Electronic Structure and Surfaces of Sulfide Minerals: Density Functional Theory and Applications

By Jianhua Chen, Zhenghe Xu and Ye Chen

Electronic Structure and Surfaces of Sulfide Minerals: Density Functional Theory and Applications examines the mineral structure and electronic properties of minerals and their relationship to mineral floatability by density functional theory (DFT). This pragmatic guide explores the role of minerals in flotation by focusing on the mineral surface structure, electronic properties, and the adsorption of flotation agents through the study of the microscopic mechanism of reagents from the structure and properties of minerals. The flotation mechanism is explained from the point-of-view of solid physics, which is of great significance for both theoretical research and practical applications.

The study of the structure and properties of the minerals can reveal the essential nature of mineral flotation, hence why minerals have floatability, the mechanism of response of different minerals to different chemicals, and the origin of the selectivity of flotation agents.

• Discusses the relationship between mineral properties and floatability in terms of crystal structure, atomic coordination structure and electronic properties
• Covers the influence of the surface structure of the mineral on surface charge distribution, reactivity and electron density, including a quantitative calculation method for the atomic reactivity of the mineral surface
• Includes research on the microstructure and mechanism of reagent molecules adsorption on the surface of minerals, focusing on the interactions between water molecules, oxygen molecules and reagents

• Language: English
• Publisher: Elsevier Science
• Release date: Apr 23, 2020
• ISBN: 9780128179758

Jianhua Chen

Dr. Jianhua Chen graduated with a PhD in Minerals Processing Engineering from Central South University, Changsha, P.R. China. He is currently the Director of Guangxi Key Laboratory of Processing for Non-ferrous Metal and Featured Materials and Professor in the College of Resources, Environment and Materials at Guangxi University in Nanning, P. R. China. Dr. Chen’s main research area includes Minerals Processing, Solid Physics of Mineral, Quantum Simulation of Minerals and Molecular design of new flotation reagent.

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Electronic Structure and Surfaces of Sulfide Minerals

Density Functional Theory and Applications

Jianhua Chen

Zhenghe Xu

Ye Chen

Table of Contents

Cover image

Title page

Copyright

Preface

Prologue

Chapter 1. Introduction of density functional theory

1.1. Introduction

1.2. Thomas−Fermi model

1.3. Hohenberg−Kohn theorem

1.4. Kohn−Sham equation

1.5. Exchange-correlation energy functional

1.6. Energy band theory

Chapter 2. Electronic properties of sulfide minerals and floatability

2.1. Crystal structure and electronic properties of copper sulfide minerals

2.2. Crystal structure and electronic properties of iron sulfide minerals

2.3. Crystal structure and electronic properties of lead–antimony sulfide minerals

2.4. Electronic and chemical structures of pyrite and arsenopyrite

2.5. Electronic structure and flotation behavior of monoclinic and hexagonal pyrrhotite

2.6. Galvanic interaction between pyrite and galena

Chapter 3. Surface relaxation and electronic properties of sulfide minerals

3.1. Development of surface electronic states

3.2. Surface relaxation and surface states: foundation

3.3. Surface relaxation and surface state of sulfide minerals

3.4. Density of states of sulfide minerals surface

3.5. Effect of surface structure on the electronic properties

3.6. Surface atomic reactivity on sulfide minerals

Chapter 4. Interaction of water and oxygen with sulfide mineral surface

4.1. Effect of water molecule on surface relaxation

4.2. Adsorption of multilayer water molecules on galena and pyrite surfaces

4.3. Interaction of water and oxygen on the pyrite surface

4.4. Coadsorption of water and oxygen on the galena surface

Chapter 5. Structure and reactivity of flotation reagents

5.1. Density states of collector molecules

5.2. Structure–activity of chelating collectors

5.3. Azo compound depressants

5.4. Frothers adsorption at water–gas interface

Chapter 6. Interaction of flotation reagents with mineral surface

6.1. Interaction of xanthate on galena and pyrite surfaces

6.2. Adsorption of xanthate, dithiophosphate, and dithiocarbamate on galena and pyrite surfaces

6.3. Copper activation of sphalerite and pyrite surfaces

6.4. Interaction of lime with pyrite surface

6.5. The adsorption of cyanide on pyrite, marcasite, and pyrrhotite

6.6. Effect of water molecules on the thiol collector interaction on galena and sphalerite surfaces

Chapter 7. Electronic structures and surface adsorption of impurity-bearing sulfide minerals

