PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Condensed matter physics → Crystal structure and lattice ↓


Band theory in crystal structure and lattices


Band theory is a fundamental concept in condensed matter physics that helps explain the electronic behaviour of solids. It is particularly important in understanding the conductivity of various materials such as metals, insulators and semiconductors. This theory is based on the background of quantum mechanics and the periodic structure of crystalline solids.

Introduction to band theory

To fully understand band theory, we need to start with the basics of crystal structures and lattices. In condensed matter physics, a crystalline solid is a substance whose components, such as atoms or molecules, are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions.

The repeating structure of the lattice is known as the unit cell. The entire crystal is built by repeating the unit cell in three-dimensional space. This regularity is important for creating energy bands.

Formation of energy bands

The concept of energy bands originates from the quantum mechanical model of atoms. In an isolated atom, electrons have discrete energy levels. However, when atoms are brought together to form a crystal, these discrete levels expand into energy bands due to interactions between the atoms.

Discrete level interaction Energy Bands

Energy levels in a crystal are so close that they form a band of allowed energies for electrons. The valence band is the band containing the outermost electrons that are involved in bonding. Above this is conduction band, which can accept electrons and allows electrical conduction.

Band gaps

An important concept within band theory is the idea of the band gap. This is the energy difference between the top of the valence band and the bottom of the conduction band. The band gap classifies materials into conductors, semiconductors, and insulators:

  • Conductors: These materials have overlapping valence and conduction bands, or very small band gap, making it easy for electrons to move freely and conduct electricity.
  • Semiconductors: These have a medium band gap, which can be overcome by thermal or optical energy, giving controlled conductivity.
  • Insulators: These have a large band gap which electrons cannot easily cross, thereby impeding electrical conduction.
valence band conduction band band gap

Kroenig–Penney model

The Kronig-Penney model is a simplified model that gives information about the formation of bands in a potential field. Imagine electrons in a periodic lattice model with a sinusoidal potential. In reality, the potential is more complicated, but this model helps to demonstrate how energy bands are formed.

The Kronig-Penney model shows that in a periodic potential, only specific energy values are allowed, resulting in allowed and forbidden energy bands. Solutions to this model are obtained by solving the Schrödinger equation:

-ħ²/(2m) * d²ψ/dx² + V(x)ψ = Eψ

Where:

  • ħ is the reduced Planck constant
  • m is the mass of an electron
  • ψ is the wave function
  • E is the energy
  • V(x) is the periodic potential

Brillouin zone

To further understand band structures, we examine Brillouin zones. These are uniquely defined regions in reciprocal space (momentum space) that are related to the periodicity of the crystal lattice. They play an important role in characterizing the symmetry and electronic states within the crystal.

1 Brillouin Area second Brillouin zone

The electronic states and associated energy levels within each Brillouin zone can be mapped, allowing investigation of the band structure of a material.

Applications of band theory

There are many important applications of band theory, which significantly impact technology and materials science:

Semiconductors and electronics

In semiconductors, control over the band gap through doping drives electronics and optoelectronics. Doping involves manipulating the electrical properties of a semiconductor by adding impurities to it. These properties are the basis of modern electronic devices such as diodes, transistors, and solar cells.

Metals and conductors

In metals, where the valence and conduction bands overlap, there is high electronic mobility. This explains the wide range of conductivity in metals, which is important for electrical and thermal applications.

Conclusion

Band theory is an essential component of condensed matter physics, providing profound insights into the electronic properties of materials. From explaining fundamental physical features to driving advances in technology, this underscores the importance of understanding energy bands in crystalline solids.


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Band theory in crystal structure and lattices

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