Epitaxial growth of Epitaxial growth of graphene on 6H-silicon graphene on 6H-silicon

carbide substrate by carbide substrate by simulated annealing methodsimulated annealing method

Yoon Tiem LeongYoon Tiem LeongSchool of Physics, Universiti Sains MalaysiaSchool of Physics, Universiti Sains Malaysia

Talk given at the Theory Lab, School of Physics,Talk given at the Theory Lab, School of Physics,USMUSM

24 Jan 201424 Jan 2014

AbstractAbstract

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

Single layer graphene formationSingle layer graphene formation

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.

Figure 1: Snapshot from http://cst-www.nrl.navy.mil/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 ≠ c ; α = β = 90 degree, γ = 120 degree.

The snapshots from the above webpage

Figure 2: Snapshot from http://cst-www.nrl.navy.mil/lattice/struk.xmol/6

h.pos

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).

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.

Lattice parameters

( ) ( )

( ) ( )

2 22 2 1/2 1/2

2 2 2 2

2 22

2 2

2 2

1 1 1 1(1) (2) 3 3

2 2 2 2

1 32 1.54035000 2.66796446 2

4 4

1.54035000 2.66796446

3.08

(3) ( )

15.11740000

15.11

a a a a a a

a a

a

a

a c

c

c

b a

+ = − + + ÷ ÷ + = + ÷

= +=

==

=

=

X Y X Y

Z

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.

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:

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.

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.

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

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.

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)

2.52 Å< 1 Å = 0.63 Å

Conjugate gradient minimization

Simulated annealing

2.0 Å

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.

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.

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

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).

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.

Figure 6: carbon-rich unit cell of SiC

11 atoms per unit cell left as one Si atom (No. 9) has been removed.

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

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.

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.

Figure 7 (b)

Figure 7 (c)

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.

Annealing procedure

Figure 8. Annealing procedure based on suggestion by Prof. S.K. Lai.

Monitor output here

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.

5 x 1013 K/s

5000

1

1K

5000 steps

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).

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

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.

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 1013

K/s. 4. Cooling rate: From target temperatures -> 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 -> target annealing temp, 60,000 steps.

Essentially, all the parameters used are the same as that used by the NCU group.

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.

Results from TEA force field

Configuration of the carbon-rich substrate before and after equilibration at T = 1.0 K for single-layered

graphene formation0.624 Å

1.896 Å0.2276 Å

1.9948 Å

Before minimisation

After minimisation

0.63Å1.89

Å

0.22Å

1.99Å

After minimization but before simulated annealing

As comparison, this figure shows the geometry obtained by the NTCU group before and after minimisation

Graphene before and after formation at Tanneal = 1200 K

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:

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.

Definition of d34 for single layer graphene formation

Top carbon-rich layer, (labeled as cr3)

SiC substrate

d34 = average distance between the carbon-rich layer and the substrate just below it

Si atoms (labeled as Si4)

Figure 10(i): Tanneal = 1100 K.

No graphene formed

Figure 10(ii): Tanneal = 1200 K.

Graphene is formed

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

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.

Anneal temp binding energy

400 -5.880470833333334500 -5.872215972222222600 -5.8549182291666595700 -5.853736979166666800 -5.844029131944445900 -5.82532534722222551000 1100 -5.8103161805555541200 -6.6477012847222211300 -6.8301505555555521400 -6.8446080555555531500 -6.7136649999999951600 -6.8331126388888931700 -6.7473708333333321800 -6.8770489930555551900 -6.709565694444447

data\singlelayer\TEA\bdvstemp.dat

Binding energy vs anneal temperature

Figure 11

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.

Anneal temp average nn

400 1.750833424567148500 1.746625140692055 600 1.7589214030309763 7001.7419868390032442 800 1.7414136922144865900 1.7589380688936933100 1.7417389709279334 110 1.7344180027393463 1200 1.5027327808093742 1300 1.4693218689432666 1400 1.4764060075537178

Average nearest neighbour (nn) vs anneal temperature

Figure 12

Anneal temp average distance cr3-Si6

400 2.1048033680555562500 2.1063726736111175 600 2.105993090277779 700 2.105879791666668 800 2.1031867708333323 900 2.12025628472222661000 2.1176251041666685 1100 2.1241626041666751200 2.4697990625000052 1300 2.560621458333336 1400 2.547773645833331500 2.61784770833333271600 2.60550965277778431700 2.7227105208333411800 2.81834159722221771900 2.6691606597222215

Average distance between cr1 and Si6 vs anneal temperature

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.

Outcome from Tersoff

• We have also simulated with Tersoff force field.

• The outcome are summarised in the next slide.

Tersoff vs. TEATersoff

TEA

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.

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

14

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.

Two-layered carbon-rich substrate with thickness z = 1

for double-layered graphene formation

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

0.61Å

1.89 A

1.59 A

0.51 A

1.35 A

Figure 15. z = 1.

Note: we note that the substrate get distorted significantly after energy minimisation.

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?v=7rGk1yTBp7A&feature=youtu.be&hd=1

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

second carbon-rich layer,cr2

SiC substrate

d12, average distance between the two carbon-rich layers

d25Si5

Tanneal = 500 KNo graphene is formed

Tanneal = 600 K

Graphene is formed

• 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.

{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.

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.

Two-layered carbon-rich substrate with thickness z = 2

for double-layered graphene formation

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.

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.

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

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

Snapshot of carbon-rich layers at various temperatures

Summary of temperautre dependence of double-layered

greaphene formation, z = 2

Binding Energy

Top carbon-rich layer, cr1 second carbon-rich layer, cr2

Average Nearest Neighbour

Top carbon-rich layersecond carbon-rich

layer

Average Distance of Two Graphene Layers

Distance Between Graphene and Buffer

Layer

Double-layered graphene formation with substrate

thickness z = 3

Transition temperautre

• http://www.youtube.com/watch?v=5RC8Gj8JqaM&feature=youtu.be&hd=1

• We found the transition temperature T ff occurs at = 1100 K

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

for z = 2

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

Conjugate gradient minimization

Simulated annealing

15

Slide adopted from Prof. Lai

http://www.youtube.com/watch?v=7oZzjXqtpi4&feature=youtu.be&hd=1

TRILAYER GRAPHENE FORMED ON Z=2 SUBSTRATE

Temperature dependence ofbd, nn, distances

for trilayer graphene formation, z = 2

Binding Energy

Average Nearest Neighbour

Average Distances between atomic layers

Average distance between top layer graphene and middle layer grapheneAverage distance between middle layer graphene and bottom

layer graphene

Average distance between bottom layer graphene and buffer layer

First layer graphene layer at 300K First layer graphenelayer at 1200K

Second layer graphene layer at 300K Second layer graphene layer at 1200K

Third layer graphene layer at 300K Third layer graphene layer at 1200K

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