aY ÷
2
2
2
 2

3 
2
2
1
2 ( 1.54035000 ) + ( 2.66796446 )  = 2  a 2 + a 2 ÷


4 
4
2
2
a 2 = ( 1.54035000 ) + ( 2.66796446 )
2
2
a = 3.08
a (3) 2 = ( cZ) 2
15.117400002 = c 2
c = 15.11
b=a


The lattice constants, as obtained from the above
calculation, are a = 3.08 nm, b=3.08 nm, c = 15.11 nm.
Since the 6H-SiC belongs to a hexagonal class, α = β =
90 degree, γ = 120 degree.

11. Translation of lattice parameters
into LAMMPS-readable unit



We refer to the instruction manual from
the LAMMPS website in order to feed
in the information of the lattice
parameters into LAMMPS:
http://lammps.sandia.gov/doc/Section_
howto.html#howto_12, section 6.12,
Triclinic (non-orthogonal) simulation
boxes
In LAMMPS, the units used are {lx, ly,
lz; xy, xz, yz}. We need to convert {a,
b, c; α, β, γ} into these units. This could
be done quite trivially, via the
conversion show in the right:

12. Raw unit cell of 6H-SiC



Based on the procedures described in
previous slides, we constructed a
LAMMPS data file for a raw 6H-SiC unit
cell.
It represents a unit cell of 6H-SiC
comprises of six hexagonal layers
repeating periodically in the z-direction.
The resultant data file, named
dataraw.xyz, is include in Figure 4. It is to
be viewed using xcrysdens or VMD.

13. Figure 4
Sublayer of
Si and C
Sublayer of
Si and C
Sublayer of
Si and C
Sublayer of
Si and C
Sublayer of
Si and C
Figure 4: Visualization of the original unit cell’s atomic configuration as specified in
dataraw.xyz. The coordinates of the atoms are also shown. There is a total of 12 atoms in
the unit cell.

14. Raw unit cell of 6H-SiC



Each hexagonal layer consists of two
sublayers. Each of these sublayers is
comprised of Carbon and Silicon atoms
(see Figure 4).
Note that the topmost atom is a Carbon.
This means the (0001) surface of the 6HSiC is Carbon terminated.
There is a total of 12 atoms it the original
unit cell.

15. Modification for carbon-rich
layer


Next, we shall modify the unit cell
dataraw.xyz via the following procedure:
The Si atom (No. 9) is removed. The atom
C (No. 5) is now translated along the z–
direction to take up the z-coordinate left
vacant by the removed Si atom (while the
x- and y-coordinate remains unchanged).

16. Simulation method of graphene growth (one layer)
Simulated Annealing
2.52 Å
< 1 Å = 0.63 Å
2.0 Å




Conjugate gradient minimization

Simulated annealing


Timestep = 0.5 fs
Increase the temperature
slowly until it attains 300 K
at approximately 5˟1013
K/s.
Equilibrating the system at
300 K for 20000 MD steps.
Raise the temperature of
the system slowly to the
desired T at approximately
1013 K/s.
Equilibrating the system at
T for 30000 MD steps.
Cool down the system until
0.1 K at 5x1012 K/s
16
Extracting the result.

17. Content of data.singlelayer.xyz
11
Atoms
C 0.00000000 0.00000000 1.89572196
C0.00000000 0.00000000 9.45442196
Si 0.00000000 0.00000000 0.00000000
Si 0.00000000 0.00000000 7.55870000
C 1.54035000 0.88932149 10.07877058
C 1.54035000 -0.88932149 4.41276906
C 1.54035000 0.88932149 6.92981616
C 1.54035000 -0.88932149 -0.62888384
Si 1.54035000 -0.88932149 2.52007058
Si 1.54035000 0.88932149 5.03711768
Si 1.54035000 -0.88932149 -2.52158232

The content of
dataraw.xyz is now
modified and
renamed as
data.singlelayer.xyz,
which content is
shown in Figure 5,
and visualised in
Figure 6.
Figure 5: The content of data.singlelayer.xyz, detailing the
coordinates of the atoms in a carbon-rich SiC substrate
unit cell. Note that now only 11 atoms remain as one Si
atom (atom 5).

