Abstract
Our theoretical interpretation shows that the electron density is condensed at 150°orientation and perhaps the reason for the high mobility of electron in graphene. To determine the above result, the constructed monolayer of graphene contains 40 carbon atoms with 48 bonds in between them and 16 hexagonal polyhedra and the bilayer of graphene has 80 atoms with 96 bonds with 32 polyhedra. The hexagon of carbon lattice has an area of 5.24 Å2. One unit-cell has the volume of 314.807763 Å3. Hexagonal lattice structures have the angle of α = β= 90˚ and γ= 120˚. The result concentrates on mono and bilayer of graphene by varying the scattering resolution in the form of 0.1Å to 1.1Å in the order of 0.1Ǻ to the maximum occupation of electron density in the lattice. This lattice constants of the layer are ‘a’ and ‘b’ is 2.456Å and ‘c’ is 6.696Å. The electron density attraction to the bilayer of graphene shows that the fraction of electron density is concentrated in-between two layers, towards the center of the lattices, in the monolayer of graphene the electron density is concentrated above and below the lattice. The electron density slices are to study the electron densities in two-dimensional planes. From their conformation by theoretical powder X-ray diffraction delivers for the mono, bilayer & tri-layer (graphite) and compare with experimental data. Here we are focused on model electron density and Patterson density for the above work.
Introduction
The scientific world is always curious about the one and two-dimensional materials and its properties with applications. In 1947 Wallace first studied the electronic band structure of graphite [1]. Then it was first separated as a single or monolayer and studied in 2004 by Novoselov [2]. Researchers are more concentrate on the most popular graphene sheets from 2004 onward and for their hexagonal lattice strength is more than steel and natural structure of the honeycomb with carbon atoms are separated by 1.42Å [1], and the orbital Sp2 hybridization [3]. Graphene has the highest theoretical surface area of 2630 m2/g when compared to its allotropes [4]. Graphene elasticity is approximately 1Tpa [5].
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Figure 1 gives the basic structure of graphene with the bond length of 1.42 Å and the bond angle of 120°. The possible pathway of electron density in the lattice orientation (A –B, C – D, E – F).
The hexagonal lattice structure of carbon atoms allows valence electrons were free to move within the 2D sheet. The mobility of electron in the graphene have two dimensional structures are higher than almost all the materials in the world. Comparing graphene and silicon, graphene dissipates heat very much faster than Silicon, because of the high thermal conducting property [Alexander A. Balandin et. al][13].
The graphene, two-dimensional sheet has one valence electron per atom were free at the end of structure, that gives high charge mobility. The large surface area of graphene are allows electrons form the electronic clouds and it moves freely in the lattice. As per the Report high thermal conductivity at room temperature of suspended single layer graphene is 5300 W/mK [13]. In the view of mobility, scientists ever observed highest mobility from two-dimensional electron gas, which has the value of 35,000,000 cm2/Vs [6]. The second material is one-dimensional carbon nanotubes, which has the mobility 300,000 cm2/Vs [7]. At the third place is graphene having the mobility in the order of 250,000 cm2/Vs in room temperature [6].
In quantum mechanics, Electrons in a crystal or molecule are condensed in certain space around the nucleus of the atoms. These are called Electron densities and it plays a major rule in the material properties. The electron densities of the region are delocalized and it is seen in bonds structure of the molecules. The shape of the molecule may change or depend on the formation in their ground state [8]. Similar to this work Egor et al. [9] to study the band structure of mono and bilayer of graphene by using the DFT and AB INTIO software.
Table 1. Contain the structure of monolayer,bilayer and tri-layer of graphene in the crystal axis a, b, c.
Layers/Direction of view
Along the direction of “a”
Along the direction of “b”
Along the direction of “c”
Monolayer
Bilayer
Trilayer
The Visualization of Electron and Structural Analysis (VESTA) software helps to study the monolayer and bilayer of graphene in view of Model electron Densities and Patterson model of electron densities [10]. This software is also used to create crystal structures and generate to visualize of the DFT output files. The Model electron densities were obtained by Fourier transformation of structure factors & Patterson made changes in the density function by adding the experimental data like X-ray diffraction values to locate heavy atoms in the lattice. Patterson densities are also known as the heavy atom densities. The images simulated by varying the scattering resolution of electron densities in the order of 10-10m or Å. The equations of Electron density of the figures are shown below. Vesta simulates the density images based on the following equations.