7.1. Effect of impurities on the floatability of sulfide minerals

7.2. Effect of impurities and defects on the lattice constants of sulfide minerals

7.3. Effect of impurities and defects on the band gap

7.4. Impurities contribution on the properties of sulfide mineral: the frontier orbital coefficient studies

7.5. Occurrences and correlation of Au and As in pyrite

7.6. Effect of impurities on the band structure and oxidation of galena

7.7. Activation and collecting of impurity-bearing sphalerite

7.8. Effect of impurities on the interaction between galena and xanthate

Subject index

Author index

Copyright

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Preface

Flotation is a process of separating fine valuable mineral particles from their associated gangues. In flotation, hydrophobic minerals of interest are attached to air bubbles, floated under the buoyancy force of bubbles to the top of pulp, and collected as products referred to as concentrate, leaving hydrophilic particles in the pulp as tailings. Flotation has been used in large-scale mineral processing industry since the 1920s. In 1921, Perkins first patented the slightly soluble thiocarbamate as a nonoil chemical collector for sulfide mineral flotation, followed by Keller who invented in 1925 water-soluble xanthates and Whitworth who developed in 1926 dithiophosphate, which revolutionized flotation. During 1930s, the application of soaps and cationic amine collectors industrialized processing of nonmetallic ores. Due to its high efficiency and low cost, flotation is currently used to process annually over billions of tons of ores, in addition to its widely expanding application to wastewater treatment and recycling of various types of valuables from electronic wastes, metallurgical slugs, used batteries, etc. The importance of the froth flotation to the economy of the industrial world has been considered to be enormous.

Many early efforts at understanding flotation were directed toward explaining differential flotation in terms of the relative occlusion of gases. In 1916, bubbles were considered to be at the heart of flotation science.

The role of interfaces in flotation had been considered by Sulman by 1912. The first direct application of thermodynamics to systems similar to flotation was that of von Reinders, who deduced how fine solid particles would be distributed between oil and water phases based on Maxwell's capillarity equations in 1913. In 1915, Ralston suggested that flotation might result from the electrical attraction between negatively charged air bubbles and positively charged mineral particles. In 1917, Taggart and Beach fairly lucidly applied thermodynamics concepts directly to flotation. At present, thermodynamics has become a fairly widely used tool for the analysis of flotation phenomena.

In 1917, Anderson suggested that adsorption might play a dominant role in flotation and used the Gibbs adsorption equation to discuss the frother adsorption at the air–water interface. In 1920, Langmuir found the correlation between adsorption and hydrophobicity. He reported that oleic acid created large contact angles on cleaved calcite and galena but only small angles on clean glass and cleaved mica. In 1928, Taggart described the results of adsorption tests on sulfide minerals that related the structure of the adsorbate to its ability to act as a flotation collector. Taggart formulated the definition of the molecular structure needed for a soluble flotation collector, namely, that it must possess both a polar group that binds it to the surface and a nonpolar group that can orient away when adsorbed at a mineral–water interface. Although such early researchers as Fahrenwald, Sulman, and Taggart carried out a number of experiments to elucidate flotation phenomena, the founder of the scientific basis of flotation was A.M. Gaudin and his colleagues, who opened the beginning of the modern approach to research in flotation chemistry. Major advances, particularly starting in the 1950s, were achieved to flotation systems through better understanding and application of the fundamental principles of surface and colloid chemistry, particularly electrical double-layer phenomena.