18. Figure 6: carbon-rich unit cell
of SiC
11 atoms per unit cell left
as one Si atom (No. 9) has
been removed.
Figure 6. : Visualization of the unit cell’s atomic configuration as specified in
data.singlelayer.xyz. This is the carbon-rich substrate to be used for single layer
graphene growth.

19. Generating supercell



We then generated a supercell comprised
of 12 x 12 x 1 unit cells as specified in
data.singlelayer.xyz.
This is accomplished by using the
command
replicate 12 12 1

20. Periodic BC



Periodic boundary condition is applied
along the x-, y- and z-directions via the
command:
 boundary p p p
We created a vacuum of thickness 10 nm
(along the z-direction) above and below
the substrate.
The 12 x 12 x 1 supercell constructed
according to the above procedure is
visialised in Figure 7.

21. Figure 7 (a)
Figure 7: A 1584-atom supercell mimicking a carbon-rich SiC substrate. It is made
up of 12 x 12 x 1 unit cells as depicted in Figure 6. 7(a) Top view, 7(b) side view
and 7(c) a tilted perspective are presented. Yellow: Carbon; Blue: Silicon.

22. Figure 7 (b)

23. Figure 7 (c)

24. Visualisation of the 12 x 12 x 1
supercell





There is a total of 1584 atoms in the simulation box.
Coordinates of all the atoms in the supercell can be
obtained from LAMMPS’s trajectory file during the
annealing process.
These coordinates are simply the atomic coordinates of
the first step output during the MD run.
View the structure file 10101.xyz using VMD.
The 12 x 12 x 1 unit cell mimicking a Carbon-rich
substrate will be used as our input structure to LAMMPS
to simulate epitaxial graphene growth.

25. Annealing procedure

Once the data file for the Carbon-rich SiC
substrate is prepared, we proceed to the next
step to growth a single layer graphene via
the process described in Figure 8 below.
1K
1
Monitor
output here
5 x 1013 K/s
5000
5000 steps
Figure 8. Annealing procedure based on suggestion by
Prof. S.K. Lai.

26. Implementation
To implement the above procedure, a
fixed value of target annealing
temperature was first chosen, e.g.
Tanneal = 900 K.
 We ran the LAMMPS input script
(in.anneal) using a (fixed) target
Tanneal.
 We monitor the LAMMPS output while
the system undergoes equilibration at
the target annealing temperature
(after the temperature has been
ramped up gradually from 1 K).


27. Figure 9
Temperature profile

A typical temperature profile that
specifies how the temperature of the
system being simulated changes as a
function of step is illustrated, with the
target temperature at Tanneal = 1200
K).
Figure 9

28. Implementation (cont.)





If graphene is formed at a given target annealing
temperature, the following phenomena during
equilibrium (at that annealing temperature) will be
observed:
(i) An abrupt formation of hexagonal rings by the
carbon rich layer (visualize the lammps trajectory
file using VMD in video mode),
(ii) an abrupt drop of biding energy,
(iii) an abrupt change of pressure.
In actual running of the LAMMPS calculation, we
repeat the above procedure for a set of selected
target annealing temperature one-by-one, Tanneal
= 400 K, 500K, 1100K, 1200 K …, 2000 K.

29. Numerical parameters








The essential parameters used in annealing the substrate
for single layered graphene growth:
1. damping coefficient: 0.005
2. Timestep: 0.5 fs.
3. Heating rate from 300 K -> target temperatures, 5 x 10 13
K/s.
4. Cooling rate: From target temperatures -> 1 K, 1 x 10 13
K/s.
5. Target temperatures: 700 K, 800 K, …, 2000 K.
6. Steps for equilibration: (i) At 1K, 5000 steps. (ii) At 300
K, 20,000 steps, (iii) target annealing temp -> target
annealing temp, 60,000 steps.
Essentially, all the parameters used are the same as that
used by the NCU group.

