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Mathematical Fundamentals of Nanotechnology

R. Marchiori Interdisciplinary Department of Science and Technology, Federal University of Rondônia, Ariquemes, Rondônia, Brazil

Abstract

This chapter succinctly presents the mathematical foundation that is necessary to understand the mechanical, chemical, thermal, and electrical properties that characterize materials at the nanometric scale. This approach is necessarily based on the use of quantum theory, which describes the properties of materials at the atomic and molecular scales. This subject is further elaborated at a basic theoretical level to understand the need for the use of quantum theory to describe material properties at the nanometric scale. The final portion of the chapter is dedicated to several examples of particularly relevant nanostructured materials with the intention of showing the mathematics that are generally used to deduce some of the properties of these materials.

Keywords

nanometric scale

quantum mechanics

wave-material interaction

crystal lattice

electronic structure

energy levels

molecular orbitals

tight binding simulation

carbon nanotubes

graphene

8.1. Introduction

Classical mechanics, with its deterministic description of the behavior of matter, does not provide the correct mathematical structure to describe the physics and molecular interactions of matter at the atomic level. This representation is an average of a sufficiently large number of molecular interactions that is valid at the macroscopic scale. At the microscopic level, the laws of physics that describe the interactions between atoms, molecules, or nanoparticles can no longer be described by the deterministic laws of classical mechanics because they require a more appropriate description, which is offered by the mathematical formalism of quantum mechanics. This formalism modifies the deterministic perspective from the classical description to a probabilistic perspective of the behavior of matter. In the mathematical construction of quantum mechanics, time and space are interconnected, which is different from classical mechanics, in which they are independent. In the space-time universe of quantum mechanics, this interdependence between the spatial and temporal dimensions has drastic consequences in the interpretation of reality at the atomic level [1,2]. The laws of quantum physics deviate from those that we are accustomed to observing, which thereby conflicts with intuitive comprehension and provides an extremely unconventional description of reality [3].

Several theories consider that quantum mechanics is also valid at a macroscopic scale, but the direct interaction of the observer with the object, which occurs at this scale, causes the “collapse” of the possible energetic configurations of the object into a specifically determined configuration, which represents the highest probability configuration. Thus, any object, such as the Moon, is visible and exists where we see it because we are looking at it. If that was not the case, the speculations of some quantum physicists would lead to the determination that it is not there and that it occupies all possible energetic configurations with some probability; that is, any position within space-time in the universe!

The essential characteristic in the advent of nanotechnology around 1985 was the development of knowledge and technology that allowed us to manipulate and analyze physical phenomena and material properties at the atomic scale, which made it possible to access knowledge about material properties based on interactions between particles at the nanometric scale. At this scale, which is the order of magnitude of interatomic distances, the energy levels that are accessible to a physical system are no longer continuous as in classical mechanics, in which energy can assume any value in a continuous manner. At the nanometric scale, wave–particle duality is manifest, which causes the discretization of energy. In other words, energy can acquire only specific, discrete values to respect the boundary conditions that emerge from the wave nature of matter; these conditions exclude certain configurations of the system and, consequently, certain relative energy values because the system is defined by atomic or molecular interactions. Thus, the mathematical formalism of quantum mechanics is needed to calculate the energy of the system at this scale.

At the nanometric scale, the classical formalism becomes an approximation that loses validity. To handle nanomaterials such as carbon nanotubes or graphene, which are currently the nanostructured materials with the greatest potential for use, quantum mechanics provides the mathematical background that describes the energy levels that are accessible to these materials.

When a material has properties due to its nanometric-scale structure and morphology, quantum theory is the only theory that can appropriately describe these properties, which indicates that the wave behavior of elementary particles becomes important at this scale. This behavior is described directly by the de Broglie equation, λ = h/p, which shows the dual wave–particle character of matter by relating the momentum p, which is a classical property that is directly proportional to the mass and velocity of the particle, to the wave property of matter, which is given by the corresponding wavelength λ, and Planck’s constant h is the proportionality factor. The wavelength λ only assumes detectable values for extremely small masses, such as in the case of elementary particles and nanostructured materials.

Boundary conditions can be used to maintain a physical system in equilibrium at the nanometric scale; these conditions are related to the characteristics of the material and determine its electrical, optical, and other properties.

Therefore, quantum mechanics must be used in nanotechnology and can be summarized in the following essential points:

• The nanometric scale, which is on the order of the size of an atom, involves the wave behavior of matter and requires the use of a wave function in the Schrödinger equation to describe the movement of a particle at this scale.

• The interaction of particles with matter at the nanoscale, such as the movement of an electron in electrical conduction by a nanostructured material, is limited by the boundary conditions. These conditions limit particle movement in the crystal lattice of the material, which forces the electrons into trajectories that are defined by the periodic interaction potential of the Bravais lattice of the material. This determines the discretization of the energy, whose possible levels are calculated by the mathematical formalism of quantum mechanics.

8.2. Classical Mechanics

8.2.1. The Classical Formalism at the Nanometric Scale

Particle dynamics can be represented from the classical formalism that describes the dynamics of a physical system. However, the particles of microscopic systems follow wave movement laws; that is, at the nanometric scale, elementary particles behave like waves, and their description requires the introduction of a wave function ψ(r,t) to describe their movement.

In general, the function ψ(q) is a complex function, with q = (q₁, q₂, … q_n) being the generalized coordinates that represent the “state” of a dynamic system with n degrees of freedom. Due to the objectives of this chapter, a more extensive formalism of a quantum system will not be addressed. This chapter will only cover the aspects that are necessary to understand the relationship between quantum mechanics and nanotechnology.

In general, the evolution of a system, such as the simplest case of a particle, must be consistent with the mechanics equation [4,5]:

$E = p^{2} / 2 m + V$ $E = p^{2} / 2 m + V$

(8.1)

which represents the energy E of a particle of mass m and momentum p as a function of the kinetic energy p²/2m and the potential energy V. The quantum mechanics equation that corresponds to Eq. 8.1 and involves the wave function will also have to satisfy the postulates of Einstein (1905), de Broglie (1921), and de Jammer (1966), (1974) [4,6–8]:

$E = h ν = ℏ ω and p = ℏ k$ $E = h ν = ℏ ω and p = ℏ k$

(8.2)

and be valid for wave movement, which can be written in the form:

$ℏ ω = ℏ^{2} k^{2} / 2 m + V (x, t)$ $ℏ ω = ℏ^{2} k^{2} / 2 m + V (x, t)$

(8.3)

where ħ = h/2π.

A general mathematical procedure is available to handle the behavior of any microscopic system of particles: the Schrödinger theory of quantum mechanics [9–14].

8.3. Quantum Mechanics

The Schrödinger theory is based, among others phenomenological processes, on the laws of wave movement to express the space-time position of elementary particles in terms of probability, which the particles of any microscopic system obey; when this description is required, the system is called a quantum system.

