Graduate
  1. Physical Chemistry  1.1. Thermodynamics  1.1.1. Laws of Thermodynamics  1.1.2. Enthalpy and heat capacity  1.1.3. Entropy and free energy  1.1.4. Phase Equilibrium  1.1.5. Statistical thermodynamics  1.1.6. Thermodynamic cycle  1.1.7. Fugacity and activity  1.2. Quantum Chemistry  1.2.1. Principles of quantum mechanics  1.2.2. Schrödinger Equation  1.2.3. Particle in a box  1.2.4. Operators and eigenvalues  1.2.5. Atomic orbitals  1.2.6. Molecular orbital theory  1.2.7. Perturbation theory  1.2.8. Valence bond theory  1.2.9. Hybridization and chemical bonding  1.3. Chemical kinetics  1.3.1. Rate laws and reaction mechanisms  1.3.2. Collision theory  1.3.3. Transition state theory  1.3.4. Enzyme Kinetics  1.3.5. Chain Reactions and Polymerization  1.3.6. Photochemical reactions  1.4. Statistical mechanics  1.4.1. Partitioning functions  1.4.2. Molecular distribution functions  1.4.3. Boltzmann Distribution  1.4.4. Bose–Einstein and Fermi–Dirac statistics  1.5. Spectroscopy  1.5.1. Rotational Spectroscopy  1.5.2. Vibrational Spectroscopy  1.5.3. Electronic Spectroscopy  1.5.4. Nuclear magnetic resonance spectroscopy  1.5.5. Mass Spectrometry  1.5.6. Raman Spectroscopy  1.5.7. Electron paramagnetic resonance spectroscopy  1.6. Surface and colloidal chemistry  1.6.1. Absorption isotherm  1.6.2. Surface tension and wetting  1.6.3. Colloidal Stability  1.6.4. Catalysis  1.6.5. Emulsions and micelles  1.7. Electrochemistry  1.7.1. Nernst equation  1.7.2. Electrochemical cells  1.7.3. Conductivity and mobility  1.7.4. War  1.7.5. Fuel cells  1.7.6. Electrolysis
  2. Organic chemistry  2.1. Reaction mechanism  2.1.1. Nucleophilic substitution reactions  2.1.2. Electrophilic addition reactions  2.1.3. Elimination reactions  2.1.4. Rearrangement reactions  2.1.5. Radical reactions  2.1.6. Pericyclic reactions  2.2. Spectroscopy and structural determination  2.2.1. UV-Vis Spectroscopy  2.2.2. IR Spectroscopy  2.2.3. NMR Spectroscopy  2.2.4. Mass Spectrometry  2.2.5. X-ray crystallography  2.3. Stereoscopic  2.3.1. Chirality and optical activity  2.3.2. Structural analysis  2.3.3. Geometrical isomerism  2.3.4. Dynamic Stereochemistry  2.4. Organometallic Chemistry  2.4.1. Organolithium and organomagnesium reagents  2.4.2. Palladium-catalyzed cross-coupling reactions  2.4.3. Transition Metal Complexes  2.4.4. Metal–carbon bond  2.5. Polymer chemistry  2.5.1. Polymerization mechanism  2.5.2. Polymer Characteristics  2.5.3. Biodegradable Polymer  2.5.4. Conducting polymer  2.5.5. Supramolecular polymers  2.6. Medicinal chemistry  2.6.1. Drug design and development  2.6.2. Pharmacokinetics and pharmacodynamics  2.6.3. Structure-activity relationships  2.6.4. Molecular docking and drug screening
  3. Inorganic chemistry  3.1. Coordination chemistry  3.1.1. Crystal field theory  3.1.2. Ligand field theory  3.1.3. Spectrochemical Series  3.1.4. Chelation and stability  3.2. Organometallic Chemistry  3.2.1. Metal Carbonyls  3.2.2. Catalysis by organometallic complexes  3.2.3. Metallocenes  3.3. Bio-inorganic chemistry  3.3.1. Metalloproteins and enzymes  3.3.2. The role of metals in biological systems  3.3.3. Metal ion transport and storage  3.4. Solid state chemistry  3.4.1. Crystal Structures  3.4.2. Band theory of solids  3.4.3. Superconductors  3.4.4. Defects in the crystal  3.5. Lanthanides and Actinides  3.5.1. Electronic configuration in lanthanides and actinides  3.5.2. Coordination chemistry of the lanthanides  3.5.3. Magnetic Properties of Lanthanides
  4. Analytical chemistry  4.1. Chromatography  4.1.1. Gas Chromatography  4.1.2. High Performance Liquid Chromatography  4.1.3. Thin Layer Chromatography  4.2. Spectroscopic Techniques  4.2.1. Atomic absorption spectroscopy  4.2.2. X-ray diffraction  4.2.3. Inductively coupled plasma spectroscopy  4.3. Electroanalytical methods  4.3.1. Potentiometry  4.3.2. Voltammetry  4.3.3. Colometry  4.4. Mass Spectrometry  4.4.1. Ionization Technique  4.4.2. Fragmentation Pattern  4.5. Chemometrics  4.5.1. Multidisciplinary analysis  4.5.2. Machine Learning in Chemistry
  5. Theoretical and Computational Chemistry  5.1. Molecular dynamics simulation  5.1.1. Force fields and energy minimization  5.1.2. Monte Carlo simulation in molecular dynamics simulations  5.2. Quantum chemical methods  5.2.1. Hartree–Fock theory  5.2.2. Density functional theory  5.2.3. Semi-empirical methods  5.3. Computational drug design  5.3.1. Molecular Docking  5.3.2. QSAR Modeling  5.3.3. Virtual Screening
  6. Biochemistry  6.1. Enzyme Kinetics  6.1.1. Michaelis-Menten kinetics  6.1.2. Inhibition mechanism  6.2. Metabolism and Bioenergetics  6.2.1. Glycolysis  6.2.2. Citric acid cycle  6.2.3. Electron transport chain  6.3. molecular Biology  6.3.1. DNA replication and repair  6.3.2. Protein synthesis  6.3.3. Gene Regulation  6.4. Structural Biochemistry  6.4.1. Protein Folding  6.4.2. Membrane Biophysics
  7. Environmental Chemistry  7.1. Atmospheric Chemistry  7.1.1. Greenhouse gases  7.1.2. Ozone depletion  7.1.3. Air pollutants and smoke  7.2. Water Chemistry  7.2.1. pH and water hardness  7.2.2. Wastewater treatment  7.2.3. Aquatic Chemistry  7.3. Soil Chemistry  7.3.1. Heavy metal contamination  7.3.2. Soil pH and Buffering  7.3.3. Nutrient cycling  7.4. Toxicology and Chemical Safety  7.4.1. Toxicokinetics  7.4.2. Risk Assessment in Toxicology and Chemical Safety in Environmental Chemistry  7.4.3. Effects of pollutants on the environment