Since most of natural minerals are originally not sufficiently hydrophobic for effective attachment to air bubbles, a key for separating valuable minerals from gangue minerals is therefore to render the target minerals hydrophobic by selective adsorption of added chemicals known as collectors in mineral processing. Collectors can adsorb on mineral surfaces by electrostatic attraction, electrochemical reaction, chemical binding, hydrogen bonds, etc. To achieve selective adsorption of collectors on target minerals requires designing special structures of collectors to suit particular mineral surfaces. A number of theories have been proposed to explain the selectivity of collector adsorption at mineral surfaces in flotation. In 1930, for example, the solubility product theory of solutions was suggested by Taggart that the smaller the solubility product of the compounds formed by reagent and metallic ions is, the stronger is the adsorption of the reagents on the corresponding minerals, hence the more effective flotation. This theory has provided satisfactory explanation on some flotation phenomena, such as strong flotation of calcium minerals and weaker or negligible collecting power of silicate minerals by oleic acids. Similarly, the product solubility theory has been used to explain selective flotation of metal sulfide minerals from gangue minerals by xanthate. Other theories based on chemical reactions, chelation, and coordination of collectors with metal ions in solutions have also been proposed to explain selective flotation. In these theories, the potency of metal ions to react with collectors in bulk solutions is used to infer their interactions on mineral surfaces. Clearly the influence of the mineral surface structure and the properties of adjacent coordination atoms on their interactions with collectors are not considered in these theories. Therefore, they cannot explain why copper, lead, and iron sulfides can be effectively floated by xanthate, but not their corresponding oxidized minerals. The atoms (ions) in mineral crystal structure and on mineral surfaces are known to interact with each other, which greatly affects their reactivity with collectors. For example, the reactivity of iron in hematite (Fe2O3) and in pyrite (FeS2) with collectors is drastically different, leading to the use of completely different collectors for hematite flotation and pyrite flotation. Pradip suggested that the selectivity of flotation reagents depends greatly on the structural/stereochemical compatibility between the molecular architecture of the adsorbing collector and the specific structure of mineral surface. More accurate prediction of interactions between the flotation reagent molecule and the mineral surface requires better understanding of spatial effect of mineral surfaces. Although Langmuir noticed the effect of solid surface structure on adsorption as early as 1917, actual effect of mineral surface structure on collector adsorption remains unknown. M.C. Furstenau et al. wrote in The Froth Flotation Century that We are now at a stage where the further improvement of the flotation process requires a deeper understanding of its fundamental theory.

In 1925, Schrödinger put forward the wave function equation of electrons to describe the behavior of microscopic particles. In 1927, Hitler and London made the first attempt to describe the structure of hydrogen molecules using the Schrödinger equation, which laid the foundation of modern quantum chemistry. In 1991, K. Takahashi calculated the electronic properties of reagent molecules by adopting extended Hükel molecular orbital (EHMO) method to predict their reactivity. D. Z. Wang focused on the structure–activity relationship of the reagent by using semiempirical LCAO-MO method. However, the calculation using atomic orbital-based MO method is very difficult. For a molecule of 100 electrons, for example, resolving the RHF equation using MO methods requires integration of 100 million double-electron equations. It is therefore almost impossible to calculate mineral surfaces by MO approach. Density functional theory (DFT) using electron density distribution as a basic variable is a new revolutionary approach for studying the ground state properties of multi-particle systems, which greatly reduce the intensity of calculations. With the recent development of supercomputing power, a wide range of software for DFT calculation has been developed and available for various applications. As a result, DFT is now being rapidly used to calculate the crystal structure of minerals, lattice impurity, mineral surface and interface properties, and reagent adsorption.