A wave function Ψ(r,t) for the general three-dimensional case of propagation with time t in the direction r can be represented using a sine wave [10]:

$Ψ (r, t) = \sin 2 π (\frac{r}{λ} - ν \cdot t)$ $Ψ (r, t) = \sin 2 π (\frac{r}{λ} - ν \cdot t)$

(8.4)

where Ψ(r,t) is the wave function, λ is the wavelength, and ν is the oscillation frequency. To write the wave function more easily, the following variables are introduced: k = 2π/λ and ω = 2πν, where k is the angular wave number, and ω is the angular oscillation frequency. The wave function then becomes:

$Ψ (r, t) = s e n (k r - ω t)$ $Ψ (r, t) = s e n (k r - ω t)$

(8.5)

The Schrödinger equation must also satisfy the linearity in Ψ(r,t), which guarantees that wave functions can produce the constructive and destructive interferences that are characteristic of wave behavior. Remember that because the Schrödinger equation [9] is consistent with Eq. 8.1, it is not valid when it is applied to particles with relativistic velocities. In the case of a free particle, a potential function must satisfy the condition:

$V (r, t) = V_{0} = const.$ $V (r, t) = V_{0} = const.$

(8.6)

This guarantees that no force is applied to the particle because the force F is equal to:

$F = - \frac{\partial V (r, t)}{\partial x} = 0 .$ $F = - \frac{\partial V (r, t)}{\partial x} = 0 .$

(8.7)

These conditions and concepts form the basis of the description of particles in matter and, in particular, the behavior of elementary particles, such as electrons in the crystal lattice of a material.

8.3.1. The Energy of Quantum Systems—Stationary Conditions

To describe the properties of nanostructured materials, it is necessary to obtain the eigenvalues of the energy of the system [9–14]. The Hamiltonian formulation of the time-independent Schrödinger equation is used to find the eigenvalues for the energy of a quantum system, as is shown in the example with carbon nanotubes at the end of this chapter. The majority of atomic and molecular systems show time-independent conditions; this means that the potential energy does not depend explicitly on time, or V(r,t) ≡ V(r). Under these conditions, the wave function in Eq. 8.4 can be written by separating the variables:

$Ψ (r, t) = ψ (r) \cdot ϕ (t)$ $Ψ (r, t) = ψ (r) \cdot ϕ (t)$

(8.8)

Thus, the Hamiltonian formulation of the time-independent Schrödinger equation is [10]:

$\hat{H} ψ (r) \equiv H ({\hat{q}}^{0}, {\hat{p}}^{0}) ψ (x) = E ψ (r)$ $\hat{H} ψ (r) \equiv H ({\hat{q}}^{0}, {\hat{p}}^{0}) ψ (x) = E ψ (r)$

(8.9)

where

\hat{H}

$\hat{H}$

is the Hamiltonian operator that is used in the quantum formalism, and

{\hat{q}}^{0}

${\hat{q}}^{0}$

and

{\hat{p}}^{0}

${\hat{p}}^{0}$

are the sets of operators that express the position and velocity of the particle, respectively. Acceptable solutions for this equation exist only for specific energy values E, which are called the eigenvalues of the potential energy V(r) that is generated by the crystal lattice of the system. An eigenfunction ψ(r) for the energy corresponds to each eigenvalue; using the Bohr interpretation, these eigenfunctions are wave functions whose energy is the energy of the stationary states of the system. The quantization of energy [10,11,13] emerges naturally in the Schrödinger theory due to the wave nature of matter. Only certain discrete energy values, which are obtained from the time-independent Schrödinger equation, permit solutions for the wave function. Considering the discrete set of eigenvalues E_i of the system, the Schrödinger equation can be written as [10]:

$\hat{H} ψ_{i} (r) = E_{i} ψ_{i} (r)$ $\hat{H} ψ_{i} (r) = E_{i} ψ_{i} (r)$

(8.10)

This equation allows one to obtain the eigenvalues for the energy of the system and is used for stable molecular aggregates at the nanometric scale. The analysis of the solutions to this equation provides information about the mechanical, thermal, and electrical properties of these nanomolecules, as is shown in the following paragraphs.

8.3.2. Periodic Structure of a Crystal Lattice—The Bloch Theorem

The main nanomolecules that are studied and used in nanotechnology applications have periodic structures due to their crystal lattice; therefore, the interaction potential of the electrons in the molecule is of the type [15,16]:

$V (\vec{r}) = V (\vec{r} + \vec{R})$ $V (\vec{r}) = V (\vec{r} + \vec{R})$

(8.11)

where

\vec{R}

$\vec{R}$

is the vector that represents the interatomic distance in the molecule. The vector notation that is used here is necessary to describe the crystal lattice of these nanomolecules. The electrons of these molecules occupy discrete energy states that are defined through quantum mechanics by the presence of a periodic potential that is related to the crystal structure of the nanomolecule. To find the eigenvalues of the energy of the wave functions of the electrons, appropriate approximations are used to simplify the problem without significantly altering the description of the system. This allows these molecular systems to be studied using various mathematical modeling techniques, such as the “tight binding” (TB) approximation. When this approximation is applied to the method known as the “linear combination of atomic orbitals” (LCAO), the monoelectronic states in a crystal approximately maintain their atomic behavior because the interactions between the electronic states of the neighboring atoms are relatively small and thus only represent system perturbations. In this approximation, the Hamiltonian operator H is equal to [17]:

$H = H_{atom} + H_{crystal} ≅ H_{atom}$ $H = H_{atom} + H_{crystal} ≅ H_{atom}$

(8.12)

This operator results from the interaction of the atom with the electron (H_atom) and the effect due to the interaction with the crystal (H_crystal), which is considered to be a perturbation; thus, the eigenfunction

ψ (\vec{r})

$ψ (\vec{r})$

of the electron becomes the eigenfunction of H_atom with the eigenvalue E₀. The reason for this is that these valence electrons are found in deep potential wells that are generated by the crystal lattice of the molecule and do not have the properties of free electrons. In other words, the potential energy that acts on the electrons due to the interaction with other electrons is much smaller than the atomic energy potential in the crystal lattice (Bravais lattice). Therefore, the Hamiltonian operator assumes the following form [18]:

$\hat{H} \propto \sum_{i} {\hat{T}}_{i} + \sum_{R} \hat{V} (\vec{r} - \vec{R})$ $\hat{H} \propto \sum_{i} {\hat{T}}_{i} + \sum_{R} \hat{V} (\vec{r} - \vec{R})$

(8.13)

where

{\hat{T}}_{i}

${\hat{T}}_{i}$

is the kinetic energy operator for all of the electrons e_i, and

\hat{V} (\overset{⇀}{r} - \vec{R})

$\hat{V} (\overset{⇀}{r} - \vec{R})$

is the complete potential of the crystal lattice at the atomic site

\vec{R}

$\vec{R}$

. The wave function ψ(r) of the electron in the crystal lattice for a defined value of the wave vector

\vec{k}

$\vec{k}$

is:

$ψ_{k} (r) = \sum_{\overset{⇀}{R}} C_{k R} ϕ (\overset{⇀}{r} - \vec{R})$ $ψ_{k} (r) = \sum_{\overset{⇀}{R}} C_{k R} ϕ (\overset{⇀}{r} - \vec{R})$

(8.14)

in which the positions

\vec{R}

$\vec{R}$

of the sites that are occupied by the atoms determine the periodicity of the Bravais lattice. The wave vector

\vec{k}

$\vec{k}$

has the direction and orientation of the wave propagation and the magnitude /k/ = 1/λ. The wave function

