Formulation of atomic positions and carbon–carbon bond length in armchair graphene nanoribbons: an ab initio study
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Abstract
In this paper, we investigate the atomic positions of single layer armchair graphene nanoribbon for two cases, with and without hydrogen-passivate edges, accurately and propose a formula which either removes the need of structural relaxation generally or decreases its time extremely (up to seven times). We also propose a general pattern (hyperbolic) for these positions. On the other hand, we show that edge effect influences several atoms near the edge not just one. These results can be used in software, which compute atomic positions and can increase their efficiency. In addition, we prove that the C–C bond distance depends on dimer number and differs in length and width directions, especially for narrow AGNRs. The maximum value of these differences is about 0.017 Å.
Keywords
Armchair graphene nanoribbon (AGNR) Structural relaxation Atomic position Dimer C–C bond lengthIntroduction
Graphene is a one-atom-thick crystal of sp^{2}-bonded carbon atoms ordered in a two-dimensional (2D) hexagonal lattice [1]. Graphene and graphene nanoribbon (GNRs) have attracted much recent attention for their unique two-dimensional structures and physicochemical properties [2], as well as their wide potential applications in electronics [3, 4], photonics, and optoelectronics [5], energy storage and conversion [6], and chemical-bio sensing [7].
Regarding the broad usage of GNR in nano-electronics, physics and chemistry, this substance is under wide investigations at present. Most of these investigations are done using some software based on density functional theory (DFT), and they need structural relaxation of nanoribbons. It takes much time, especially if the length of nano-transistors being investigated is more than 10 nm. In this case, the number of atoms will increase. If the method for simulation of the structure is based on order-N3, the time it needs will be much more even for supercomputers. In the case of investigating multilayer graphene nanoribbon, the situation will deteriorate.
The idea is proposing a method having the dimer number of GNR and its length to acquire the structural relaxation and the exact position of atoms. In this case, either the general need for structural relaxation would be unnecessary or even by doing this, the time will be extremely decreased.
To use graphene nanoribbon as a channel, we should first determine the approximate position of atoms. Then these positions should be given to applications such as SIESTA, VASP and ATK [8, 9, 10] which change them using structural relaxation and spending a long time, until these atoms are placed in their precise positions. The running time for relaxation depends on several parameters: the accuracy of initial positions given to the software, the method used in the software for structural relaxation, the parameters that the user sets in the software, and the value of the requested accuracy to end run.
Except for the initial positions, the rest of the parameters are usually specified by the researcher at the beginning, and do not change [10, 11, 12]. Given that the initial position of the atoms is of paramount importance in comparison to the rest of the parameters, it is under way in the present study. It is natural that if we relax an atomic structure and set the obtained positions to the software as new initial positions and repeat the structural relaxation, the run time would be reduced several times. This time for very large structures, can be reduced from months to days. So acquiring a way to determine the precise initial atomic positions sounds completely logical.
Researchers obtain these initial positions from some specific ways: downloading from the Internet, using simple formulas available in books and articles [13, 14], and the use of applications that calculate these positions for a number of particular structures such as carbon nanotube [15]. It is natural that the atomic structures found on the Internet are limited. Using the simple formula readily accessible in the books and articles, we can calculate the positions with low accuracy, and so much time is required to run structural relaxation.
Applications that are available in this field are also very limited. For example, Nano Tube Modeler software, which is used for CNT [15, 16]. Such applications use simple and ideal formulas to calculate the atomic positions too, which make the run time very long. So by determining the exact formulas for the positions in GNRs in this paper, researchers can utilize these formulas.
In all applications and simple formulas, C–C bond lengths are considered constant for all GNRs in different directions [13, 14]. As we calculate the exact atomic position for all atoms, it will be obvious that the D_{cc} (carbon-carbon bond length) is not equal for them. We consider the average value of middle atoms in all GNRs in two directions and compare them.
The paper is organized as follows: in Sect. 2, we present a basic formula for atomic positions for all carbon atoms in ideal (slab cut of graphene) and real (relaxed) AGNR. In Sect. 3, we derive all atomic positions in x and y directions and energy gap, and then present a generic formula for them and show the fact that the C–C bond lengths are different in two directions. Finally, Sect. 4 summarizes our findings.
Problem statement
Results and discussion
Electronic structures and geometry relaxations are calculated based on the DFT [19], using the Spanish initiative for electronic simulations with thousands of atoms package (SIESTA) [20, 21] with local-density approximation (LDA) [22]. A double-ζ plus polarization basis set is employed to describe the localized atomic orbitals and an energy cutoff for real-space mesh size is set to be 380 Ry. All nanostructure geometries are fully relaxed with a force tolerance, 0.01 eV/Å [11].
