Epitaxial growth of graphene on 6H-silicon carbide substrate by simulated annealing method

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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

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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 re?ned 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.

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(No Transcript)
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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.mi
l/lattice/struk/6h.html. The 6H-SiC 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 ¹ c
a b 90 degree, g 120 degree.
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Figure 2 Snapshot from http//cst-www.nrl.navy.mi
l/lattice/struk.xmol/6h.pos
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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).

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Procedure to construct rhombus-shaped 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.

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Lattice parameters

• The lattice constants, as obtained from the above
  calculation, are a 3.08 nm, b3.08 nm, c
  15.11 nm.
• Since the 6H-SiC belongs to a hexagonal class, ?
  ? 90 degree, ? 120 degree.

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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.htmlho
  wto_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 ???????
  into these units. This could be done quite
  trivially, via the conversion show in the right

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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.

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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
cells 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.
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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 6H-SiC is Carbon
  terminated.
• There is a total of 12 atoms it the original unit
  cell.

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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 zdirection to
  take up the z-coordinate left vacant by the
  removed Si atom (while the x- and y-coordinate
  remains unchanged).

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Simulation method of graphene growth (one layer)

• 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
• Extracting the result.

2.52 Å
lt 1 Å 0.63 Å
2.0 Å
Conjugate gradient minimization
Simulated annealing
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Content of data.singlelayer.xyz

• 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.

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Figure 6 carbon-rich unit cell of SiC
11 atoms per unit cell left as one Si atom (No.
9) has been removed.
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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

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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.

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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.
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Figure 7 (b)
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Figure 7 (c)
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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 LAMMPSs 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.

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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
5 x 1013 K/s
5000
5000 steps
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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).

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Figure 9Temperature 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
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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.

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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 -gt target
  temperatures, 5 x 1013 K/s.
• 4. Cooling rate From target temperatures -gt 1 K,
  1 x 1013 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 -gt target annealing temp, 60,000
  steps.
• Essentially, all the parameters used are the same
  as that used by the NCU group.

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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.

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Results from TEA force field
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Configuration of the carbon-rich substrate before
and after equilibration at T 1.0 K for
single-layered graphene formation
Before minimisation
After minimisation
As comparison, this figure shows the geometry
obtained by the NTCU group before and after
minimisation
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Graphene before and after formation at Tanneal
1200 K
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Formation of single layered graphene with
thickness z1 substrate, with TEA at 1200 K

• http//www.youtube.com/watch?vklkg2Rlf7Gk
• Agrees with what Hannon and Tromp measured

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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.

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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
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Figure 10(i) Tanneal 1100 K.
No graphene formed
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Figure 10(ii) Tanneal 1200 K.
Graphene is formed
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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 50008500040(temp-300)

s
s
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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.

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Binding energy vs anneal temperature
data\singlelayer\TEA\bdvstemp.dat
Anneal temp binding energy 400 -5.8804708333
33334 500 -5.872215972222222 600
-5.8549182291666595 700 -5.853736979166666 800 -5.
844029131944445 900 -5.8253253472222255 1000
1100 -5.810316180555554 1200 -6.647701284722221 1
300 -6.830150555555552 1400 -6.844608055555553 150
0 -6.713664999999995 1600 -6.833112638888893 1700
-6.747370833333332 1800 -6.877048993055555 1900
-6.709565694444447
Figure 11
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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.

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Average nearest neighbour (nn) vs anneal
temperature

• Anneal temp average nn
• 400 1.750833424567148
• 500 1.746625140692055
• 600 1.7589214030309763
• 1.7419868390032442
• 800 1.7414136922144865
• 900 1.7589380688936933
• 100 1.7417389709279334
• 110 1.7344180027393463
• 1200 1.5027327808093742
• 1300 1.4693218689432666
• 1400 1.4764060075537178

Figure 12
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Average distance between cr1 and Si6 vs anneal
temperature

• Anneal temp average distance cr3-Si6
• 400 2.1048033680555562
• 500 2.1063726736111175
• 2.105993090277779
• 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

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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.

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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
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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
  Tersoffs three body interactions are fitted with
  lesser accuracy.

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Double-layered graphene formation(Only for TEA
force field)
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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 z1.

Conjugate gradient minimization
Simulated annealing
Conjugate gradient minimization
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14
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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.

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Two-layered carbon-rich substrate with thickness
z 1 for double-layered graphene formation
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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
1.89 A
Note we note that the substrate get distorted
significantly after energy minimisation.
Figure 15. z 1.
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Visualising graphene formation for 15 x 15 x 1
supercell at Tanneal 1100 K, z 1

• We found that for substrate thickness z 1,
  double-layered 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?v7rGk1yTBp7Afeature
  youtu.behd1

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Output for double-layered graphene formation

1. Average binding energies (BE) for the top (cr1)
   and the second graphene layer (cr2) vs. step at
   a fixed target annealing temperature.
2. Average nearest neighbours (bound length) for the
   top (cr1) and the second graphene layer (cr2)
   vs. step at a fixed target annealing temperature.
3. 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).
4. 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).

Top carbon-rich layer, cr1
d12, average distance between the two carbon-rich
layers
second carbon-rich layer,cr2
d25
Si5
SiC substrate
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Tanneal 500 K
No graphene is formed
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Tanneal 600 K
Graphene is formed
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• 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.

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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.
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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.

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Two-layered carbon-rich substrate with thickness
z 2 for double-layered graphene formation
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Figure 15 Substrate with thickness z2

• 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.

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Visualisation of two carbon rich layer substrate
and graphene formation for z2

• http//www.youtube.com/watch?v7rGk1yTBp7Afeature
  youtu.behd1
• http//www.youtube.com/watch?v9PAvX_BEsNkfeature
  youtu.behd1

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Snapshot of carbon-rich layers at various
temperatures
Top layer carbon at 300 K
Top layer graphene formation at 1200 K
Bottom layer graphene formation at 1200 K
Bottom layer carbon at 300 K
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Summary of temperautre dependence of
double-layered greaphene formation, z 2
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Binding Energy
second carbon-rich layer, cr2
Top carbon-rich layer, cr1
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Average Nearest Neighbour
second carbon-rich layer
Top carbon-rich layer
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Average Distance of Two Graphene Layers
Distance Between Graphene and Buffer Layer
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Double-layered graphene formation with substrate
thickness z 3
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Transition temperautre

• http//www.youtube.com/watch?v5RC8Gj8JqaMfeature
  youtu.behd1
• We found the transition temperature Tf occurs at
  1100 K

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

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Three-layered carbon-rich substrate with
thickness z 2 for trilayered graphene
formation (TEA only)
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Simulation method of graphene growth (three
layers)
1.9 Å
Slide adopted from Prof. Lai
Conjugate gradient minimization
Simulated annealing
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15
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TRILAYER GRAPHENE FORMED ON Z2 SUBSTRATE
http//www.youtube.com/watch?v7oZzjXqtpi4feature
youtu.behd1
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Temperature dependence ofbd, nn, distances for
trilayer graphene formation, z 2
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Binding Energy
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Average Nearest Neighbour
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Average Distances between atomic layers
Average distance between middle layer graphene
and bottom layer graphene
Average distance between top layer graphene and
middle layer graphene
Average distance between bottom layer graphene
and buffer layer
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First layer graphenelayer at 1200K
First layer graphene layer at 300K
Second layer graphene layer at 300K
Second layer graphene layer at 1200K
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Third layer graphene layer at 1200K
Third layer graphene layer at 300K
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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