ϕ (\overset{⇀}{r} - \vec{R})

$ϕ (\overset{⇀}{r} - \vec{R})$

describes the electron that is close to the atomic site in

\vec{R}

$\vec{R}$

. The sum is extended to all of the sites of the crystal lattice, so this is a LCAO. The coefficient C_kR is obtained from the Bloch theorem, which allows the expression of the eigenfunction of the electron in the periodic potential of the crystal lattice. The probability density of the wave function

ϕ (\overset{⇀}{r})

$ϕ (\overset{⇀}{r})$

of the electron in the periodic potential is also periodic and is equal to:

${|ϕ (\overset{⇀}{r} + \vec{R})|}^{2} = {|ϕ (\vec{r})|}^{2}$ ${|ϕ (\overset{⇀}{r} + \vec{R})|}^{2} = {|ϕ (\vec{r})|}^{2}$

(8.15)

Using the normalization of the electronic distribution in the unit cell, it is equal to:

$\int {|ϕ (\vec{r} + \vec{R})|}^{2} d V = \int {|ϕ (\vec{r})|}^{2} d V = 1$ $\int {|ϕ (\vec{r} + \vec{R})|}^{2} d V = \int {|ϕ (\vec{r})|}^{2} d V = 1$

(8.16)

Rewriting Eq. 8.16 yields the Bloch theorem [17]:

$ϕ_{k} (\overset{⇀}{r} + \vec{R}) = e^{i k R} ϕ_{k} (\overset{⇀}{r})$ $ϕ_{k} (\overset{⇀}{r} + \vec{R}) = e^{i k R} ϕ_{k} (\overset{⇀}{r})$

(8.17)

This equation represents the boundary conditions for solutions to the Schrödinger equation for a periodic potential. In this case, the eigenfunction of the electron in the Hamiltonian can be written in the form of a plane wave e^ikR with the periodicity of the Bravais lattice multiplied by the wave function

ϕ_{k} (\overset{⇀}{r})

$ϕ_{k} (\overset{⇀}{r})$

. In Eq. 8.14 for the wave function of the electron in the crystal lattice, the coefficients C_kR can be written as:

$C_{k R} = N^{- 1 / 2} e^{i k R}$ $C_{k R} = N^{- 1 / 2} e^{i k R}$

(8.18)

where N is the number of atoms in the crystal. This term is used because the crystal is finite, which determines the finite number of translation vectors of the Bravais lattice, which consists of N = N_xN_yN_z unit cells.

The normalization condition of the wave function in the expression of the density in the occupied space determines the application of the square root of the number of atoms N in the crystal (Eq. 8.18). Thus, Eq. 8.18 becomes:

$Ψ_{k} (\overset{⇀}{r}) = N^{- 1 / 2} \sum R e^{i k R} ϕ (\overset{⇀}{r} - \vec{R})$ $Ψ_{k} (\overset{⇀}{r}) = N^{- 1 / 2} \sum R e^{i k R} ϕ (\overset{⇀}{r} - \vec{R})$

(8.19)

8.3.3. Crystal Lattice and Reciprocal Lattice

The crystalline structure of a material can be accurately represented by considering only the electrons around it, which allows the formation of an indirect microscopic image of the material. Because electrons behave as waves, they can suffer reflections due to the periodicity of the crystal lattice of the material; these reflections are called Bragg reflections. The discussion of Bragg reflections leads to the notions of a “reciprocal lattice” or “reciprocal space” and “Brillouin zones.” The study of X-ray diffractions that are generated by the crystal lattices of materials permits the acquisition of crystal lattice parameters from the diffraction angles θ, which form a spectrum of lines that correspond to the constructive interference of the incident rays. The reflection of the incident X-rays that is defined by Bragg’s law is shown in Fig. 8.1 [19,20]:

Figure 8.1 Reflection of X-ray radiation that is incident on the atomic planes of a material that are separated by a distance d.

$2 d \sin θ = n λ$ $2 d \sin θ = n λ$

(8.20)

where d is the distance between two parallel atomic planes, n is a positive whole number, and λ is the wavelength of the incident electromagnetic wave. Bragg’s law provides the conditions for the occurrence of constructive interference of the waves that are scattered by point charges at points in the lattice, which permits the calculation of the lattice parameters, or the spacing d between the crystal planes as a function of the diffraction angles θ. If the path difference 2d sinθ between two parallel rays is a multiple of the wavelength of the incident rays, the interference will be constructive.

To determine the intensity of each scattering that is produced by a spatial distribution of electrons within each cell of the crystal lattice, it is necessary to consider the numerical density of the electrons n(r) because the intensity of the incident light that is scattered by a volume element is proportional to this density. The electronic density is naturally periodic for the translation operation that is defined by the vector

\vec{T}

$\vec{T}$

in the crystal lattice of the material. A diffraction figure of a crystal can be considered to be a representation of the reciprocal lattice of the crystal in contrast to the direct image of the real crystal lattice. Each crystal structure has two lattices: the crystal lattice and the reciprocal lattice. The concept of a reciprocal lattice is essential to describe nanomolecules, such as, carbon nanotubes or graphene. The definition of the vectors of the reciprocal lattice contributes to the calculation of the spectrum of eigenvalues of the energy of these nanostructures.

Using the concepts that will be discussed in the following paragraphs to define the “Brillouin zone,” it is possible to define the structure of the valence and conduction energy bands in nanostructures, the number of electrons that belong to the Brillouin zone and, thus, the occupation of these bands by electrons. These factors characterize the material as a conductor, semiconductor or insulator and describe the behavior of the material in the presence of external fields.

8.3.3.1. Reciprocal Lattice

\vec{a_{1}}

$\vec{a_{1}}$

\vec{a_{2}}

$\vec{a_{2}}$

and

\vec{a_{3}}

$\vec{a_{3}}$

are primitive vectors of the crystal lattice, the translation vectors of the reciprocal lattice b_1, b_2, and b₃ must satisfy the condition [17]:

$a_{i} \cdot b_{j} = 2 π δ_{i j}$ $a_{i} \cdot b_{j} = 2 π δ_{i j}$

(8.21)

where δ_ij is the Kronecker delta (δ = 1 for i = j, δ = 0 for i ≠ j). The unit cell of the reciprocal lattice, with all of its symmetry properties, is called the first Brillouin zone, and its points are the wave vectors

\vec{k}

$\vec{k}$

8.3.3.2. First Brillouin Zone

The Brillouin formulation is the most important diffraction condition for solid-state physics and is the only one that is used to describe the energy bands of electrons in addition to expressing the elementary excitations in crystals. The first Brillouin zone is the unit cell of the reciprocal lattice; this provides a geometric interpretation of the diffraction condition that is generated by the crystal lattice of the material [17]. The boundaries of the Brillouin zone of the linear lattice in one dimension are at k = ± π/a, where a is the parameter of the crystal lattice of the material.

8.3.4. Electronic Structure

As was already emphasized, a material’s electronic structure can be analyzed to study its properties and structure. This analysis of materials, primarily nanostructured materials, is an area of theoretical investigation in which quantum mechanics is applied to describe the spatial distribution and energy levels of electrons that make up the studied system. Through the analysis of the density of states (DOS) of the electrons, the electronic structure determines the electrical conduction properties of the material, which define whether it is an electrical insulator, semiconductor, or conductor.

