**2. Structural and energetic properties**

To give insights about the structural stability of nanostructures, firstly, it is suggested to evaluate if the proposal unit cell may array forming a stable crystalline state. Usually, a structural analysis is carried out computing the cohesive energy per atoms o per unit cell. The cohesive energy (*E*C) is the energy required to disassemble a molecular system into its constituent parts. From a physical point of view, a bound (stable) system has a positive value of *E*C, which represents the energy gained during the formation of the bound state. To calculate the *E*C of ACNRs, it is necessary to obtain the optimized energy of the unit cell being aware of the well converged energy with respect to the *k*-points and the cutoff energy for the planewave basis set, evaluating the impact of the exchange-correlation functional used and its ability to accurately describe both the atom and bulk phase.

Although *E*C is a reference to know the stability of bulk materials, it differs from a nanoparticle. [28–30] At nanoscale, size effects on the cohesive energy of nanoparticles has been demonstrated, which decreases with decreasing the particle size. [31] However, slight differences of *E*C are found when nuclei radii of constituent are similar, which do difficult to analyze or find a trend, e. g., the effect of the relative position of dopants along the NRs. For example, **Table 2** shows the calculated values of *E*C of armchair carbon nanoribbons (ACNR) doped with boron atoms in randomly (ACNR-R) and forming one B nanoisland (ACNR-I) arrangements compared with those pristine ACNRS of size 16x2, 20x2, 16x2 and 20x4 respectively. [32] The arrangement of the nanoisland (ACNR-I) explained in this section is shown in part (a) of **Figure 1** numbering from 1 to 6 the C atoms are substituted for impurities. Note that, B doping slightly reduces the cohesive energy of ACNRs compared with the pristine ones with similar *E*C values mainly found in the largest B-ACNRs. However, at lower doping concentrations, i. e. in the case of the largest ACNR (20x4) very close values of *E*C are obtained which makes difficult to observe a trend.

Because of these CNRs has 3 different chemical species, *E*C does not provide a suitable way to evaluate the relative stability. **Table 2** also shows in brackets the calculated values of the Gibbs free formation energy to take into account the chemical composition of ACNRs. The relative thermodynamic stability that is considered to evaluate the relative stability of multicomponent systems. This approach has been used in binary and tertiary phase thermodynamics and nanostructures other than NRs. [25, 33, 34] it can be calculated by using the following expression:

$$\delta \mathcal{G} = E\left(\boldsymbol{\omega}\right) + \sum\_{i=1}^{n} \mathbf{x}\_i \boldsymbol{\mu}\_i \tag{1}$$

**309**

*Calculation of the Electronic Properties and Reactivity of Nanoribbons*

components (H, N, B, C) which satisfies ∑*xi* <sup>=</sup> <sup>1</sup> , where <sup>=</sup> *<sup>i</sup>*

respect to the conforming constituents, whereas negative values of

where *E x*( ) is the binding energy per atom of the B-ACNR for the example

*i*

*x*

µ

*T n*

*<sup>i</sup>* ) can be approximate as the

δ

, being *ni* the

*G* refer to

*Na a* ~ *w*<sup>−</sup> ∆ . [19, 23, 37]

*n*

*G* represent a metastable structure with

shown in **Table 1**, *xi* corresponds to the molar fraction of the conformant

number of atoms of specie *i* in the unit cell and *nT* the total number of atoms

stable structures in accordance with their constituents. As we can observe in

energetically the pristine carbon nanoribbons more than the randomly cases.

of quantum confinement gaps, which can be characterized by <sup>1</sup>

binding energy per atom of the singlet ground state of the H2, the triplet ground state of the B2 molecule and the cohesive energy per atom of the graphene sheet

δ

*G* suggests that the arrangement of B-nanoisland leads to stabilize

The electronic properties of nanoribbons can be inferred from the band structure and total and local density of states (DOS and LDOS) respectively. For the case of NRs, these calculations are relatively simple because they are computed sampling the Brillouin zone only in one direction, i. e., the grown direction from 0 to gamma point. We recommend to use a denser grid than the case of the total energy calculations, including a Gaussian smearing (of width 0.01 eV) to improve the convergence of the integrals of the energy levels for the band structure calculations, for DOS calculations, to use the tetrahedron method with Blöchl corrections. [35, 36] Pristine CNRs with hydrogen passivated armchair edges passivated are direct bandgap semiconducting, which decreases as their width increases. The edges of ACNRs play an important role on their electronic properties and reactivity because

In order to evaluate the electronic nature of nanoribbons, firstly, spin-polarized and non-spin polarized solutions of the Kohn-Sham equations must be taken into account to evaluate possible magnetic configurations, as found in zigzag carbon nanoribbons, [38] that implies the magnetic state is the most stable. For armchair ribbons, the non-magnetic state is always the most stable [22] even for ACNRs doped with boron atoms, [32] so that, for simplicity, we consider the armchair

The electronic behavior of ACNRs can be tuned for the influence of substitutional dopants. To illustrate this fact, we think about a unit cell containing one pristine CNR with even number of electrons of valence. If we replace only one carbon atom (with 4 valence electrons) for B (3 valence electrons) or N (5 valence electrons) such change gives one unit cell with odd number of valence electrons, in such cases is necessary to search for spin polarized solutions of the Kohn Sham equation, i. e, to evaluate if there are significant differences with respect to the non-

