**3. Electronic properties of nanoribbons**

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 of quantum confinement gaps, which can be characterized by <sup>1</sup> *Na a* ~ *w*<sup>−</sup> ∆ . [19, 23, 37]

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 topology as a case of study.

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 nonspin polarized solution.

**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

**Figure 2.** *Band structure and DOS of (a) pristine, (b) B-doped and (c) N-doped ACNRs of size 12x2.*

to higher energies with respect to the pristine one. In both cases, the closest energy bands to EF are partially unoccupied and occupied respectively giving rise to metallic behavior in both cases.

### **4. Reactivity of nanoribbons**

In order to explore the reactivity of 1D nanomaterial's, such as nanoribbons, it is mandatory to use appropriate reactivity descriptor. However, there is not a well stablish criteria to accomplish this task without prior knowing of an adsorption mechanism or experimental evidence, particularly for doped nanoribbons.

This is why, it is suggested the employment of two reactivity descriptors that are able to cover covalent and non-covalent interactions. The first one is the electrostatic potential, defined as:

$$V(\mathbf{r}) = \sum\_{A} \frac{Z\_A}{|\mathbf{R}\_A - \mathbf{r}|} - \int \frac{\rho(\mathbf{r}') \, d\mathbf{r}'}{|\mathbf{r}' - \mathbf{r}|} \tag{2}$$

**311**

*Calculation of the Electronic Properties and Reactivity of Nanoribbons*

functional. **Figure 3** was built in the software VESTA [40] plotting the charge file and then, adding the cube file containing the local potential. The color scheme used in **Figure 3** represents in blue, regions where a positive charge may repel each other, unlike in red, it represents regions where a positive charge, ion or

*Molecular electrostatic potential of the nanoribbons (a) pristine (b) B-doped and (c) N-doped of size 12x2.*

We can observe from **Figure 3a** that, hydrogens are weak positive meanwhile the delocalized charge is distributed along the carbon atoms, particularly found in the edged carbons, which is in agreement with the DOS of pristine ACNRs. The lacking of π electrons of the boron atoms is particularly observed in **Figure 3b**, which influences over their neighbor carbon atoms finding localized charge in such region. On the other hand, the N doping influences over the edges with more negative

The second reactivity descriptor is the Fukui or frontier functions (Ffs), helpful chemical reactivity descriptors for process controlled by electron transfer. Ffs were introduced for the first time by Parr and Yang, [41], which is convinced from the area of research so-called conceptual Density Functional Theory given by Geerlings in a comprehensive way. [42] Fukui functions play an important role linking the Molecular Orbital Theory with the HSAB principle, [43] they are defined as the change of the electronic density with respect to number of electrons (*N*), consider-

( ) ( )

Due to the discontinuity of the above equation with respect to *N*, in a finite

, ( ) ,1 , ( ) ( ) <sup>+</sup> *vN vN* = − <sup>+</sup> *v N fr r r* ρ

*f r <sup>N</sup>* ρ

 <sup>∂</sup> <sup>=</sup> <sup>∂</sup> **r**

( )

(3)

(4)

*v r*

 ρ

electrostatic potential than the pristine carbon nanoribbon.

ing the nuclei position fixed, i.e. constant external potential *v*( *r* ):

difference approximation three functions can be defined as:

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

chemical group can interact.

**Figure 3.**

Where *ZA* and *RA* are the atomic number of nucleus A and its position respectively, *R r <sup>A</sup>* − is its distance from the point *r* and ρ (*r*′) is the electron density in each volume element. This descriptor provides the response of electron density when a positive unit charge is approaching, which is commonly plotted in a color scheme. Because of the electrostatic potential *V* (**r**) , is a local property, it has one

value for each **r** point in the space surrounding a molecule or unit cell, so that, depending the nature of the ions (for instance positive or negative nature), the electrostatic potential will depend on the radial distance **r** from the nucleus. Commonly is followed that the contour of the electrostatic potential is plotted on the isovalue of the molecular electron density, for example, see the Bader's suggestion. [39] Be aware that, the chosen outer electron density contour depends on the Van der Waals radii of involving ions, which should reflect the molecular properties we want to observe, e. g., lone pairs, strained bonds, conjugated π systems, etc.

To illustrate, **Figure 3** shows the electrostatic potential of pristine, B-doped and N-doped carbon nanoribbons of size 12x2 plotted on the electron density surface of value 0.001 au, computed by using the generalized gradient approximation (GGA) in the form proposed by Perdew et al. for exchange-correlation

*Calculation of the Electronic Properties and Reactivity of Nanoribbons DOI: http://dx.doi.org/10.5772/intechopen.94541*

*Nanofibers - Synthesis, Properties and Applications*

lic behavior in both cases.