Energy gap in AGNRs vs m
m = 0 | m = 1 | m = 2 | |
---|---|---|---|
With H | |||
\(A_{0}\) | 0.1206 | 0.0741 | 0.1585 |
\(A_{1}\) | 6.1866 | 9.9777 | 1.0715 |
MSE | 3.7e−05 | 8.2e−05 | 4.1e−05 |
Without H | |||
\(A_{0}\) | 0.1475 | 0.2286 | 0.1329 |
\(A_{1}\) | 3.9968 | 8.0067 | 2.7275 |
MSE | 2.9e−05 | 8.0e−04 | 2.1e−05 |
In Table 1, each column of data is related to a specific category of AGNRs, which has been shown by the value m = 0, 1, and 2. In MSE section of the table, it can be seen clearly that the value of error for m = 1, and for the case without hydrogen is more than the other considerably.
As a result, the diagram of simulation and nonlinear regression (for m = 1) in Fig. 3, are slightly different from each other. Although there exist some minor errors in one branch, simulation results show enough good matches with that of proposed formula.
Computing ∆x
The average of ∆x in middle carbons
m = 0 | m = 1 | m = 2 | |
---|---|---|---|
With H | |||
\(A_{0}\) | −0.0082 | −0.0374 | 0.0618 |
\(A_{1}\) | −2.7524 | −2.5069 | −4.1218 |
MSE | 7.4e−08 | 3.3e−08 | 3.0e−07 |
Without H | |||
\(A_{0}\) | −0.0204 | 0.0075 | −0.1548 |
\(A_{1}\) | −9.1735 | −9.9429 | −6.8449 |
MSE | 1.2e−07 | 7.2e−09 | 6.4e−07 |
Peak to Peak of ∆x in middle carbons
m = 0 | m = 1 | m = 2 | |
---|---|---|---|
With H | |||
\(A_{0}\) | 0.0972 | −0.0222 | 0.0613 |
\(A_{1}\) | 1.5239 | 3.5494 | 2.2642 |
MSE | 1.9e−07 | 8.3e−08 | 6.0e−08 |
Without H | |||
\(A_{0}\) | −0.3470 | −0.0671 | −0.2856 |
\(A_{1}\) | 18.0746 | 12.3439 | 16.6161 |
MSE | 3.3e−06 | 3.7e−08 | 1.8e−06 |
We have achieved all these diagrams for both cases, with and without hydrogen, and for four atoms near the edge (j = 1, 2, 3, 4), but considering the high number of figures, we do not draw them all. In all these diagrams, the general shape of hyperbolic can be seen. Thus, we use the formula (9). In this formula f = ∆x and x = N_{y}.
∆x of four first carbons in AGNRs with H
m = 0 | m = 1 | m = 2 | |
---|---|---|---|
Carbon 1 | |||
\(A_{0}\) | −1.7607 | −1.7344 | −1.6468 |
\(A_{1}\) | −5.1316 | −5.5395 | −7.3415 |
MSE | 3.5e−08 | 3.0e−08 | 6.4e−07 |
Carbon 2 | |||
\(A_{0}\) | −0.4238 | −0.3344 | −0.3516 |
\(A_{1}\) | −0.1986 | −2.6628 | −1.5550 |
MSE | 3.6e−09 | 1.5e−08 | 7.1e−08 |
Carbon 3 | |||
\(A_{0}\) | −0.0203 | 0.0647 | 0.0694 |
\(A_{1}\) | −3.7301 | −5.6972 | −5.3053 |
MSE | 3.3e−08 | 3.6e−08 | 4.6e−07 |
Carbon 4 | |||
\(A_{0}\) | 0.1402 | 0.1537 | 0.1080 |
\(A_{1}\) | −1.6357 | −2.2305 | −0.7365 |
MSE | 3.9e−09 | 8.6e−09 | 1.7e−08 |
∆x of four first carbons in AGNRs without H
m = 0 | m = 1 | m = 2 | |
---|---|---|---|
Carbon 1 | |||
\(A_{0}\) | −6.0231 | −5.9924 | −5.9971 |
\(A_{1}\) | −18.244 | −19.098 | −19.027 |
MSE | 3.3e−07 | 7.1e−08 | 1.1e−07 |
Carbon 2 | |||
\(A_{0}\) | 0.9351 | 0.9999 | 0.9759 |
\(A_{1}\) | −2.9057 | −5.1337 | −4.1839 |
MSE | 1.8e−08 | 1.4e−08 | 2.9e−08 |