**Figure 2** presents the band structure, total density of states and local density of states of dopants (shown in line red) of the 12x2 ACNR pristine, B-doped and N-doped substituting two dopants on positions 3 and 4 using the numbering shown in **Figure 1**. Note that, the pristine ACNR is a semiconducting in agreement with the literature [22] and the positive doping caused for the B moves the Fermi level (EF) to lower energies meanwhile the negative doping related with the N moves the EF

*DOI: http://dx.doi.org/10.5772/intechopen.94541*

respectively. Note that positive values of

**3. Electronic properties of nanoribbons**

**Table 2**,

δ

topology as a case of study.

spin polarized solution.

conforming the unit cell. The chemical potential (


#### **Table 2.**

*Cohesive per atom (Gibbs free) energy in eV of pristine and B-doped ACNRs of randomly (ANCR-R) and forming a B-nanoisland (ACNR-I) [32].*

*Nanofibers - Synthesis, Properties and Applications*

**2. Structural and energetic properties**

To give insights about the structural stability of nanostructures, firstly, it is suggested to evaluate if the proposal unit cell may array forming a stable crystalline state. Usually, a structural analysis is carried out computing the cohesive energy per atoms o per unit cell. The cohesive energy (*E*C) is the energy required to disassemble a molecular system into its constituent parts. From a physical point of view, a bound (stable) system has a positive value of *E*C, which represents the energy gained during the formation of the bound state. To calculate the *E*C of ACNRs, it is necessary to obtain the optimized energy of the unit cell being aware of the well converged energy with respect to the *k*-points and the cutoff energy for the planewave basis set, evaluating the impact of the exchange-correlation functional used and its ability to accurately describe both the atom and bulk

Although *E*C is a reference to know the stability of bulk materials, it differs from a nanoparticle. [28–30] At nanoscale, size effects on the cohesive energy of nanoparticles has been demonstrated, which decreases with decreasing the particle size. [31] However, slight differences of *E*C are found when nuclei radii of constituent are similar, which do difficult to analyze or find a trend, e. g., the effect of the relative position of dopants along the NRs. For example, **Table 2** shows the calculated values of *E*C of armchair carbon nanoribbons (ACNR) doped with boron atoms in randomly (ACNR-R) and forming one B nanoisland (ACNR-I) arrangements compared with those pristine ACNRS of size 16x2, 20x2, 16x2 and 20x4 respectively. [32] The arrangement of the nanoisland (ACNR-I) explained in this section is shown in part (a) of **Figure 1** numbering from 1 to 6 the C atoms are substituted for impurities. Note that, B doping slightly reduces the cohesive energy of ACNRs compared with the pristine ones with similar *E*C values mainly found in the largest B-ACNRs. However, at lower doping concentrations, i. e. in the case of the largest ACNR (20x4) very close values of *E*C are obtained which makes difficult

Because of these CNRs has 3 different chemical species, *E*C does not provide a suitable way to evaluate the relative stability. **Table 2** also shows in brackets the calculated values of the Gibbs free formation energy to take into account the chemical composition of ACNRs. The relative thermodynamic stability that is considered to evaluate the relative stability of multicomponent systems. This approach has been used in binary and tertiary phase thermodynamics and nanostructures other than NRs. [25, 33, 34] it can be calculated by using the

( )

**MxN Pristine ACNR -R ACNR-I** 16x2 7.224 (0.003) 6.992 (−0.272) 7.003 (−0.291) 20x2 7.338 (0.002) 7.143 (−0.224) 7.158 (−0.239) 16x4 7.225 (0.003) 7.112 (−0.142) 7.116 (−0.147) 20x4 7.338 (0.002) 7.249 (−0.119) 7.250 (−0.130)

*Cohesive per atom (Gibbs free) energy in eV of pristine and B-doped ACNRs of randomly (ANCR-R) and* 

δ

= +∑

=1

*i*

*i i*

*G Ex x* (1)

 µ

*n*

**308**

**Table 2.**

phase.

to observe a trend.

following expression:

*forming a B-nanoisland (ACNR-I) [32].*

where *E x*( ) is the binding energy per atom of the B-ACNR for the example shown in **Table 1**, *xi* corresponds to the molar fraction of the conformant components (H, N, B, C) which satisfies ∑*xi* <sup>=</sup> <sup>1</sup> , where <sup>=</sup> *<sup>i</sup> i T n x n* , being *ni* the number of atoms of specie *i* in the unit cell and *nT* the total number of atoms conforming the unit cell. The chemical potential ( µ*<sup>i</sup>* ) can be approximate as the binding energy per atom of the singlet ground state of the H2, the triplet ground state of the B2 molecule and the cohesive energy per atom of the graphene sheet respectively. Note that positive values of δ*G* represent a metastable structure with respect to the conforming constituents, whereas negative values of δ*G* refer to stable structures in accordance with their constituents. As we can observe in **Table 2**, δ*G* suggests that the arrangement of B-nanoisland leads to stabilize energetically the pristine carbon nanoribbons more than the randomly cases.