8.3.4.1. Density of States—Discretization of Energy in Quantum Systems

The DOS is a property that is used extensively in quantum systems in condensed matter physics; it refers to the energy level of the electrons, photons, or phonons in a solid crystal. The electronic DOS quantifies how “packed” the electrons in a quantum mechanical system are in energy levels. The DOS can vary from zero for an energy level that is inaccessible to the electrons, with no space occupied by them, to a defined occupation value at a specific energy level that is accessible to the electrons of the material. There is a direct correlation between the concept of quantized energy levels that is described by quantum mechanics and the DOS. The DOS is an energetic configuration due to the wave property of matter. In some systems, the interatomic spacing, the crystal structure, and the atomic charge of the material only allow electrons of certain wavelengths to propagate in the system, which also limits the possible directions of wave propagations. Each wave occupies a different mode or state that can have the same wavelength or the same quantized energy levels. This determines the degeneracy of states with the same energy and the absence of states in other energies that are incompatible with the system in which no space is occupied by the system. In the case of electronic states, the DOS permits the calculation of the number of electrons for each energy level, and the diagram of these states defines the electrical conduction properties of a material. DOS is usually denoted with one of the symbols g, ρ, n, or N. If the DOS is the function g(E), one can write the expression ∆N = g(E)·∆E, which represents the number of states ∆N with energies between E and E + ∆E. If the fundamental state is completely occupied by electrons (full valence band) and the first excited state overlaps the valence band (an electron-hole pair in the conduction band), the material will be a conductor. However, if the bands are separated by an excitation energy E_g, which is called a bandgap, the material will be an insulator or a semiconductor as shown in Fig. 8.2 for graphene. Thus, whether a material is an insulator or semiconductor depends on the value of the energy gap E_g between the valence and conduction bands. In general, the bandgap E_g is on the order of eV (E_g ∼ eV) for insulating materials and E_g ∼ 10⁻¹ eV for semiconductors. Several semiconducting materials, such as silicon, have a bandgap in another range of values (E_g ∼ 1.1 eV), and materials such as ZnO, GaN, and AlN have bandgaps of a few eV in the energy band that corresponds to ultraviolet radiation.

Figure 8.2 Occupation density g(E) of the valence and conduction bands.
(A) Semiconductor characteristics. (B) Metallic characteristics.

The Fermi energy is defined as the maximum energy value in the valence band that corresponds to the energy level of the last state that was occupied by the electrons in the material. The Fermi surface is the surface in the k-space that comprises the occupied electronic states. However, in the case of materials that are “doped” with other elements, other energy levels are accessible to the system, which modifies the conductive properties of the material. In the presence of defects and impurities in the material due to doping, there is a readjustment of the effective Fermi energy to a new energetic configuration that favors electronic conduction.

8.3.4.2. Quantum Confinement of Electrons in the Volume of the Material

In a quantum system under stationary conditions, the confinement of electrons limits its energy to certain specific “discrete” levels due to the confinement in the volume of the material, which excludes specific energy values. For example, this condition determines the electrical behavior of nanomaterials, such as graphene or carbon nanotubes, as will be explained in the following paragraphs.

Consider a material with the dimensions L_x, L_y, and L_z and volume V = L_xL_yL_z. The stationary conditions of the system (Bloch theorem) require that the wave functions of the electrons belong to the crystal lattice of the material to respect the boundary conditions, which means that the electronic wave functions can only have certain wavelengths λ_n that correspond to the wave numbers k_n = 2π/λ_n. The potential of the electrons within the material does not depend on the location of the particle; that is, it is constant and can be considered null for ease of calculation.

Under these conditions, the energy levels of the electrons become “quantized” at discrete values depending on the values that are allowed by the wave vector

\vec{k}

$\vec{k}$

$E_{n} = \frac{p^{2}}{2 m} = \frac{ℏ^{2} k_{n}^{2}}{2 m}$ $E_{n} = \frac{p^{2}}{2 m} = \frac{ℏ^{2} k_{n}^{2}}{2 m}$

(8.22)

which is the solution for the eigenstates of the energy of the Schrödinger equation of the system. In the formalism of the quantum theory that uses the wave vector

\vec{k}

$\vec{k}$

in the energy expression, the energy of the electrons is E = (ħk)²/2m, and the energy of the photons is E = ħck. The periodic wave function ψ(x) of the electrons, which is expressed by a sin function of the type ψ(x) = A sin (2πx/λ), will have to respect the boundary conditions of the system, which is the condition ψ(x) = ψ(0) = ψ(L) = 0 due to the confinement of electrons in the material. The stationary oscillations of the system are represented in Fig. 8.3, where n is the order of oscillation of the wave function of the electron and L is the distance between the walls. For n = 1, the wavelength is double the size of the material, and it changes phase at the edges to respect the stationary condition; n = 2 represents λ = L.

Figure 8.3 Quantum confinement condition of electrons with the resulting definition of the discrete energy values E_n.

Only certain oscillations of the specific wavelengths λ_n can respect the boundary conditions with the resulting discretization of the energy at defined values.

For ψ(L) = A·sin(2πL/λ) = 0 and 2πL/λ = nπ, the resulting values of λ_n are therefore λ_n = 2L/n and k_n = 2π/λ_n = nπ/L. Eq. (8.22) therefore becomes:

$E_{n} = \frac{p^{2}}{2 m} = \frac{ℏ^{2} {(k_{n})}^{2}}{2 m} = \frac{ℏ^{2} {(n π)}^{2}}{2 m {(L)}^{2}}$ $E_{n} = \frac{p^{2}}{2 m} = \frac{ℏ^{2} {(k_{n})}^{2}}{2 m} = \frac{ℏ^{2} {(n π)}^{2}}{2 m {(L)}^{2}}$

(8.23)

The quantum confinement of electrons in atoms and molecules is due to the stable and stationary condition of electrons in the atomic and molecular orbitals. The theory that describes the electronic density around nuclei in a material finds an important simplification in the description of molecular orbitals in the LCAO.

The DOS leads to the concept of a “fundamental state,” which is the lowest energy state of a system that contains N electrons without any type of interaction. This state is filled by the electrons of the system from the lowest energy until it is occupied by the last electron for the molecular orbital with the lowest possible energy. These orbitals represent the states that are accessible to the quantum system due to the crystal lattice, charges, and other factors. The structure of the molecular orbitals and the corresponding energies will be discussed in the following chapters. The set of orbitals that correspond to each energy level of the system determines the formation of bands of energy, the valence band and the conduction band.

8.3.5. Graphene

As an application of the concepts that were introduced in the previous paragraphs, the molecular structure of graphene is shown in real space and in reciprocal space (Fig. 8.4B) to determine the band structure that defines the electronic behavior of graphene. The Brillouin zone for graphene can be defined as the rhombus that is formed by the lattice vectors of the reciprocal space

{\vec{b}}_{1}

${\vec{b}}_{1}$

and

{\vec{b}}_{2}

${\vec{b}}_{2}$

Figure 8.4 (A) Graphene’s structure in real space. The rhombus represents the unit cell of graphene and is delimited by the network vectors ${\vec{a}}_{1}$ ${\vec{a}}_{1}$ and ${\vec{a}}_{2}$ ${\vec{a}}_{2}$ . This area involves two atoms, which are indicated as A and B. (B) Graphene structure in reciprocal space showing the unit vectors ${\vec{b}}_{1}$ ${\vec{b}}_{1}$ and ${\vec{b}}_{2}$ ${\vec{b}}_{2}$ and the Brillouin zone that is delimited by them. The shaded hexagon shows the Brillouin zone that is considered because it represents all of the symmetry operations of the unit cell of graphene [21]. (C) Representation of a graphene sheet.