Model electron density........................................ (1)
Patterson Model electron density................................... (2)
Result and Discussion
The unit cell of carbon atoms is the key to high mobility in the structure. The mono and bilayer of graphene structure has similar pockets of electron density above and below the lattice. Those pockets of planes are separated by 0.6Å. All the possibility of electron density images is having the same scattering length of 0.6Å. That particular scattering length mono layer is become a dumbbell shape of density. The dumb bell shape merges with increase in scattering resolution and it merge at 1.1Ǻ. Then it became slab like shapes.
Dumbbell shapes up to 0.6Ǻ, then the two atoms of carbon form a hexagon in a particular direction and merge together at 1.1Ǻ then it becomes a single sheet at hexagonal 150° orientation. The constructed graphene mono layer contains 40 carbon atoms with 48 bonds in between them and 16 hexagonal polyhedra. For the bilayer of graphene 80 atoms with 96 bonds and 32 polyhedra. The hexagon of carbon lattice has an area of 5.24 Å2. One unit-cell has the volume of 314.807763 Å3. Hexagonal lattice structures have the angle of α = β= 90˚ and γ= 120˚. This study starts with giving model electron density for constructed structures. Atoms and bonds removed to see the clear look of electron densities. The following table contains the images of the density changes with the scattering resolution.
In the bilayer of graphene also generated as scattering resolution from 0.1 to 1.1Ǻ, here the electron density formed toward the center of the layers.
Table 2. This table shows the variation in the monolayer of graphene’s electron density by changing the scattering resolution.
S. no
Resolution in Å
Electron Density Output Images
1
0.1
2
0.2
3
0.3
4
0.4
5
0.5
6
0.6
7
0.7
8
0.8
9
0.9
10
1.0
11
1.1
12
2.24
Table 2 shows the electron density images were obtained by the software using equation (1). The monolayer electron density increases with the increasing scattering resolution. At a certain 0.6Å, the EDIs (Electron Density Images) is giving us a pattern that suggests orientation mobility in graphene without any perturbation. Other values of scattering resolution the EDIs are following the pattern and confirming the orientational density in graphene. Limitation of density in monolayer, where the density is occupies the entire layer of graphene at 1.1Å. When the resolution of 2.24Å is applied, the density is fully occupied the constructed volume area. This is because of the electron cloud formation.
Table 3. This table shows the electron density of bilayer graphene with the inter-lattice distance is 6.69Å
S.no
Resolution in Å
Electron Density Output Images
axis c
Electron Density Output Images
Three dimensional view
1
0.1
2
0.2
3
0.3
4
0.4
5
0.5
6
0.6
7
0.7
8
0.8
9
0.9
10
1.0
11
1.1
12
3.35
Table 3 shows the EDIs in the bilayer of graphene. From this study, the limitation for the electron density occupation of the lattice is the same as monolayer graphene. But the occupation of the full structure is not as same as the monolayer. For the bilayer of graphene, at 3.35Å the Electron density is fully occupied the constructed region.
Table 4. This table shows the Patterson density of the bilayer of graphene.
S.no
Resolution in Å
Electron Density Output Images
axis c
Electron Density Output Images
Axis b
1
0.1
2
0.2
3
0.3
4
0.4
5
0.5
6
0.6
7
0.7
8
0.8
9
0.9
10
1.0
11
1.1
12
3.34
Table 3 shows the Patterson density images of the bilayer of graphene. Patterson density is based on equation (2) that Patterson derived from equation (1). From Patterson density images, the similar changes occurred in 0.6Å. Density becomes orientational and continues. But the limitation of lattice and constructed surface is 1.1Å and 3.34Å.
Table.5. This tables shows the Electron density images of Trilayer of Graphene.
S. No.
Resolution In Å
Electron Density Output Images
axis c
Electron Density Output Images
Axis b
1
0.1
2
0.2
3
0.3
4
0.4
5
0.5
6
0.6
7
0.7
8
0.8
9
0.9
10
1.0
11
1.05
12
1.34
In this table 5 gives the EDI of three layers of graphene shows some orientational electron density like at scattering resolution of 0.5 and 0.6Å. Then the orientation starting to collapse at 0.7 and continues. At 1.05Å it is completely not present in the constructed structure.
Table.6. This tables shows the Patterson Electron density images of Trilayer of Graphene.
S. No.
Resolution In Å
Electron Density Output Images
axis c
Electron Density Output Images
Axis b
1
0.1
2
0.2
3
0.3
4
0.4
5
0.5
6
0.6
7
0.7
8
0.8
9
0.9
10
1.0
11
1.05
12
1.34
From this table, Patterson density shows that there is a directional like density at 0.5Å. But in the other scattering angles, there is no orientation. This results also imply that more than two layers of graphene act as graphite and it has not exact orientation at 0.6Å, unlike monolayer and bilayer. It shows us that graphite is a good conductor, but not as good as mono and bilayer graphene. From trilayer of graphene (graphite) tables ( 5 & 6 ), there is no electron density at 1.1Å and it is not covering the plane like in the mono and bilayer of graphene. Trilayer covers constructed structure at 1.34Å. But monolayer and bilayer cover all constructed structure at 2.24 and 3.34Å respectively.
