Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Solid state physics → Crystal structure ↓


Band theory


Band theory is an important concept in solid state physics that helps explain the behavior of electrons in solids. This theory is essential for understanding the electrical conductivity and properties of materials ranging from metals to insulators and semiconductors. In this article, we will explore the fundamentals of band theory, its historical development, and how it applies to different types of materials. We will also introduce some simple visual aids in SVG format to illustrate important concepts.

Basics of band theory

Band theory describes the energy levels of electrons in a solid. It is based on the quantum mechanical model of electrons and considers how these particles behave when they are in a periodic potential, such as the lattice structure of a crystal. Here's the basic premise:

  • Electrons in solids do not have discrete energy levels like isolated atoms. Instead, they have bands of energy that they can occupy.
  • The band structure of a material determines its electrical properties. In general, band theory divides energy levels into two main bands: the valence band and the conduction band.
  • The energy gap between these bands plays an important role in determining whether a material is a conductor, semiconductor, or insulator.

Energy bands and Brillouin zones

To understand band theory it is important to consider the concept of Brillouin zones. These zones are defined in the reciprocal space of a crystal, which is an abstract mathematical space used to analyze the wave vectors of particles.

In a one-dimensional crystal, we can imagine the lattice as a series of equally spaced points. Electrons move in this periodic structure, and their possible energy levels are known as block waves. The allowed energy levels are grouped into bands, which are separated by forbidden regions known as band gaps.

valence bandconduction bandband gap

The diagram above shows the concept of the valence and conduction bands, as well as the band gap that lies between them. Electrons in the valence band do not have enough energy to move freely throughout the material, while electrons in the conduction band can do so, contributing to electrical conductivity.

Historical development

The development of band theory was a gradual process. Let us briefly review the major historical milestones that led to the current understanding of this theory:

Initial theories

Before the development of quantum mechanics, early theories of electrical conductivity relied on classical physics. These theories were unable to explain why some substances conduct electricity and others do not.

With the advent of quantum mechanics in the early 20th century, physicists began using wave functions to describe the behavior of electrons. This new approach laid the groundwork for the development of band theory.

Block theorem

An important contribution to band theory was by Felix Bloch in 1928. He proposed that electrons in a periodic potential can be described by wave functions called Bloch functions. These functions have the property of being periodic along the lattice.

The block theorem states mathematically that the wave function of an electron in a periodic potential can be written as:

ψ_k(r) = e^(ik⋅r) * u_k(r)

Here, k is the wave vector, r is the position vector, and u_k(r) is a function of the periodicity of the lattice.

Nearly free electron model

The nearly free electron model, developed in the 1930s, was an extension of Block's ideas. It considers the electrons in a crystal to move around nearly freely, their motion being slightly disturbed by interactions with the lattice.

This model helped physicists better understand how energy bands form and how their shape depends on the properties of the crystal.

Types of materials

Band theory helps us classify substances based on band structures. In general, substances can be divided into three broad categories: metals, semiconductors, and insulators.

Metals

Metals have overlapping valence and conduction bands, or they have no band gap. This means their electrons can easily move from the valence band to the conduction band, allowing them to conduct electricity efficiently.

valence bandconduction band

In the above diagram the bands overlap each other, meaning there is no band gap, which is common in metals.

Semiconductors

Semiconductors have a small band gap between the valence and conduction bands. This allows them to be conductors under certain conditions, such as when they are exposed to light or heat.

Semiconductors are often alloyed with impurities to enhance their conducting properties. This process involves adding small amounts of other elements to a semiconductor to increase the number of charge carriers (electrons or holes).

valence bandconduction bandSmall band gap

In the above figure, small band gap represents semiconductor material.

Insulator

Insulators have a very large band gap, which means that very few electrons can move from the valence band to the conduction band. This makes them poor conductors of electricity.

valence bandconduction bandLarger band gap

In the above figure, large band gap is the characteristic of insulator material.

Band theory and conductivity

One of the primary applications of band theory is to explain the electrical conductivity of materials. By examining the band structure, we can predict how a material will conduct electricity and how it will behave under various external influences, such as changes in pressure or temperature.

Conductor

  • In conductors such as metals, the valence and conduction bands are either the same or overlap, allowing electrons to flow freely.
  • This structural feature means that conductors have high electrical conductivity, and electrons can move relatively easily.

Semiconductors

  • Semiconductors have a moderate band gap that is affected by temperature or impurities, affecting their conductivity.
  • These materials are important in electronic devices because they allow control over electron flow, which is evident in transistors and diodes.

Insulator

  • Insulators have large band gaps, which prevent the electrons in the valence band from gaining enough energy to move into the conduction band.
  • This makes them poor conductors of electricity, and suitable for use as insulators in electrical circuits to prevent unwanted electric current flow.

Conclusion

Band theory is a foundational theory in solid state physics, providing insight into the electronic properties of materials. By studying energy bands within crystal structures, we gain a deeper understanding of why materials behave the way they do, which is crucial to the development of technology from computer chips to solar cells.

The purpose of this explanation is to introduce you to the basics of band theory, its historical context, and its practical implications. By considering a variety of materials, we have seen how band gaps affect conductivity and how modern technology uses these principles. As new materials and techniques emerge, band theory will continue to be a major area of study in physics and engineering.


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Band theory

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