8.3.5.1. Electronic Structure of Graphene

The electronic structure of a graphene molecule that is defined by the LCAO theory is composed of a valence band and a conduction band. Fig. 8.5 shows the structure of the valence and conduction bands that are occupied by the electrons as calculated using the TB approximation and the corresponding electron cloud in the π valence orbitals (bonding) and π* conduction orbitals (antibonding). These bands determine the electrical conduction properties of graphene as shown in the DOS diagram for graphene (Fig.8.5C). The π and π* bands that are shown in Fig. 8.5A are also shown in three dimensions in Fig. 8.6A. These bands are close to the Fermi energy level at points of high symmetry in the Brillouin zone of graphene (points K, K′,Γ and M in Figs. 8.4A–B and 8.5). These points are due to the symmetry of the hexagonal structure of graphene and represent special conditions in the description of the electrical properties of graphene [22].

Figure 8.5 Electronic properties of graphene.
(A) Structure of the electronic bands: band π (last occupied band) and band π* (conduction band, first unoccupied) touch each other at the K points of the first Brillouin zone. The Fermi energy is set to zero. (B) and (D) Electronic configuration of π states at the K points in bonding orbitals. (C) and (E) σ states at Γ points far from the Fermi energy that are occupied by the electrons of the graphitic sp₂ bonds plan [23].

Figure 8.6 (A) Diagram showing the conduction and valence bands in the first Brillouin zone of hexagonal graphene [21]. The points of high symmetry Γ, K, K’, M, M’, and M” are shown in the hexagonal projection. (B) Valence and conduction bands in the high-symmetry directions ΓK, KM, and MΓ of the Brillouin zone of graphene. (C) Density of states (DOS) of graphene [23].

The valence and conduction bands degenerate at the points K and K′ of the reciprocal space, which are found at the Fermi level (Fig. 8.6C), which determines the electrical conduction properties of graphene. The energy gap between the valence and conduction bands is zero, which characterizes graphene as a zero-gap semimetallic material.

8.3.6. Carbon Nanotubes

8.3.6.1. Structure of Carbon Nanotubes

Carbon nanotubes [24] can be considered as one or several sheets of graphene that are rolled into a perfect cylinder [25,26]. The size of these materials is on the order of magnitude of an atom [diameters from 0.7–2 nm for single-walled carbon nanotubes (SWNTs) to 10 nm for multiwalled carbon nanotubes (MWNTs)]. Quantum mechanics is used to describe these nanomolecules because the movement of electrons is “confined” in the direction of the circumference of the nanotubes in a very limited number of configurations to respect the stationary conditions of the wave functions as was explained in the preceding paragraphs. A continuous number of electronic states is permitted in the direction of the tube’s length because the length of the tube is much greater than the wavelength of the electronic wave functions. Placing the permitted values for the wave vector

\vec{k}

$\vec{k}$

in the first Brillouin zone of graphene generates a series of parallel lines called “cutting lines.” The length, number, and orientation of these lines depend on the chiral indices (n, m) of the nanotubes, which are called “chiral vectors.” For many properties, the carbon nanotube can be considered as a unidimensional crystal with a translation vector

\vec{T}

$\vec{T}$

along its main axis and a small number of carbon atoms on its circumference. The structure of a carbon nanotube can be determined unambiguously by the chiral vector

\vec{C_{h}}

$\vec{C_{h}}$

(Fig. 8.7A), which, when rolling the graphene sheet, coincides with the circumference of the nanotube. The chiral vector can be written as a function of the lattice vectors

\vec{a_{1}}

$\vec{a_{1}}$

and

\vec{a_{2}}

$\vec{a_{2}}$

of graphene as follow:

Figure 8.7 (A) Projection of an open nanotube on a graphene layer. When the graphene sheet is closed to form a nanotube, the chiral vector ${\vec{C}}_{h}$ ${\vec{C}}_{h}$ becomes the circumference of the cylinder (nanotube), and the translation vector $\vec{T}$ $\vec{T}$ is aligned parallel to the nanotube’s axis. The vector $\vec{R}$ $\vec{R}$ is the vector of symmetry, θ indicates the chiral angle, and ${\vec{a}}_{1}$ ${\vec{a}}_{1}$ and ${\vec{a}}_{2}$ ${\vec{a}}_{2}$ are the unit vectors of graphene. The unit cell of the nanotube is defined by the rectangle that is defined by the two vectors ${\vec{C}}_{h}$ ${\vec{C}}_{h}$ and $\vec{T}$ $\vec{T}$ . (B) Helical arrangement in achiral and chiral nanotubes. (C) FEG microscopy image of carbon nanotubes [27].

${\vec{C}}_{h} = n {\vec{a}}_{1} + m {\vec{a}}_{2}$ ${\vec{C}}_{h} = n {\vec{a}}_{1} + m {\vec{a}}_{2}$

(8.24)

8.3.6.2. Lattice Vectors in Real Space

To specify the symmetry properties of carbon nanotubes as a 1D system, it is necessary to express the lattice vector, which is the translation vector

\vec{T}

$\vec{T}$

(Fig. 8.7) that has the direction of the primary axis of the nanotube. The translation vector

\vec{T}

$\vec{T}$

of a nanotube can be written as a function of n and m as

\vec{T} = t_{1} {\vec{a}}_{1} + t_{2} {\vec{a}}_{2}

$\vec{T} = t_{1} {\vec{a}}_{1} + t_{2} {\vec{a}}_{2}$

, where t₁ = (2m + n)/d_R, and t₂ = (2n + m)/d_R. The length of the translation vector is

\vec{T} = \sqrt{3} \vec{C_{h}} / d_{R}

$\vec{T} = \sqrt{3} \vec{C_{h}} / d_{R}$

, where d_R is the maximum common divisor of (2n + m, 2m + n). By defining d as the greatest common divisor of (n, m), the values of d and d_R are related by [28]:

$d_{R} = \{\begin{array}{c} d if (n - m) is not a multiple of 3 d \\ 3 d if (n - m) is a multiple of 3 d \end{array}$ $d_{R} = \{\begin{array}{c} d if (n - m) is not a multiple of 3 d \\ 3 d if (n - m) is a multiple of 3 d \end{array}$

(8.25)

For the nanotube (6, 2) that is shown in Fig. 8.7, d_R = d = 2, and (t₁, t₂) = (5, 7). For achiral zigzag and armchair nanotubes,

T = \sqrt{3 a}

$T = \sqrt{3 a}$

and T = a, respectively. The area of the unit cell of the nanotube can be found by calculating the vector product of the two vectors

\vec{T}

$\vec{T}$

and

{\vec{C}}_{h}

${\vec{C}}_{h}$

| C_{h} \times \vec{T} | = \sqrt{3} a^{2} (n^{2} + n m + m^{2}) / d_{R}

$| C_{h} \times \vec{T} | = \sqrt{3} a^{2} (n^{2} + n m + m^{2}) / d_{R}$

. Dividing this value by the area of the unit cell of graphene

|{\vec{a}}_{1} \times {\vec{a}}_{2}| = \sqrt{3} a^{2} / 2

$|{\vec{a}}_{1} \times {\vec{a}}_{2}| = \sqrt{3} a^{2} / 2$

results in the number of hexagons in the unit cell of the nanotube:

$N = \frac{2 (n^{2} + n m + m^{2})}{d_{R}}$ $N = \frac{2 (n^{2} + n m + m^{2})}{d_{R}}$

(8.26)

In the case of nanotube (6, 2), N is equal to 26, so the unit cell of this nanotube has 52 carbon atoms because there are two carbon atoms in the unit cell of graphene. For achiral nanotubes, N = 2n.