The solid physical properties of minerals are of particular importance for sulfide flotation. First, sulfide minerals have semiconducting properties, and the flotation of sulfide mineral is an electrochemical process. Electrochemistry is the basic feature of sulfide mineral flotation. Secondly, electrochemical reactions or electrochemical interactions are common in the sulfide ore processing from grinding to flotation separation, such as galvanic interaction between sulfide minerals and grinding media, the electrochemical interactions between flotation reagents and sulfide mineral surfaces, the galvanic corrosion between different sulfide mineral particles, and the electrochemical reaction between the sulfide mineral surface and the oxygen and water medium, all of which involve the semiconductor band structure and electronic properties of sulfide minerals. Therefore, the semiconducting properties of sulfide minerals are the foundation of electrochemistry of sulfide mineral flotation. Studying the solid physical properties of sulfide minerals (energy band structure, electron state, and electron transfer) could provide theoretical explanation to the electron transfer mechanisms during sulfide mineral flotation. This book systematically summarizes the research results of the authors in recent years and expounds the relationship between the crystal properties and the floatability of sulfide minerals from solid physics, crystal chemistry, surface science, and quantum mechanics.

The research works of this book have been funded by the National Natural Science Foundation of China (50864001, 51164001, 51864003, 51304054). The authors are thankful for these supports. We would also like to thank Dr. Li Yuqiong, Dr. Zhao Cuihua, Dr. Lan Lihong, and others for their contributions to this book.

Prologue

This book systematically studies the electronic structure of sulfide minerals, surface properties, and interaction of reagents with mineral surfaces. The book is structured in seven chapters. The first chapter introduces density functional theory (DFT), and some important concepts of solid state physics are also introduced. The second chapter deals with the crystal structure and electronic properties of sulfide minerals and their applications in flotation. The relationships between the floatability and their crystal structure, band structure, density of states, and frontier orbitals are provided. The third chapter presents the surface relaxation and electronic structure of sulfide minerals surfaces. The difference in charge distribution between surface atoms and bulk atoms, as well as the correlation between surface atomic coordination and reactivity is discussed. In the fourth chapter the adsorption of flotation reagents on mineral surfaces at the solid–liquid interface was studied. In addition, the effect of water and oxygen molecule on the surface properties and reagent adsorption are discussed. The fifth chapter explores the electronic properties of flotation reagents by DFT, and structure–activity of reagents is discussed. In the sixth chapter the mechanism of flotation reagent interacting with mineral surfaces was studied by DFT calculation and microcalorimetry tests. The seventh chapter reports the effects of lattice defects on the properties of sulfide minerals, surface structure, and adsorption behaviors of reagents.

Chapter 1

Introduction of density functional theory

Abstract

In this chapter, the history of quantum theory and the development of density functional theory (DFT) are briefly reviewed. Foundation theorems of DFT, such as Thomas–Fermi model, Hohenberg–Kohn theorem, and Kohn–Sham Equation are introduced in the chapter. The frequently used approximations for the exchange-correlation energy functional are reviewed. In addition, some important concepts of energy band theory, including Bloch's theorem and the first Brillouin zone, are introduced.

Keywords

Density functional theory; Energy band theory

1.1. Introduction

In 1926 and 1927, physicists Schrodinger and Heisenberg, respectively, put forward the Schrodinger equation and uncertainty principle, which marked the birth of quantum mechanics. After that, a new world that is completely different from classical physics was shown in front of the physicist. Meanwhile, a new theoretical tool for understanding the chemical structure of matter was provided for the chemist too. In 1927 the physicists Heitler and London applied the approach of quantum mechanics to atomic structure to study H2 molecule [1], successfully explaining the bonding mechanism in a homonuclear molecule. Their success marked the interdisciplinary science of quantum mechanics and chemistry: the birth of quantum chemistry.

After Heitler and London, chemists have also begun to apply quantum mechanics theory to study. On the basis of the study of hydrogen molecule by the two physicists, three theories of molecular structure were established by chemists, namely valence bond theory, molecular orbital theory, and ligand field theory. Pauling developed the valence bond theory on the basis of the earliest hydrogen molecular model [2] and won the Nobel Prize in Chemistry in 1954. In 1928, the physicist Mulliken put forward the earliest molecular orbital theory [3–5]. In 1931, Hückel developed the molecular orbital theory of Mulliken and applied it to conjugated and aromatic hydrocarbons [6]. In 1929, Bethe proposed the theory of ligand field and applied it to the theoretical research on the transition metal complexes [7]. Later, the theory of ligand field and molecular orbital theory developed into a modern ligand field theory. The valence bond theory, molecular orbital theory, and ligand field theory are the three basic theories of quantum chemistry used to describe molecular structure. In the early stages, due to the limitation of calculation method and relatively small calculation amount, the more intuitive valence bond theory dominated the study of quantum chemistry. After the 1950s, with the invention and rapid development of the computer, a huge amount of computation became an easy task. The advantages of molecular orbital theory were highlighted at this background, which gradually replaced the valence bond theory.