Figure 2 Figure 3
Figure 2. Shows the structure of Graphene inner structure of zig-zag and arm chair which is drawn line black and red respectively. Figure 3 shows the zigzag pattern of honeycomb structure has maximum density.
From those tables, it shows the variation in the electron density in mono and bilayer of graphene by varying the scattering resolution by the order of 0.1Å, 0.2 Å... and to the maximum occupation of electron density in the lattice and the constructed surface. The lattice constants of this layer are ‘a’ and ‘b’ is 2.456Å and ‘c’ is 6.696Å.To study the electron density attraction to the bilayer of graphene shows that the fraction of electron density is concentrated in-between the layers. Even in the single layer of graphene, the electron density is concentrated above and below the layer. The electron density slices are sliced by the VESTA software to study the electron densities in two-dimensional planes [9],[11].
(a) (b)
(c)
Figure 4. The lattice planes view of mono (a), bi (b) and (c) trilayer of graphene
Figure 4 (a) and (b) show us the similar structure in electron densities is distributed similarly from the 100 and 100 crystal planes of mono and bilayer of graphene respectively. Even in the single layer of graphene show us the some of the electron densities distributed outside the of the lattice plane. For (c) shows the three layer of graphene and its electron density at the plane 0 -1 0 at 0.6Å.
Figure 5. This large circle contain a square shaped electron density surrounding the carbon atom. Other small circles are indicating the minimal electron density in the above and below the atomic plane. At some distance from the monolayer of graphene, the electron densities are not diminishing; it all converges in small packs like structures because of the wave property of atoms or electrons.
Figure 6. Red area confirms the high density of free electrons around the carbon atoms.
Fig 7. Mono layer model electron density
Fig 8. Bilayer model electron density
Fig 9. Bilayer Patterson electron density
Figure 8-10. By comparing these images at 0.6Å, the results shows zigzag atoms concentrates in the valance electron density by themselves. This allows the electrons moves in the particular orientation or the direction of electron flow.
Figure 10. Theoretical Powder X-ray diffraction data for the constructed mono and bilayer of graphene.
Figure 11. Experimental X-Ray Diffractions pattern of (a) graphene oxide and (b) graphene from Madzlan Aziz et. al (2014)[12].
Theoretical Powder X-ray diffraction data for the constructed mono and bilayer of graphene gives similar peaks. Because of the lack of inter lattice distance is absent in mono layer graphene for theoretical approach. This powder X-ray diffraction data compared with the Madzlan Aziz et.al JurnalTeknologi (2014) and AMCSD data. To confirm simulated graphene has same property as the synthesised one. The comparison gives at 26.6° and 54.7° of 2θ values gives exact same peak as theoretical plot. Hkl planes of the similar peaks are 002 and 004. This confirms our theoretical graphene structure is graphene.
Figure.12. Shows the Theoretical powder X-Ray Diffraction Pattern
Figure.13. Shows the X- Ray diffraction pattern of Ball-Milled Graphite and Commercial Graphite from C.H. Manoratne et al. (2017)[14]
From figure 12 and 13, the powder X-ray spectrum of both theoretical and experimental matches at three main peaks with same hkl plane. This confirms the constructed three layer structure is acted as a graphite as expected. The graphite from the ball milled and commercial has similar peaks with the some absence of peak. But in the constructed three layers has most of the graphite peaks. The planes 002, 004, 006 peaks are reported as graphite in so many literatures.
IV. Conclusion:
From the study of monolayer and bilayer of graphene (Table 2, 3 and 4), the distribution of electron density is very similar. The unit cell of carbon atoms is the key to high mobility in the structure. At the 0.6 Å scattering resolution, electron density is starting to accumulate in the unit cell. At 150°orientation all the electron densities converge and form a path. The structure has pockets of electron densities are above and below the lattice for monolayer and in-between for bilayer. Those pockets of planes are separated by 0.6Å. That particular scattering length mono layer is become a dumbbell shape of density, bilayer gave the flow of the electron density that might be the possible reason for the high mobility in graphene. In the trilayer of graphene most likely have the properties of graphite. The structural analysis shows that the mono and bilayers are graphene layers and trilayer belongs to graphite.
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