8.3.6.3. Application of Quantum Theory to Carbon Nanotubes

The description of carbon nanotubes is based on the analysis of graphene, whose electrons make strong covalent σ-type bonds along the graphitic plane that do not significantly affect the electronic properties of the nanotubes because they are far from the Fermi energy. The electrons that occupy the p_z atomic orbitals, which are perpendicular to the planes, interact with the p_z orbitals of neighboring atoms and generate weak interactions with the creation of π bonding molecular orbitals and antibonding π* molecular orbitals. The bond energy of these orbitals is close to the Fermi energy in the first Brillouin zone and is responsible for the weak interaction between the SWNTs in bundles in the same manner that it acts in graphite to generate weak bonds between the planes. The relatively simple Hamiltonian formalism is normally used to describe the π and π* electronic bands, which is known as the TB approximation.

8.3.6.4. Application of the TB Model

By applying the TB formalism to carbon nanotubes [18], it is possible to write the Bloch states for the p_z orbital of the electron in each carbon atom in the Brillouin zone of graphene. Rewriting Eq. 8.10 yields:

$ψ_{k}^{A} (r) = \frac{1}{{(N)}^{1 / 2}} \sum_{R_{A}} \exp (i k \cdot R_{A}) χ (r - R_{A})$ $ψ_{k}^{A} (r) = \frac{1}{{(N)}^{1 / 2}} \sum_{R_{A}} \exp (i k \cdot R_{A}) χ (r - R_{A})$

(8.27)

and

$ψ_{k}^{B} (r) = \frac{1}{{(N)}^{1 / 2}} \sum_{R_{B}} \exp (i k \cdot R_{B}) χ (r - R_{B})$ $ψ_{k}^{B} (r) = \frac{1}{{(N)}^{1 / 2}} \sum_{R_{B}} \exp (i k \cdot R_{B}) χ (r - R_{B})$

(8.28)

The A and B indices refer to the two carbon atoms that occupy the primitive cell. The sums are extended to all of the sites of the crystal lattice in relation to the position of the electron in A, where the wave function is a linear combination of the wave functions relative to the atomic sites, which also applies to the electron of atom B. χ(r) represents the wave function of the p_z atomic orbital normalized to an isolated atom, while the wave function of the electron in the main cell is the linear combination of the wave functions for each atom:

$Ψ = ψ_{k}^{A} (r) + λ ψ_{k}^{B} (r)$ $Ψ = ψ_{k}^{A} (r) + λ ψ_{k}^{B} (r)$

(8.29)

Considering the general expression for the Hamiltonian operator [10]:

$H_{μ ν} (k) = \int Ψ^{μ} * (k, r) \hat{H} Ψ^{ν} (k, r) d^{3} r$ $H_{μ ν} (k) = \int Ψ^{μ} * (k, r) \hat{H} Ψ^{ν} (k, r) d^{3} r$

(8.30)

substituting Eq. 8.29 into HΨ = EΨ (Eq. 8.9) [10], the following components are obtained:

$H_{AA} + λ H_{AB} = E$ $H_{AA} + λ H_{AB} = E$

(8.31)

$H_{BA} + λ H_{BB} = λ E$ $H_{BA} + λ H_{BB} = λ E$

(8.32)

The coefficient λ is eliminated using the variational principle [29], which obtains the most appropriate eigenvalues for the energy. The H_ii terms are:

$H_{AA} = H_{BB} = α$ $H_{AA} = H_{BB} = α$

(8.33)

With the parameter α that is defined from the expression for the Hamiltonian that is valid in the TB approximation:

$α (r) = \int {|χ_{μ} (r)|}^{2} [\sum_{R \neq 0} V (r - R)] d^{3} r$ $α (r) = \int {|χ_{μ} (r)|}^{2} [\sum_{R \neq 0} V (r - R)] d^{3} r$

(8.34)

where α represents the effect of the distant potentials, or the crystal field, on the electron of the atom under consideration. The terms

H_{AB} = H_{BA}^{*} = \int ψ_{A}^{*} (r) \hat{H} ψ_{B} (r) d^{3} r

$H_{AB} = H_{BA}^{*} = \int ψ_{A}^{*} (r) \hat{H} ψ_{B} (r) d^{3} r$

are obtained from Eqs. 8.27 and 8.28, which represent the wave functions for the p_z atomic orbitals of the electrons of the primitive cell:

$H_{AB} = β [\exp (i k_{x} \frac{1}{\sqrt{3}}) + 2 \exp (- i k_{x} \frac{a}{2 \sqrt{3}}) \cos (\frac{k_{y} a}{2})]$ $H_{AB} = β [\exp (i k_{x} \frac{1}{\sqrt{3}}) + 2 \exp (- i k_{x} \frac{a}{2 \sqrt{3}}) \cos (\frac{k_{y} a}{2})]$

(8.35)

The parameter β is defined as the Hopping term and represents the quantity that is related to the overlap between the p_z atomic orbitals via the potential:

$β (R) = \int χ_{μ}^{*} (r) [\sum_{R} V (r - R)] χ_{μ} (r - R^{l l}) d^{3} r$ $β (R) = \int χ_{μ}^{*} (r) [\sum_{R} V (r - R)] χ_{μ} (r - R^{l l}) d^{3} r$

(8.36)

This term defines the width of the energy bands. The components k_x and k_y of the wave vector

\vec{k}

$\vec{k}$

are shown in Fig. 8.7A. A 2 × 2 matrix is thus obtained, and setting the determinant equal to zero leads to the secular Schrödinger equation:

$Det (\begin{array}{c} H_{AA} - ɛ H_{AB} \\ H_{BA} H_{BB} - ɛ \end{array}) = 0$ $Det (\begin{array}{c} H_{AA} - ɛ H_{AB} \\ H_{BA} H_{BB} - ɛ \end{array}) = 0$

(8.37)

The spectrum of eigenvalues of the graphene sheet is obtained by calculating the determinant of the matrix as a function of the components k_x and k_y of the wave vector

\vec{k}

$\vec{k}$

$ɛ (k_{x}, k_{y}) = α \pm β {[1 + 4 \cos (\frac{\sqrt{3} k_{x} a}{2}) \cos (\frac{k_{y} a}{2}) + 4 \cos^{2} (\frac{k_{y} a}{2})]}^{1 / 2}$ $ɛ (k_{x}, k_{y}) = α \pm β {[1 + 4 \cos (\frac{\sqrt{3} k_{x} a}{2}) \cos (\frac{k_{y} a}{2}) + 4 \cos^{2} (\frac{k_{y} a}{2})]}^{1 / 2}$

(8.38)

where a is the length of the vectors

\vec{a_{1}}

$\vec{a_{1}}$

and

\vec{a_{2}}

$\vec{a_{2}}$

of the primary lattice. Eq. 8.38, which was obtained for graphene, determines the electronic structure of the carbon nanotube and defines the energy dispersion to the energy bands that is related to the π orbitals of graphene as shown in Fig. 8.5A.