In 1928, Hartree proposed the Hartree equation [8], which assumed that the charge distribution of each electron was the solution of the Schrödinger equation for an electron in a potential ν(r), derived from the field. In 1930, Hartree's students, Fock and Slater, proposed a self-consistent field iterative equation considering the Pauli principle, called the Hartree-Fock equation, which further improved the Hartree equation [9–11]. To solve the Hartree-Fock equation, in 1951, Roothaan further proposed that molecular orbitals could be expressed as the linear combination of atomic orbitals that composed the molecule and developed the famous Roothaan-Hartree-Fock (RHF) equation [12]. This equation along with the method based on the further development of this equation is the fundamental method of modern quantum chemistry.

In 1952, Japanese chemist Kenichi Fukui proposed a frontier molecular orbital theory [13]. In 1965, the American organic chemist Woodward and the quantum chemist Hoffmann jointly proposed the theory of conservation of molecular orbital symmetry in organic reactions. The theories proposed by Fukui, Woodward, and Hoffman use simple models that are based on the simple molecular orbital theory to avoid complex mathematical operations and apply quantum chemistry theory to qualitative treatment of chemical reactions in an intuitive form. Through their theory, experimental chemists can intuitively understand the abstract concepts of molecular orbital wave functions. In 1981, Fukui and Hoffman won the Nobel Prize in Chemistry for their contributions.

Although the quantum theory had been established as early as the 1930s, the Schrodinger equation is very complex and still difficult to obtain the exact solution. Even for the approximate solution by molecular orbital, the required computations are enormous. For example, for a small molecule with 100 electrons, there are over 100 million of the double-electron integrals in the process of solving the RHF equation. This calculation is obviously impossible to complete by humans. Hence, in the next decades, quantum chemistry progressed slowly, and was even rejected by experimental chemists. In the study of solid state physics, it is almost impossible to calculate the crystal and the surface from the classical molecular orbital due to the periodic structure of the crystal and 10²³ order magnitude of nuclei and electrons per cubic centimeter, thus the theoretical calculation of solid physics has been developing slowly. It was not until the 1990s that the maturity of density functional theory (DFT) and the development of computer hardware provided an effective theoretical tool for the calculation of solids and their surfaces.

DFT is one of the solutions based on quantum mechanics and the ab initio method of Born–Oppenheimer approximation. Distinguished with many methods based on molecular orbital theory, which constructs wave functions of multielectron systems (e.g., Hartree–Fock methods), this method is based on electron density function and solves the single-electron many-body Schrodinger equation by Kohn–Sham self-consistent field (KS-SCF) iteration to obtain the electron density distribution. This operation reduces the number of free variables and the degree of systematic oscillation, thus improving the rate of convergence.

In 1964, Hohenberg and Kohn put forward an important computational idea and proved that the electron energy was determined by the electron density [14]. Thus, the electronic structure can be obtained by electron density without dealing with complex many-body electron wave functions. The electronic structure can be described by only three spatial variables. This method is called as density functional theory (DFT). According to this theory, the Hamiltonian of the particle is determined by the local electron density, and the local density approximation (LDA) method is derived. This method has achieved great success in the simulation of solid materials such as metal and semiconductors through the combination of metal electron theory, periodic boundary condition, and energy band theory. LDA was later extended to several other fields, in particular to study the property of molecules and condensed matter. Now it is one of the most commonly used methods in the field of condensed matter physics and computational chemistry. Walter Kohn received the Nobel Prize in Chemistry for the great contributions in the developments of density functional theory. In view of the extensive application and great achievements of DFT, this theory is taken as the second revolution of quantum chemistry. At present, DFT is the main method to calculate the structure and electronic properties of solids, and the self-consistent calculation based on this method is called the first principle method.