**Figure 2.**

static potential, defined as:

**4. Reactivity of nanoribbons**

to higher energies with respect to the pristine one. In both cases, the closest energy bands to EF are partially unoccupied and occupied respectively giving rise to metal-

*Band structure and DOS of (a) pristine, (b) B-doped and (c) N-doped ACNRs of size 12x2.*

In order to explore the reactivity of 1D nanomaterial's, such as nanoribbons, it is mandatory to use appropriate reactivity descriptor. However, there is not a well stablish criteria to accomplish this task without prior knowing of an adsorption mechanism or experimental evidence, particularly for doped nanoribbons.

This is why, it is suggested the employment of two reactivity descriptors that are able to cover covalent and non-covalent interactions. The first one is the electro-

> ( ) ( ´ ´ ) ´ = − − − <sup>∑</sup> <sup>∫</sup> **r r**

Where *ZA* and *RA* are the atomic number of nucleus A and its position respec-

*A*

*A*

each volume element. This descriptor provides the response of electron density when a positive unit charge is approaching, which is commonly plotted in a color scheme. Because of the electrostatic potential *V* (**r**) , is a local property, it has one value for each **r** point in the space surrounding a molecule or unit cell, so that, depending the nature of the ions (for instance positive or negative nature), the electrostatic potential will depend on the radial distance **r** from the nucleus. Commonly is followed that the contour of the electrostatic potential is plotted on the isovalue of the molecular electron density, for example, see the Bader's suggestion. [39] Be aware that, the chosen outer electron density contour depends on the Van der Waals radii of involving ions, which should reflect the molecular properties we want to observe, e. g., lone pairs, strained bonds, conjugated π systems, etc. To illustrate, **Figure 3** shows the electrostatic potential of pristine, B-doped and N-doped carbon nanoribbons of size 12x2 plotted on the electron density surface of value 0.001 au, computed by using the generalized gradient approximation (GGA) in the form proposed by Perdew et al. for exchange-correlation

**R r rr**

*Z d*

ρ

ρ

(*r*′) is the electron density in

(2)

**r**

*V*

tively, *R r <sup>A</sup>* − is its distance from the point *r* and

*A*

**310**

**Figure 3.** *Molecular electrostatic potential of the nanoribbons (a) pristine (b) B-doped and (c) N-doped of size 12x2.*

functional. **Figure 3** was built in the software VESTA [40] plotting the charge file and then, adding the cube file containing the local potential. The color scheme used in **Figure 3** represents in blue, regions where a positive charge may repel each other, unlike in red, it represents regions where a positive charge, ion or chemical group can interact.

We can observe from **Figure 3a** that, hydrogens are weak positive meanwhile the delocalized charge is distributed along the carbon atoms, particularly found in the edged carbons, which is in agreement with the DOS of pristine ACNRs. The lacking of π electrons of the boron atoms is particularly observed in **Figure 3b**, which influences over their neighbor carbon atoms finding localized charge in such region. On the other hand, the N doping influences over the edges with more negative electrostatic potential than the pristine carbon nanoribbon.

The second reactivity descriptor is the Fukui or frontier functions (Ffs), helpful chemical reactivity descriptors for process controlled by electron transfer. Ffs were introduced for the first time by Parr and Yang, [41], which is convinced from the area of research so-called conceptual Density Functional Theory given by Geerlings in a comprehensive way. [42] Fukui functions play an important role linking the Molecular Orbital Theory with the HSAB principle, [43] they are defined as the change of the electronic density with respect to number of electrons (*N*), considering the nuclei position fixed, i.e. constant external potential *v*( *r* ):

$$f(r) = \left(\frac{\partial \rho(\mathbf{r})}{\partial \mathbf{N}}\right)\_{v(r)}\tag{3}$$

Due to the discontinuity of the above equation with respect to *N*, in a finite difference approximation three functions can be defined as:

$$f\_{\boldsymbol{\nu},N}^{\*}\left(\boldsymbol{r}\right) = \boldsymbol{\rho}\_{\boldsymbol{\nu},N\*1}\left(\boldsymbol{r}\right) - \boldsymbol{\rho}\_{\boldsymbol{\nu},N}\left(\boldsymbol{r}\right) \tag{4}$$

$$f\_{\boldsymbol{\nu},N}^{-}\left(\boldsymbol{r}\right) = \boldsymbol{\rho}\_{\boldsymbol{\nu},N}\left(\boldsymbol{r}\right) - \boldsymbol{\rho}\_{\boldsymbol{\nu},N-1}\left(\boldsymbol{r}\right) \tag{5}$$

$$f\_{\boldsymbol{\nu},N}^{\boldsymbol{0}}\left(\boldsymbol{r}\right) = \mathbf{0}.\mathsf{5}\left(\boldsymbol{\rho}\_{\boldsymbol{\nu},N+1}\left(\boldsymbol{r}\right) - \boldsymbol{\rho}\_{\boldsymbol{\nu},N-1}\left(\boldsymbol{r}\right)\right) \tag{6}$$

