Transcript Z - School of Physics Official Website
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
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.
Single 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.
The snapshots from the above webpage 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 ; a = b = 90 degree, g = 120 degree.
Figure 2: Snapshot from http://cst www.nrl.navy.mil/lattice/struk.xmol/ 6h.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 c
: ,
b
(=
a
), 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
parameter (1) and adding it to
a a
(2) squared, we could easily obtain the value for the lattice , which is also equal to
b
by definition of the crystallographic group.
Lattice parameters
a
(1) 2
a
(2) 2 1 2
a
X
1 2 3 1/2
a
Y
2 1 2
a
X
1 2 3 1/2
a
Y
2
a
2 1.54035000
2.66796446
2 2.66796446
2 2 1 4
a
2 3 4
a
2
a
3.08
a
(3) 2 (
c
Z
) 2 15.11740000
2
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, 90 degree, g = 120 degree.
a = b =
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
; a b g } 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
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: V isualization 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.
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).
Simulation method of graphene growth (one layer) 2.52
Å 2.0 Å Conjugate gradient minimization Simulated annealing < 1 Å = 0.63 Å Simulated Annealing Timestep = 0.5 fs Increase the temperature slowly until it attains 300
K
at approximately 5˟10 13
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 10 13
K
/
s
.
Equilibrating the system at T for 30000 MD steps.
Cool down the system until 0.1
K
at 5x10 12
K
/
s
Extracting the result.
16
11 Content of data.singlelayer.xyz
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
The content of dataraw.xyz is now modified and C 1.54035000 0.88932149 10.07877058
C 1.54035000 -0.88932149 4.41276906
renamed as data.singlelayer.xyz
, C 1.54035000 0.88932149 6.92981616
C 1.54035000 -0.88932149 -0.62888384
which content is Si 1.54035000 -0.88932149 2.52007058
Si 1.54035000 0.88932149 5.03711768
shown in Figure 5, and visualised in Si 1.54035000 -0.88932149 -2.52158232
Figure 6.
Figure 5: The content of atom (atom 5). 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
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.
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
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
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 10 13 K/s 5000 5000 steps Figure 8. Annealing procedure based on suggestion by Prof. S.K. Lai.
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 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
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 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.
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 formation 0.624 Å 0.2276 Å 1.896 Å 1.9948 Å Before minimisation After minimisation 1.89
Å 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
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) d 34 = average distance between the carbon-rich layer and the substrate just below it Si atoms (labeled as Si4) SiC substrate
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.
Binding energy vs anneal temperature
data\singlelayer\TEA\bdvstemp.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
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.
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
Average distance between cr1 and Si6 vs anneal temperature Anneal temp average distance cr3-Si6 400 500 2.1048033680555562
2.1063726736111175 600 2.105993090277779 700 2.105879791666668 800 900 1000 2.1031867708333323 2.1202562847222266
2.1176251041666685 1100 1200 1300 1400 2.124162604166675
2.4697990625000052 2.560621458333336 2.54777364583333
1500 2.6178477083333327
1600 2.6055096527777843
1700 1800 1900 2.722710520833341
2.8183415972222177
2.6691606597222215
Data and results for single layer graphene formation with TEA From the data generated, we conclude that: Graphene formation is observed only when Tanneal= T f (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
Tersoff vs. TEA
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)
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
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 1.59 A 0.51 A 1.35 A 0.61Å 1.89 A Note: we note that the substrate get distorted significantly after energy minimisation.
Figure 15. z = 1.
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
1.
2.
3.
4.
Output for double-layered graphene formation 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 Top carbon-rich layer, cr1 second carbon-rich layer,cr2 d25 Si5 SiC substrate
Tanneal = 500 K No 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 shown.
z
= 2 substrate unit cell is • • • 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
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=7rGk1y TBp7A&feature=youtu.be&hd=1 • http://www.youtube.com/watch?v=9PAvX_ BEsNk&feature=youtu.be&hd=1
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
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 layer second 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=5RC8Gj 8JqaM&feature=youtu.be&hd=1 • We found the transition temperature T f 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)
Simulation method of graphene growth (three layers) 1.9 Å Slide adopted from Prof. Lai Conjugate gradient minimization Simulated annealing 74
15
TRILAYER GRAPHENE FORMED ON Z=2 SUBSTRATE
http://www.youtube.com/watch?v=7oZzjXqt pi4&feature=youtu.be&hd=1
Temperature dependence of bd, 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 graphene Average 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