Since 1970, DFT has been widely used in the calculation of solid state physics. In most cases, DFT with LDA gives very satisfying results compared with other methods of solving the many-body problem of quantum mechanics, and the computational cost is less than that of the experiment. It was generally considered that quantum chemistry calculations cannot give sufficient precise results, until the 1990s, when the approximation used in the DFT was refined into a better exchange correlation model.

However, DFT is still not perfect. DFT is mainly achieved through the Kohn-Sham method. In the framework of Kohn-Sham DFT, the intractable many-body problem (due to the interacting electrons in a static external potential) is simplified to a tractable problem of non-interacting electrons moving in an effective potential. The effective potential includes the external potential and the effect of Coulomb interactions between the electrons, e.g., the exchange and correlation interactions. Modeling the latter two interactions becomes the difficulty within Kohn-Sham DFT. At present, there is no precise solution to calculate the exchange correlation energy; the simplest approximation is the local-density approximation (LDA). LDA approximation uses a simple homogeneous electron gas model to calculate the exchange energy of the system, and the correlation energy is treated by fitting free electron gas. Although DFT has been greatly improved, it is hard to accurately describe the intermolecular interactions, especially van der Waals forces and in calculations of the band gap in semiconductors. For example, the experimental band gap of zinc sulfide (ZnS) is of 3.6   eV, but the calculated result based on DFT is only 2.0   eV, which is far away from the experimental value.

1.2. Thomas−Fermi model

As early as 1927, Thomas and Fermi first realized that statistical methods could be used to approximate the distribution of electrons in an atom [15,16]. They proposed a homogeneous electron gas model based on the kinetic energy as an electron density functional expression, which is called as the Thomas–Fermi model.

According to the Thomas−Fermi model, the total kinetic energy of the electrons (TTF) can be expressed as shown in Eq. (1.1):

(1.1)

.

is a functional. For the many-electron system, in considering only the interactions between nuclei and electrons and between electrons and electrons, the total energy of the electrons can be expressed as shown:

(1.2)

Eq. (1.2) needs to be solved under equivalent periodic conditions:

(1.3)

The Thomas–Fermi model does not consider the atomic exchange energy, so the calculation accuracy is lower than other methods. Although the treatment of molecules by the Thomas–Fermi method is not successful, the Thomas–Fermi method opens a new method for DFT. Since then, the calculation accuracy of the model has been the focus of research in this field, but the results are not unsatisfactory. This situation keeps unchanged until the emergence of Hohenberg–Kohn's theorem.

1.3. Hohenberg−Kohn theorem

. This theory proposed that the energy of the system is the functional of the electron density distribution function, and the ground state is the minimum [14].

(1.4)

where

(1.5)

(1.6)

(1.7)

) is the main part of it.

is as follows:

(1.8)

(1.9)

Substituting Eq. (1.4) into Eq. (1.9) yields this:

(1.10)

. If we can find its approximate form, the Euler–Lagrange equation can be applied to any system. Therefore, Eq. (1.10) is the basic equation of the DFT.

. It was not until the Kohn–Sham equation was proposed in 1965 that DFT was introduced into practical application.

1.4. Kohn−Sham equation

Kohn and Sham proposed in 1965 that the electron density function of a multiparticle system can be obtained by a simple single-particle wave equation [17]. This simple single-particle equation is the Kohn–Sham equation (K–S equation for short).

In the Kohn–Sham equation, the electron density function of the system can be expressed by the sum of the squares of the single electron wave functions:

(1.11)

and the Kohn–Sham equation can be written as:

(1.12)

(1.13)

The problem of the ground state eigenvalues of the multi-electron system can be transformed into a single electron problem. The Kohn−Sham equation finds its self-consistent solution obtained through an iterative equation.