Where ρ*v N*, 1<sup>+</sup> (*r* ) , ρ*v N*, (*r* ) , and ρ*v N*, 1<sup>−</sup> (*r* ) , are the electronic densities of the system with N + 1, N, and N–1 electrons, respectively, all with the ground state geometry of the N electron system. Expressions 4–6 concern the Fukui function for: nucleophilic attack, the chemical change where a molecule gains an electron; electrophilic attack, when a molecule loses charge; and for free radical attacks. [44]

Although, the finite difference approximation to the Fukui functions works for a specific set of configurations whilst for others is worthless to implement (i.e., full configuration interaction), [45] in most cases they are considered a reliable descriptor to indicate how the electron (incoming or outcoming) is redistributed in regions of the molecules. [46] Chemical reactivity is based on the assumption that, when molecules A and B interact in order to form a product AB, occurs a molecular densities-perturbation. [47] As the electronic density contains all sort of information, the chemical reactivity has to be reflected within its sensitivity to infinitesimal electron changes at constant external potential *v r*( ) . Calculation of the frontier orbitals (HOMO or LUMO) are unambiguously defined. Within the frozen orbital approximation, [48] Ffs can be written in terms of the Kohn-Sham orbitals as follows:

$$f^\*\left(r\right) = \left|\phi\_{v,N}^{\text{LUMO}}\left(r\right)\right|^2 + \sum\_{i=1}^N \left(\frac{\hat{\sigma}\left|\phi\_i\left(r\right)\right|}{\hat{\sigma}N}\right) \approx \rho\_{v,N}^{\text{LUMO}}\left(r\right) \tag{7}$$

$$\left|f^{-}\left(r\right) = \left|\phi\_{v,N}^{HOMO}\left(r\right)\right|^2 + \sum\_{i=1}^{N} \left|\frac{\partial \left|\phi\_i\left(r\right)\right|}{\partial N}\right| \approx \rho\_{v,N}^{HOMO}\left(r\right) \tag{8}$$

In molecules, the relaxation term is usually very small for the discrete nature of Kohn Sham orbitals. So that, if Eqs. 7 and 8 neglect the second-order variations in the electron density, Ffs may approximate to the electron densities of its frontier orbitals.

On the other hand, referring to periodic systems, it is difficult to identify one frontier state because of the continuous character of the Blöch states, which makes difficult to compute the Fukui functions in DFT of the solid state. Although there is scarcely literature on this topic, a very useful reference for the numerical calculation of the condensed Ffs in periodic boundary conditions within the DFT applied to oxide bulk and surfaces is found here. [49]

One qualitative way to obtain Ffs for delocalized periodic systems, such as, the carbon nanoribbons is to extract its electron density and evaluate it by using the Eq. (7) and (8) respectively. From the electronic structure of these nanomaterials we can observe that only one occupied electronic band contributes below and above the Fermi level.

**Figure 3** depicts the Ffs evaluated for electrophilic attacks respectively for B-doped and N-doped armchair carbon nanoribbons of size 12x2 with doping made on positions 3 and 4 using the numbering shown in **Figure 1**.

**313**

**Figure 4.**

contributing near the Fermi level.

*Calculation of the Electronic Properties and Reactivity of Nanoribbons*

We observe from **Figure 4** that the B atoms contributes to form regions where an electrophilic attack can occur on the doped nanoribbons, i. e. larges values of <sup>−</sup> *f* mean regions where the ACNR will lose charge to stabilize it in a chemical change. The electrostatic potential and the Fukui functions provide information on the local selectivity for donor-acceptor interactions. In here, the electrostatic potential describes the long-range non-covalent interactions. [50] The evaluation of the incoming charge distribution on nanoribbons states that "The Fukui function is strong while regions of a molecule are chemically softer than the regions where the Fukui function is weak. By invoking the hard and soft acid and bases (HSAB) principle [51] in a local sense, it is possible to establish the behavior of the different sites as function of hard or soft reagents (adsorbates)". [32, 52–54] **Figure 4** shows the Fukui functions for electrophilic attack, calculated by using Eq. (8), we observe the contribution of doping particularly on the neighboring carbon atoms. Indeed, from parts (b) and (c) of **Figure 2** is observed the electronic states of dopants

*Fukui functions for nucleophilic attack of (a) B-doped and (b) N-doped ACNRs of size 12x2.*

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

*Calculation of the Electronic Properties and Reactivity of Nanoribbons DOI: http://dx.doi.org/10.5772/intechopen.94541*

*Nanofibers - Synthesis, Properties and Applications*

Where

attacks. [44]

follows:

orbitals.