By considering the structure of the carbon nanotube, two new vectors of the reciprocal lattice, the vector K_⊥ in the direction of the circumference and the vector K_// along the axis of the nanotube (Fig. 8.8), can be defined from the relationship:

Figure 8.8 Representation of the wave vectors in the directions of the translational vector $\vec{T}$ $\vec{T}$ (K_||) and of the chiral vector $\vec{C_{h}}$ $\vec{C_{h}}$ along the circumference (K_⊥).

$R_{i} \cdot K_{j} = 2 π δ_{i j}$ $R_{i} \cdot K_{j} = 2 π δ_{i j}$

(8.39)

which is translated as:

$C_{h} \cdot K_{⊥} = 2 π, T \cdot K_{⊥} = 0;$ $C_{h} \cdot K_{⊥} = 2 π, T \cdot K_{⊥} = 0;$

(8.40)

$C_{h} \cdot K_{/ /} = 0, T \cdot K_{/ /} = 2 π,$ $C_{h} \cdot K_{/ /} = 0, T \cdot K_{/ /} = 2 π,$

(8.41)

Using the expressions for the vectors t₁ and t₂ (Section 8.3.6.2), obtained due to the condition C_h · T = 0; the following is valid:

$K_{/ /} = 1 / N (- t_{2} b_{1} + t_{1} b_{2})$ $K_{/ /} = 1 / N (- t_{2} b_{1} + t_{1} b_{2})$

(8.42)

$K_{⊥} = 1 / N (m b_{1} - n b_{2})$ $K_{⊥} = 1 / N (m b_{1} - n b_{2})$

(8.43)

where b₁ and b₂ are the vectors of the reciprocal lattice of graphene, and N is the number of hexagons per unit cell.

The wave vector K_⊥, which is associated with the chiral vector

\vec{C_{h}}

$\vec{C_{h}}$

, is now quantized with a finite number of states k, while the wave vector K_//, which is associated with the translation vector

\vec{T}

$\vec{T}$

along the axis of the tube, is continuous if the difference ∆k between the two contiguous values of K_// is infinitesimal (∆k = 2π/T → 0), which is respected for a nanotube of length T → ∞. The energy eigenvalues can therefore be written in the form:

$ɛ_{p}^{nanotube} (k) = ɛ^{graphene} [k K_{/ /} + p K_{⊥}]$ $ɛ_{p}^{nanotube} (k) = ɛ^{graphene} [k K_{/ /} + p K_{⊥}]$

(8.44)

The indices k and p can assume the values:

$- π / T < k < π / T; p = 0, 1, \dots, N - 1$ $- π / T < k < π / T; p = 0, 1, \dots, N - 1$

(8.45)

For example, consider a nanotube (n, m), where C_h = na₁ + ma₂. The wave functions must satisfy the periodicity condition of the tube:

$C_{h} \cdot k = 2 π q$ $C_{h} \cdot k = 2 π q$

(8.46)

where q is a whole number, and

\vec{k}

$\vec{k}$

is a wave vector of graphene in the first Brillouin zone. Considering the case of an armchair-type nanotube, the indices that define the chiral vector are equal: n = m. The length C of the circumference of the nanotube is:

$C = |C_{h}| = \sqrt{C_{h} \cdot C_{h}} = a \sqrt{n^{2} + n m + m^{2}}$ $C = |C_{h}| = \sqrt{C_{h} \cdot C_{h}} = a \sqrt{n^{2} + n m + m^{2}}$

(8.47)

$C = a \sqrt{3} n$ $C = a \sqrt{3} n$

(8.48)

Considering the quantization that is defined in Eq. 8.46, the following is valid:

$C \cdot k = a \sqrt{3} n k_{x} = 2 π q$ $C \cdot k = a \sqrt{3} n k_{x} = 2 π q$

(8.49)

where k_x is the component of the wave vector

\vec{k}

$\vec{k}$

in the direction of the chiral vector C_h, and q = 1,2,...,2n.

Substituting this result into Eq. 8.38 results in:

$ɛ_{q}^{(n, n)} (k) = α \pm β {[1 + 4 \cos (\frac{q π}{n}) \cos (\frac{k a}{2}) + 4 \cos^{2} (\frac{k a}{2})]}^{1 / 2}$ $ɛ_{q}^{(n, n)} (k) = α \pm β {[1 + 4 \cos (\frac{q π}{n}) \cos (\frac{k a}{2}) + 4 \cos^{2} (\frac{k a}{2})]}^{1 / 2}$

(8.50)

where −π/a < k < π/a.

Consider a nanotube with the indices n = 5, m = 5 as an example. C_h = 5a₁ + 5a₂, and the spectrum to the eigenvalues of this type of carbon nanotube is:

$ɛ_{q}^{(5,5)} (k) = α \pm β {[1 + 4 \cos (\frac{q π}{5}) \cos (\frac{k a}{2}) + 4 \cos^{2} (\frac{k a}{2})]}^{1 / 2}$ $ɛ_{q}^{(5,5)} (k) = α \pm β {[1 + 4 \cos (\frac{q π}{5}) \cos (\frac{k a}{2}) + 4 \cos^{2} (\frac{k a}{2})]}^{1 / 2}$

(8.51)

The spectrum of the energy eigenvalues can be visualized as a function of the wave vector

\vec{k}

$\vec{k}$

as shown in Fig. 8.9. The number of hexagons N per unit cell is:

Figure 8.9 Occupation of the valence band π for carbon nanotube (5, 5). The circles represent the occupation of orbitals by the electrons for each allowed K-point [18].

$N = 2 \cdot (n^{2} + m^{2} + n m) / d_{R} = 10$ $N = 2 \cdot (n^{2} + m^{2} + n m) / d_{R} = 10$

(8.52)

Since there are 2 atoms/cell, 20 electrons occupy 10 orbitals.

The increasing energetic behavior of the orbitals in the case of the conduction band π* from the edge (k = −π/a) to the center of the Brillouin zone indicates a greater antibonding state of the electrons in the molecule. The width of each band is related to the energy dispersion in the band, which is proportional to the overlap between the orbitals of neighboring atoms. Fig. 8.9 shows that some of the orbitals are degenerate; that is, they are characterized by the same binding energy. The last orbital that is occupied in the π band and the first unoccupied orbital in the π* conduction band cross the Fermi energy, which means that there is an overlap between the orbitals in the two bands, which confers metallic conduction characteristics to the nanotube.