Graduate → Inorganic chemistry → Solid state chemistry ↓


Band theory of solids


Band theory of solids is a theoretical model that explains the behavior of electrons within various types of solids. This theory is paramount for understanding electrical conductivity, the differences between semiconductors and metals, insulators and semiconductors. Band theory evolved from the quantum mechanical model and provides a more comprehensive understanding of the distribution of electrons in solids.

Understanding the basics

In a single atom, electrons occupy discrete energy levels, often referred to as atomic orbitals. These energy levels are clearly distinct and specific for each electron. However, when atoms come together to form a solid, the situation changes significantly.

In solids, particularly crystals, atoms are brought close to each other in an orderly array. The proximity of these atoms affects the energy of the electrons. As more atoms are brought into the system, atomic orbitals overlap, and discrete energy levels in individual atoms expand into bands. These bands can accommodate electrons from multiple atoms in a way that atomic orbitals cannot.

When multiple atomic orbitals overlap, they create a new set of levels known as energy bands. The two most important bands in solid state chemistry are the valence band and the conduction band. These bands determine many of the physical properties of materials.

conduction band valence band band gap

Energy bands: valence and conduction bands

The valence band is the highest range of electron energy where electrons normally exist at absolute zero temperature. These electrons are involved in chemical bonding and determine the electrical properties of the material.

The conduction band has a higher energy than the valence band and is separated from it by a band gap. Electrons move into the conduction band when they gain enough energy, becoming conduction electrons that can move freely throughout the material and carry electric current.

Band gap: the key to material properties

The band gap is an important concept in band theory. It is the difference in energy between the top of the valence band and the bottom of the conduction band. The size of the band gap determines the electrical conductivity of the material. It works like this:

  • Conductors: In metals, the conduction band overlaps with the valence band, or the band gap is very small. This means electrons can move freely between them, making metals good conductors of electricity.
  • Insulators: Insulators have a very large band gap, which means electrons cannot easily move from the valence band to the conduction band. As a result, they do not conduct electricity well.
  • Semiconductors: Semiconductors have a medium band gap. At low temperatures, they behave like insulators, but as the temperature increases (or when doped with impurities), electrons in the band gap can be excited, making semiconductors good conductors.
Conductors (overlap) Insulator(Large Gap) Semiconductors (medium gap)

Mathematical formulation of band theory

Band theory is formulated mathematically using the principles of quantum mechanics. To describe the behavior of electrons in a solid, we use Schrödinger's equation, which sets the allowed energy levels for electrons.

Hψ = Eψ

Here, H is the Hamiltonian operator representing the total energy of the system, ψ (psi) is the wave function, and E is the energy eigenvalue associated with that wave function.

The solution to this equation in a periodic lattice is based on Bloch's theorem, which states that the wave function of electrons in a periodic potential can be expressed as:

ψ_k(r) = e^(ik⋅r)u_k(r)

In this expression, Ψ_k(r) is the wave function of the electron, e^(ik⋅r) is a plane wave factor with wave vector k, and u_k(r) is a function of the periodicity of the lattice.

Density of states

Another important concept in band theory is the density of states, which describes the number of electronic states per energy interval. It is essential for understanding the distribution of electrons in energy levels and for explaining the electrical and thermal properties of solids.

Influence of band theory on modern technology

Band theory is at the heart of many modern technological advances, including:

  • Semiconductors and Electronics: Understanding the behavior of semiconductors at different energies allows the design of transistors, diodes, and integrated circuits. These components are fundamental to nearly all modern electronics.
  • Photovoltaics: Band theory helps explain how electrons in solar cells are excited by light, producing electricity.
  • LEDs: The operation of light emitting diodes is based on the recombination of electrons and holes in the band gap.

Conclusion

Band theory is a cornerstone of solid state chemistry and physics, providing a robust framework for explaining and predicting the electronic properties of materials. Its influence extends far beyond academic study, fuelling advances in technology and helping us manipulate materials for a variety of applications.

This basic understanding will enable us to gain a deeper understanding of the complex behaviour of materials and discover future innovations in various fields.


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Band theory of solids

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