30. Force fields
• For single layer graphene formation, two
force fields are employed: TEA and
Tersoff.
• As it turns out later, TEA shows a better
results than Tersoff. We shall compare
their results later.

31. Results from TEA force field

32. Configuration of the carbon-rich substrate before and
after equilibration at T = 1.0 K for single-layered
graphene formation
0.624 Å
0.2276 Å
1.896 Å
1.9948 Å
Before
minimisation
1.89
Å
After
minimisation
0.63
Å
0.22
Å
1.9
9Å
After minimization
but before
simulated annealing
As comparison, this figure shows the
geometry obtained by the NTCU group
before and after minimisation

33. Graphene before and after
formation at Tanneal = 1200 K

34. Formation of single layered
graphene with thickness z=1
substrate, with TEA at 1200 K
• http://www.youtube.com/watch?v=klkg2Rlf7Gk
• Agrees with what Hannon and Tromp measured:

35. Data and results for single layer
graphene formation

In the following slides, the following quantities are shown:

(i) Temperature vs. step (tempvsstep.dat)

(ii) Binding energy versus step during equilibration at target
annealing temperature (bindingenergyvsstep.dat).

(iii) Average nearest neighbour (“bond length”) of the
topmost carbon atoms versus step during equilibration at
target annealing temperature (avenn_vs_step.dat).

(iv) Average distance between the topmost carbon atoms
(cr3) and the Si atom (Si4) lying just below these carbon
atoms vs step (distance34vsstep.dat). This is the distance
between the graphene and the substrate just below it.

36. Definition of d34 for single layer
graphene formation
Top carbon-rich layer,
(labeled as cr3)
d34 = average distance between the carbon-rich layer
and the substrate just below it
Si atoms (labeled as Si4)
SiC substrate

37. Figure 10(i): Tanneal = 1100 K.
No graphene formed

38. Figure 10(ii): Tanneal = 1200 K.
Graphene is formed

39. Determination of binding energy (BE) at a fixed
Tanneal



Should an abrupt change in binding energy occurs at a given
Tanneal during equilibration, such as that illustrated below (for
Tanneal = 1200 K), how do we decide the value of the binding
energy (which is step-dependent) for this annealing
temperature?
We choose the value of the BE at the end of equilibration step,
denoted as s. s is Tanneal-dependent:
s = 5000+85000+40*(temp-300)
s
s

40. BE vs. Tanneal



Based on the data shown in Figures 10, we
abstract the value of BE at step = s from
annealing temperature to plot the graph of
BE vs Tanneal.
The values of BE (at step s) vs Tanneal is
tabled in bdvstemp.dat.
The resultant curve is shown in Figure 11.

41. Binding energy vs anneal
temperature
datasinglelayerTEAbdvstemp.dat
Anneal temp
binding energy
400 -5.880470833333334
500 -5.872215972222222
600 -5.8549182291666595
700 -5.853736979166666
800 -5.844029131944445
900 -5.8253253472222255
1000
1100 -5.810316180555554
1200 -6.647701284722221
1300 -6.830150555555552
1400 -6.844608055555553
1500 -6.713664999999995
1600 -6.833112638888893
1700 -6.747370833333332
1800 -6.877048993055555
1900 -6.709565694444447
Figure 11

42. Average nearest neighbour (nn)
(a.k.a ‘bond length’) vs anneal
temperature


Based on the data shown in Figures 10, we
abstract the value of average nn at step = s
from each annealing temperature to plot the
graph of ave nn vs Tanneal.
The resultant curve is shown in Figure 12.