1.5. Exchange-correlation energy functional

. If a more accurate expression can be found, the DFT calculation will be more practical. Various approximation methods have been proposed, including LDA, LSDA (local spin density approximation), GGA (generalized gradient approximation), and BLYP (hybrid density functional). At present, LDA and GGA are widely used.

1.5.1. Local density approximation

equation:

(1.14)

is the exchange-correlation energy of each particle in a uniform electron gas of density.

The LDA potential function is the exchange correlation potential based on the local charge density in the system. The LDA has been very successful in dealing with the electronic energy bands and related physicochemical properties of metals and semiconductors, but there are also deficiencies in calculating the metal d-band and the band gap of semiconductor. Considering the electron spin state on the basis of the LDA, LSDA is developed. Its exchange-correlation energy is calculated:

(1.15)

is the exchange-correlation energy equivalent to the homogeneous electron gas single electron in the presence of spin polarization, which is related to the spin orientation.

1.5.2. Generalized gradient approximation

Based on LDA, Perdew and Wang proposed in 1986 that in addition to electron density, the exchange energy and correlation energy of the system also depend on the density gradient [18]. Based on this theory, the exchange-correlation functional can be expressed as a function of charge density and gradient:

(1.16)

Due to its rationality and accuracy, many functionals such as PBE, RPBE, and PW91 have been developed under the framework of GGA [19–23].

At present, LDA and GGA have been widely used in the calculation of solid physics and material chemistry and have achieved great success.

1.6. Energy band theory

1.6.1. Bloch's theorem

Bloch's theorem is the foundation of energy band theory of solid physics. It is based on a basic assumption that the atoms in the crystal are periodically arranged and that the potential field in the crystal is translational. In the periodic potential, the single-electron Schrodinger differential equation can be written as:

(1.17)

is the periodic potential, which is translational

(1.18)

Here, a 1, a 2, and a 3 are the three lattice basis vectors of the crystal. The Bloch theorem states that the electronic states in the crystal have the following properties:

(1.19)

where k is the real wave vector of kis also called Bloch function or Bloch wave, which is the most basic function in modern solid state physics.

In order to make the eigenfunction and eigenvalue, one-to-one correspondence, which is the electronic wave vector k and the intrinsic value of E (k), must limit the wave vector k values in an inverted primitive cell interval, and the interval is called the first Brillouin zone. The electronic wave vector number in the first Brillouin zone is equal to the primitive cell number of the crystal.

changes in the Brillouin zone, the energy of the corresponding Bloch wave, i.e., the eigenvalue E of the equation (where n is the energy band indicator), and they can be arranged in order of increasing energy:

, which can be written as follows:

(1.20)

where k is is the function with the same periodicity as the lattice:

(1.21)

The energy band formed by crystal valence electrons plays an important role in the physical properties of the crystal and the physical processes involved. The crystal has a band gap between its highest occupied energy band and the lowest unoccupied energy band. The crystal has only a small number of conductive electrons at low temperatures, which is a semiconductor or an insulator, depending on the band gap. If a crystal has no band gap between its highest occupied band and the lowest unoccupied band, there will still be a significant number of conductive electrons, even at very low temperatures, which is metal. The band theory of crystals explains the conductivity of solids well, and the hypothesis is reasonable. The band theory has been valued by solid physicists. Although there are still some problems that cannot be explained well, the band theory is still the most effective means of studying solid state physics.

1.6.2. The first Brillouin zone

Brillouin zone is a part of space centered on the origin in the reciprocal lattice. The first Brillouin zone can be obtained by bisecting with perpendicular planes nearest neighbors reciprocal lattice vectors, second nearest neighbors, and considering the smallest volume enclosed. Similarly, the second Brillouin zone is obtained by continuing the bisecting operations and delimiting the second volume enclosed. The volume adjacent to the second Brillouin zone and equal in volume to the first Brillouin zone is the third Brillouin zone. The first Brillouin area is also called as the simply Brillouin zone, referred to as the Brillouin zone (BZ). Brillouin zone is a symmetric primitive cell in wave vector space, which has all the symmetries of the point group of the reciprocal lattice.