ρ

*v N*, 1<sup>+</sup> (*r* ) ,

ρ

+

−

oxide bulk and surfaces is found here. [49]

φ

φ

*v N*, (*r* ) , and

ρ

electrophilic attack, when a molecule loses charge; and for free radical

, ( ) , ,1 ( ) ( ) <sup>−</sup> *vN vN vN* = − <sup>−</sup> *fr r r* ρ

( ) ( ( ) ( )) <sup>0</sup> *v N*, = 0.5 *v N*,1 ,1 + − − *v N fr r r* ρ

system with N + 1, N, and N–1 electrons, respectively, all with the ground state geometry of the N electron system. Expressions 4–6 concern the Fukui function for: nucleophilic attack, the chemical change where a molecule gains an electron;

Although, the finite difference approximation to the Fukui functions works for a specific set of configurations whilst for others is worthless to implement (i.e., full configuration interaction), [45] in most cases they are considered a reliable descriptor to indicate how the electron (incoming or outcoming) is redistributed in regions of the molecules. [46] Chemical reactivity is based on the assumption that, when molecules A and B interact in order to form a product AB, occurs a molecular densities-perturbation. [47] As the electronic density contains all sort of information, the chemical reactivity has to be reflected within its sensitivity to infinitesimal electron changes at constant external potential *v r*( ) . Calculation of the frontier orbitals (HOMO or LUMO) are unambiguously defined. Within the frozen orbital approximation, [48] Ffs can be written in terms of the Kohn-Sham orbitals as

> ( ) ( ) ( ) ( ) <sup>2</sup> , , 1

> ( ) ( ) ( ) ( ) <sup>2</sup> , , 1

In molecules, the relaxation term is usually very small for the discrete nature of Kohn Sham orbitals. So that, if Eqs. 7 and 8 neglect the second-order variations in the electron density, Ffs may approximate to the electron densities of its frontier

On the other hand, referring to periodic systems, it is difficult to identify one frontier state because of the continuous character of the Blöch states, which makes difficult to compute the Fukui functions in DFT of the solid state. Although there is scarcely literature on this topic, a very useful reference for the numerical calculation of the condensed Ffs in periodic boundary conditions within the DFT applied to

One qualitative way to obtain Ffs for delocalized periodic systems, such as, the carbon nanoribbons is to extract its electron density and evaluate it by using the Eq. (7) and (8) respectively. From the electronic structure of these nanomaterials we can observe that only one occupied electronic band contributes below and above

**Figure 3** depicts the Ffs evaluated for electrophilic attacks respectively for B-doped and N-doped armchair carbon nanoribbons of size 12x2 with doping made

on positions 3 and 4 using the numbering shown in **Figure 1**.

∂

*N* φ

∂

*N* φ

*r*

 <sup>∂</sup> <sup>∑</sup> *<sup>N</sup> HOMO <sup>i</sup> HOMO v N v N i*

*r*

 ρ

 ρ

(7)

(8)

 <sup>∂</sup> <sup>∑</sup> *<sup>N</sup> LUMO <sup>i</sup> LUMO v N v N i*

=

=

*fr r r*

=+ ≈

*fr r r*

=+ ≈

 ρ

> ρ

(5)

(6)

*v N*, 1<sup>−</sup> (*r* ) , are the electronic densities of the

**312**

the Fermi level.

**Figure 4.** *Fukui functions for nucleophilic attack of (a) B-doped and (b) N-doped ACNRs of size 12x2.*

We observe from **Figure 4** that the B atoms contributes to form regions where an electrophilic attack can occur on the doped nanoribbons, i. e. larges values of <sup>−</sup> *f* mean regions where the ACNR will lose charge to stabilize it in a chemical change.

The electrostatic potential and the Fukui functions provide information on the local selectivity for donor-acceptor interactions. In here, the electrostatic potential describes the long-range non-covalent interactions. [50] The evaluation of the incoming charge distribution on nanoribbons states that "The Fukui function is strong while regions of a molecule are chemically softer than the regions where the Fukui function is weak. By invoking the hard and soft acid and bases (HSAB) principle [51] in a local sense, it is possible to establish the behavior of the different sites as function of hard or soft reagents (adsorbates)". [32, 52–54] **Figure 4** shows the Fukui functions for electrophilic attack, calculated by using Eq. (8), we observe the contribution of doping particularly on the neighboring carbon atoms. Indeed, from parts (b) and (c) of **Figure 2** is observed the electronic states of dopants contributing near the Fermi level.