43. Average nearest neighbour (nn) vs
anneal temperature
Anneal temp average nn
400 1.750833424567148
500 1.746625140692055
600 1.7589214030309763
7001.7419868390032442
800 1.7414136922144865
900 1.7589380688936933
100 1.7417389709279334
110 1.7344180027393463
1200 1.5027327808093742
1300 1.4693218689432666
1400 1.4764060075537178
Figure 12

44. Average distance between cr1 and
Si6 vs anneal temperature
Anneal temp average distance cr3-Si6
400
2.1048033680555562
500
2.1063726736111175
600 2.105993090277779
700
2.105879791666668
800
2.1031867708333323
900
2.1202562847222266
1000
2.1176251041666685
1100
2.124162604166675
1200
2.4697990625000052
1300
2.560621458333336
1400
2.54777364583333
1500
2.6178477083333327
1600
2.6055096527777843
1700
2.722710520833341
1800
2.8183415972222177
1900
2.6691606597222215

45. Data and results for single layer
graphene formation with TEA
From the data generated, we conclude that:
 Graphene formation is observed only when
Tanneal= Tf (transition temperature) = 1200 K
or above for TEA potential.


46. Outcome from Tersoff
• We have also simulated with Tersoff force
field.
• The outcome are summarised in the next
slide.

47. Tersoff vs. TEA
Tersoff
TEA

48. TEA vs Tersoff
• TEA results compared better with
experiment than TERSOFF did
• TEA fitting of the three body interactions
among the Si-C atoms is more rigorous;
whereas Tersoff’s three body interactions
are fitted with lesser accuracy.

49. Double-layered graphene
formation
(Only for TEA force field)

50. Figure 13
•Prepare a two-layered carbon-rich substrate by further knocking off
two layers of Si atom, and then shift the topmost carbon atom layers
to form two carbon rich layers.
•Thickness of the substrate is z=1.
Conjugate gradient
minimization
Simulated annealing
Conjugate gradient
minimization
50
14

51. Double-layered graphene
formation
• We have simulated the graphene
formation based on three different sizes
for the thickness of the 6H-SiC substrate,
i.e., z = 1, 2, 3.
• Results of the simulation for each z will be
presented in sequence.
• Only TEA force field is used.

52. Two-layered carbon-rich
substrate with
thickness z = 1
for double-layered graphene
formation

53. Figure 14: After minimising the two-layered
carbon-rich substrate with thickness, z = 1
•
Shown here is the 15 x 15 x 1 supercell right after energy minimisation
•
The values of the z-coordinates allow us to estimate the distances between the atomic
layers, as indicated.
0.31Å
1.59 A
0.51 A
1.35 A
0.61Å
1.89 A
Figure 15. z = 1.
Note: we note that the
substrate get distorted
significantly after energy
minimisation.

54. Visualising graphene formation for 15 x 15 x 1
supercell at Tanneal = 1100 K, z = 1



We found that for substrate thickness z = 1, doublelayered graphene is formed at as low as Tanneal = 600
K. But the transition is not sharp.
It is visually inspected that the whole SiC substrate get
seriously distorted throughout the annealing process.
http://www.youtube.com/watch?
v=7rGk1yTBp7A&feature=youtu.be&hd=1

55. Output for double-layered graphene
formation
1.
2.
3.
4.
Average binding energies (BE) for the top (cr1) and the second graphene
layer (cr2) vs. step at a fixed target annealing temperature.
Average nearest neighbours (bound length) for the top (cr1) and the second
graphene layer (cr2) vs. step at a fixed target annealing temperature.
Average distances between the topmost carbon-rich layer (cr1) and the
carbon-rich layer below it (cr2) vs. step at a fixed target annealing temperature
(see figure below).
Average distances between the second carbon-rich layer (cr2) and Si5, the
silicon layer on the substrate, vs. step at a fixed target annealing temperature
(see figure below).
d12, average distance between the two carbon-rich
layers
d25
Top carbon-rich layer, cr1
second carbon-rich layer,cr2
Si5
SiC substrate

56. Tanneal = 500 K
No graphene is formed

57. Tanneal = 600 K
Graphene is formed

58. • Tanneal-dependence of nn, binding
energies, and distances between the
layers could be abstracted from the curves
obtained for each Tanneal.
• The resultant Tanneal-dependence curves
are to be displayed in the next slide.