8.3.6.5. Electronic Structure of a Carbon Nanotube

As a first approximation, the electronic structure of a carbon nanotube can be obtained from the 2D structure of graphene. In the case of a SWNT, however, the quantum confinement of the 1D electronic states along the circumference of the nanotube must be considered. Placing the permitted values for the wave vector in the first Brillouin zone of graphene generates a series of parallel lines that indicate the wave vector values

\vec{k}

$\vec{k}$

for the considered nanotube; the length, number and orientation of these “cutting lines” depend on the chiral indices (n, m) of the nanotube. In the case of armchair-type nanotubes, such as the (5, 5) nanotube that was considered in the preceding section, the boundary conditions that are defined by the Bloch functions that are related to the chiral vector C_h are valid:

$ψ_{k} (r + C_{h}) = e^{i k r} C_{h} ψ_{k} (r)$ $ψ_{k} (r + C_{h}) = e^{i k r} C_{h} ψ_{k} (r)$

(8.53)

The value of k_x is calculated as in the previous section. Rewriting Eq. 8.49 for the wave vector k_x results in [25]:

$k_{x}^{ν} = \frac{ν}{N_{x}} \frac{2 π}{\sqrt{3} a}$ $k_{x}^{ν} = \frac{ν}{N_{x}} \frac{2 π}{\sqrt{3} a}$

(8.54)

where ν = 1,...,N_x with the number of hexagons N_x = 5. This shows the presence of 5 modes that are permitted for the wave vector k_x. These possible energetic configurations are normally represented using a hexagon that reproduces the first Brillouin zone (BZ) for graphene as shown in Fig. 8.10A; inserting these lines into the diagram in Fig. 8.5A, which shows the valence and conduction bands in the first hexagonal Brillouin zone of graphene, results in the configuration that is shown in Fig. 8.10B. Fig. 8.10A–B show the wave vectors k for SWNTs in the first hexagonal BZ of graphene using the cutting lines; the discrete values of the wave vector

\vec{k}

$\vec{k}$

are indicated in the direction of the vector K₁ due to the periodic condition of the electronic wave function in the direction along the diameter of the nanotube, while in the longitudinal direction of the nanotube for the K₂ vector, the wave vector assumes continuous values because it lacks boundary conditions that limit the discrete quantum states for the wave vectors k in this direction. In Fig. 8.10C, the lines of the values of the wave vector

\vec{k}

$\vec{k}$

cut the valence and conduction bands of a carbon nanotube with indices (4, 2). Fig. 8.10A represents a metallic nanotube because the cutting line intersects a point K on the edge of the hexagon at the Fermi energy of graphite. Fig. 8.10B refers to the case of a semiconductor nanotube because the cutting line does not pass through the point K.

Figure 8.10 Illustrations of the allowed k values in the first Brillouin zone for (A) a metallic nanotube and (B) a semiconductor nanotube; (C) cutting lines in the case of a (4, 2) nanotube for illustrative purposes [21].

The zone-folding approximation is used to represent the energy bands of carbon nanotubes in a diagram; the basic idea is that the electronic structure of a specific nanotube is determined by the overlap of the electronic bands of graphene. The cutting lines that correspond to the permitted wave vectors k that are related to the nanotube under consideration result in the diagram that is illustrated in Fig. 8.11.

Figure 8.11 (A) Diagram of the molecular orbitals in the energy bands for the electronic states of the nanotube (5, 5), which was obtained using the zone-folding approximation; (B) the corresponding DOS of electrons; the Fermi energy is placed on the zero energy level [23].

The energy dispersion is obtained by substituting the values of k_x into Eq. 8.38. Fig. 8.11 shows the energy bands with degenerate and nondegenerate orbitals for 10 orbitals in the valence band and 10 in the conduction band. The DOS is shown on the right side of the figure. For zig-zag type tubes with chirality (n, 0), the permitted wave vectors are:

$k_{y}^{ν} = \frac{ν}{N_{y}} \frac{2 π}{a}$ $k_{y}^{ν} = \frac{ν}{N_{y}} \frac{2 π}{a}$

(8.55)

where ν = 1,...,N_y. For example, the tube (9, 0) has nine lines of permitted wave vectors (Fig. 8.12).

The relationship of the dispersion of the energy bands and the calculation of the electronic DOS indicate that this is a metallic nanotube because the valence and conduction bands touch at the Fermi energy (k = 0) as shown in Fig. 8.13B. When a bandgap is present between the bands (Fig. 8.13C), the nanotube behaves like a semiconductor. The DOS diagrams show how the DOS between the π and π* bands is greater than 0 for the metallic case and is equal to 0 in the semiconductor case.

Figure 8.12 Illustration of the allowed values of k (cutting lines) for an armchair nanotube (9, 0) in the first Brillouin zone.

Figure 8.13 (A–B) Energy dispersion ratios for the considered metallic nanotubes. (C) Semiconductor nanotube with a bandgap ≠ 0 [30].

Once the basic physical concepts that govern the behavior of matter at the atomic to nanometric scales have been introduced, it is possible to address the incredible properties of nanostructured materials.

Several possible applications for nanomaterials that are already being studied and used based on the exotic material properties at the nanometric scale will be described in the following chapters.

List of Symbols

E Energy

Λ Wavelength

H Planck’s constant

Ħ ħ Reduced Planck’s constant

p Momentum

σ Bonding atomic orbital

π Band of valence molecular orbitals

π* Band of conduction molecular orbitals

ψ(r, t) Wave function

r Position

t Time

m Mass

V Potential energy

ν Fequency

ω Angular frequency

k Wave number

F Force

$\hat{H}$ $\hat{H}$ Hamiltonian operator

δ_ij Kronecker delta

a Lattice parameter

g(E) Density of electronic states

E_g Gap energy

a_i Vectors of the crystal lattice

b_i Reciprocal space lattice vectors

C_h Chiral vector

p_i Atomic orbitals p (n = 2)

β Hopping term

ɛ Energy eigenvalue

SWNT Single Wall Nanotube

MWNT Multi Wall Nanotube

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Table of Contents for 8: Mathematical Fundamentals of Nanotechnology

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Abstract

Keywords

8.1. Introduction

8.2. Classical Mechanics

8.2.1. The Classical Formalism at the Nanometric Scale

8.3. Quantum Mechanics

8.3.1. The Energy of Quantum Systems—Stationary Conditions

8.3.2. Periodic Structure of a Crystal Lattice—The Bloch Theorem

8.3.3. Crystal Lattice and Reciprocal Lattice

8.3.3.1. Reciprocal Lattice

8.3.3.2. First Brillouin Zone

8.3.4. Electronic Structure

8.3.4.1. Density of States—Discretization of Energy in Quantum Systems

8.3.4.2. Quantum Confinement of Electrons in the Volume of the Material

8.3.5. Graphene

8.3.5.1. Electronic Structure of Graphene

8.3.6. Carbon Nanotubes

8.3.6.1. Structure of Carbon Nanotubes

8.3.6.2. Lattice Vectors in Real Space

8.3.6.3. Application of Quantum Theory to Carbon Nanotubes

8.3.6.4. Application of the TB Model

8.3.6.5. Electronic Structure of a Carbon Nanotube

List of Symbols

Table of Contents for
8: Mathematical Fundamentals of Nanotechnology