Carbon 3 | |||
\(A_{0}\) | −0.1980 | −0.1265 | −0.2033 |
\(A_{1}\) | −13.489 | −15.533 | −13.714 |
MSE | 2.8e−07 | 2.5e−08 | 8.5e−08 |
Carbon 4 | |||
\(A_{0}\) | 0.1357 | 0.1678 | 0.1086 |
\(A_{1}\) | −4.4469 | −5.4186 | −3.9314 |
MSE | 2.0e−08 | 3.1e−09 | 3.0e−08 |
Computing ∆y
The slope (S) in AGNRs in Y direction
m = 0 | m = 1 | m = 2 | |
---|---|---|---|
With H | |||
\(A_{0}\) | 0.0165 | 0.0140 | 0.0034 |
\(A_{1}\) | 0.3108 | 0.0344 | 0.6737 |
MSE | 1.0e−07 | 3.6e−08 | 7.9e−08 |
Without H | |||
\(A_{0}\) | 0.0296 | 0.0667 | 0.0073 |
\(A_{1}\) | −2.2085 | −3.3721 | −2.1073 |
MSE | 2.9e−08 | 2.4e−08 | 8.5e−08 |
The offset (O) in AGNRs in Y direction
m = 0 | m = 1 | m = 2 | |
---|---|---|---|
With H | |||
\(A_{0}\) | −0.2819 | −0.1644 | −0.2724 |
\(A_{1}\) | −0.8469 | 0.0477 | −2.0991 |
MSE | 8.0e−06 | 3.5e−06 | 9.4e−06 |
Without H | |||
\(A_{0}\) | 0.5908 | 0.4035 | 0.7838 |
\(A_{1}\) | 4.5999 | 11.5680 | 3.5510 |
MSE | 3.2e−06 | 6.6e−06 | 6.7e−06 |
∆Y of four first carbons in AGNRs without H
m = 0 | m = 1 | m = 2 | |
---|---|---|---|
Carbon 1 | |||
\(A_{0}\) | 7.3245 | 7.2128 | 7.4956 |
\(A_{1}\) | 7.4807 | 13.1602 | 7.0796 |
MSE | 1.8e−06 | 5.1e−06 | 3.2e−06 |
Carbon 2 | |||
\(A_{0}\) | 0.6364 | 0.6042 | 0.7627 |
\(A_{1}\) | −0.5570 | 2.3996 | −0.9311 |
MSE | 1.3e−06 | 4.1e−06 | 2.2e−06 |
Carbon 3 | |||
\(A_{0}\) | 0.6678 | 0.6608 | 0.8137 |
\(A_{1}\) | −1.5009 | 0.0079 | −3.0802 |
MSE | 1.2e−06 | 3.9e−06 | 2.4e−06 |
Carbon 4 | |||
\(A_{0}\) | 0.8059 | 0.7015 | 0.9006 |
\(A_{1}\) | −5.3071 | −0.9615 | −5.4591 |
MSE | 1.1e−06 | 3.2e−06 | 2.1e−06 |
∆Y of four first carbons in AGNRs with H
\(A_{0}\) | \(A_{1}\) | MSE1 | |
---|---|---|---|
Carbon 1 | 1.3966 | 2.0813 | 1.0e−06 |
Carbon 2 | −0.3637 | 1.3296 | 6.6e−07 |
Carbon 3 | −0.4808 | 0.8744 | 6.3e−07 |
Carbon 4 | −0.1334 | 0.5408 | 8.0e−07 |
Difference between C–C bond length in two directions
Conclusions
We proposed a general hyperbolic formula, and we proved that the way the AGNRs bandgap energy is relatable, there are lots of other parameters, which can be shown by this formula. In this way, we showed that the atomic positions of all carbon atoms in AGNRs can be calculated by the formula, and so we can perform the analysis without the need to its structural relaxations. Using obtained equations, we did the structural relaxations for AGNRs with different dimer numbers again, and could reduce the analysis times, 4–7 times. We also showed that the edge effect would not be only for the first edge atom, and this effect can be observed in the couple of atoms close the edges.
We also showed that the C–C bond length is slightly different for each various directions. This difference is about 0.017 Å for AGNRs with small dimer numbers and is reduced to zero for AGNRs with large width (graphene).
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