59. {bd,nn,distances} vs. temp
for z = 1
bd = binding energy; 1≡ topmost cr; 2≡ second layer cr from the top; 5≡ Si5,
silicon atom on the substrate right below cr2.

60. Comment on the data for the z =
1 case
• It is commented that the results for double
layer graphene formation using a
substrate with thickness z = 1 is not of
good quality.
• The transition happens rather gradually
and a sharp transition temperature is
ambiguous.

61. Two-layered carbon-rich
substrate with
thickness z = 2
for double-layered graphene
formation

62. Figure 15:
Substrate with thickness z=2
•
A 6H-SiC unit cell with
a thickness z = 2
substrate unit cell is
shown.
•
This is an z = 2 original
unit cell without any
atoms removed nor
displaced.
We shall subject this
unit cell to modification
procedure and
subsequent energy
minimization as
depicted in Figure 13.
•
•
The results of the
minimised structure is
displayed in Figure 16.

63. Transition temperature for doule-layared
graphene formation with substrate thickness
z=2

For substrate thickness z = 2, double-layered
graphene is formed at Tanneal = 1100 K.

64. Visualisation of two carbon rich
layer substrate and graphene
formation for z=2
• http://www.youtube.com/watch?
v=7rGk1yTBp7A&feature=youtu.be&hd=1
• http://www.youtube.com/watch?
v=9PAvX_BEsNk&feature=youtu.be&hd=1

65. Snapshot of carbon-rich layers at various temperatures
Top layer carbon at 300 K
Top layer graphene formation at 1200 K
Bottom layer carbon at 300 K
Bottom layer graphene formation at 1200 K

66. Summary of temperautre
dependence of double-layered
greaphene formation,
z=2

67. Binding Energy
Top carbon-rich layer, cr1
second carbon-rich layer, cr2

68. Average Nearest Neighbour
Top carbon-rich layer
second carbon-rich
layer

69. Average Distance of
Two Graphene Layers
Distance Between
Graphene and Buffer
Layer

70. Double-layered graphene
formation with substrate
thickness z = 3

71. Transition temperautre
• http://www.youtube.com/watch?
v=5RC8Gj8JqaM&feature=youtu.be&hd=1
• We found the transition temperature T f
occurs at = 1100 K

72. The results for double-layered graphene
formation with z = 3 are very similar to that
for z = 2

73. Three-layered carbon-rich
substrate with
thickness z = 2
for trilayered graphene
formation
(TEA only)

74. Simulation method of graphene growth (three layers)
1.9 Å
Slide adopted from Prof. Lai
Conjugate gradient minimization
Simulated annealing
74
15

75. TRILAYER GRAPHENE
FORMED ON Z=2 SUBSTRATE
http://www.youtube.com/watch?
v=7oZzjXqtpi4&feature=youtu.be&hd=1

76. Temperature dependence of
bd, nn, distances
for trilayer graphene
formation, z = 2

77. Binding Energy

78. Average Nearest Neighbour

79. Average Distances between atomic layers
Average distance between top layer graphene and middle layer graphene
Average distance between middle layer graphene and bottom
layer graphene
Average distance between bottom layer graphene and buffer layer

80. First layer graphene layer at 300K
Second layer graphene layer at 300K
First layer graphenelayer at 1200K
Second layer graphene layer at 1200K

81. Third layer graphene layer at 300K
Third layer graphene layer at 1200K

82. Conclusion
• Transition temperature 1200K as
predicted from the simulation for single
graphene layer formation agrees with that
of experiment
• TEA force field is better suited for
simulation epitaxial graphene formation
• We also simulated double and try-layered
graphene formation on the SiC (0001)
surface and provided additional insight
into the formation mechanism of epitaxial
graphene formation on SiC