The shape of the reciprocal lattice of crystal lattice of simple cube is still simple cube, and its shape of the Brillouin zone is still simple cube. The shape of the reciprocal lattice of crystal lattice of body-centered cube is face-centered, and its shape of the Brillouin zone is rhombic dodecahedron. The shape of the reciprocal lattice of crystal lattice of face-centered cube is body-centered, and its shape of the Brillouin zone is truncated octahedron. The volume of the Brillouin zone is equal to the volume of the primitive unit cell.

The primitive translation vectors of a two-dimensional lattice are a1   =   ai, a2   =   aj; then the primitive translation vectors of reciprocal lattice are:

There are four reciprocal points closest to the origin: b1, −b1, b2, −b2. The space enclosed by their perpendicular bisectors is the simply Brillouin zone, that is, the first Brillouin zone. As shown in Fig. 1.1, the square in this reciprocal lattice space is the first Brillouin zone of the square lattice.

By connecting the coordinate origin with the second nearest neighbor reciprocal points and drawing the vertical bisector of these lines, the space adjacent to the first Brillouin zone and equal in volume to the first Brillouin zone is the second Brillouin zone, which is the shaded area of the four isosceles right triangles as shown in Figure 1.1.

By connecting the coordinate origin with the third nearest neighbor reciprocal points and drawing the vertical bisector of these lines, the space adjacent to the second Brillouin zone and equal in volume to the second Brillouin zone is the third Brillouin zone, which is the region of the eight isosceles right triangles in Figure 1.1.

Figure 1.1 Two-dimensional square lattice Brillouin zone.

Figure 1.2 The first Brillouin zone of the face-centered cubic lattice.

The first Brillouin zone of the face-centered cubic lattice is more complex. It is a tetrakaidecahedron with eight regular hexagons and six squares, often called truncated octahedron. Fig. 1.2 shows the shape of this truncated octahedron.

The coordinates of the typical symmetry point in the first Brillouin zone of the face-centered cubic lattice are as follows:

References

[1] Heitler W, London F. Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik.  Z Phys . 1927;44(6–7):455–472.

[2] Pauling L.  The nature of the chemical bond . Cornell University Press; 1960.

[3] Mullike R.S. The assignment of quantum numbers for electrons in molecules, I.  Phys Rev . 1928;32(2):186–222.

[4] Mullike R.S. The assignment of quantum numbers for electrons in molecules. II. Correlation of molecular and atomic electron states.  Phys Rev . 1928;32(5):761–772.

[5] Mullike R.S. The assignment of quantum numbers for electrons in molecules. III. Diatomic hydrides.  Phys Rev . 1929;33(5):730–747.

[6] Hückel E. Quanstentheoretische Beiträge zum BenzolproblemII. Quantentheorie der induzierten Polaritäten.  Z Phys . 1931;72(5–6):310–335.

[7] Bethe H. Splitting of terms in crystals.  Ann Phys . 1929;3:133.

[8] Hartree D.R. The wave mechanics of an atom with a non-Coulomb central field. Part I. Theory and methods.  Proc Camb Philos Soc . 1928;24(01):89. .

[9] Fock V. Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems.  Z Phys . 1930;61(1–2):126–148.

[10] Slater J.C. Note on Hartree's method.  Phys Rev . 1930;35(2):210–211.

[11] Slater J.C. Atomic shielding constants.  Phys Rev . 1930;36(1):57–64.

[12] Roothaan C.C.J. New developments in molecular orbital theory.  Rev Mod Phys . 1951;23(2):69–89.

[13] Fukui K, Yonezawa T, Shingu H. A molecular orbital theory of reactivity in aromatic hydrocarbons.  J Chem Phys . 1952;20:722.

[14] Hohenberg P, Kohn W. Inhomogeneous electron gas.  Phys Rev . 1964;136(3B):B864–B871.

[15] Thomas L.H. The calculation of atomic