Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Condensed matter physics ↓


Band structure and transport theory


In condensed matter physics, understanding the electronic properties of materials is fundamental. The concepts of band structure and transport theory play a vital role in explaining how electrons behave in solids. This discussion aims to provide a broad understanding of these concepts in a way that is accessible without compromising on depth.

Band structure

Band structure refers to the range of energies that an electron within a solid can have, and also the range of energies that it cannot have. It is a result of the wave nature of electrons and their interaction with the periodic potentials of the crystal lattice.

Energy bands and band gap

To understand band structure, it is important to note that in an atom, electrons occupy discrete energy levels. However, when many atoms combine together to form a solid, these discrete levels expand into bands due to electron interactions. In these bands, electrons can move around freely. However, there are also energy ranges called band gaps where no electron states exist. These band gaps are important in determining the electrical characteristics of a material.

Visual example: Band structure

conduction band band gap valence band

The visual illustration above shows a simplified view of the band structure, with the conduction and valence bands separated by a band gap.

Semiconductors, insulators and conductors

The band structure directly determines whether a solid material is a conductor, insulator, or semiconductor. Conductors have overlapping conduction and valence bands, meaning that electrons can move around freely. Insulators have wide band gaps, which prevent electron movement under normal conditions. Semiconductors have small band gaps, which allow the controlled flow of electrons when energy is supplied.

Mathematical representation

Mathematically, the behavior of electrons in the band structure can be described by solutions of the Schrödinger equation:

        HΨ = EΨ

Here, H denotes the Hamiltonian operator, Ψ is the wave function, and E is the energy eigenvalue. In periodic solids, the solutions of this equation lead to the formation of energy bands.

Transportation theory

Transport theory in condensed matter physics is about understanding how electrons and other quantum particles move within a solid. This motion is important for the electrical and thermal conduction properties of materials.

Classical versus quantum transport

In classical terms, transport can be simply explained by Ohm's law, where the current density J is proportional to the electric field E through the conductivity σ.

        J = σE

However, in quantum mechanics, the transport properties depend on the scattering of electrons within the material and the quantized nature of the charge carriers. The Boltzmann transport equation is often used:

        df/dt + v * df/dx + F * df/dp = (df/dt)_collision

where f is the distribution function, v is the particle velocity, F is the external force, and the right-hand side accounts for collisions and other interactions.

Drude model

The Drude model provides a simple classical approximation of transport phenomena in metals. It treats the electron as a classical particle, providing information about electrical and thermal conductivity. The Drude model is given as:

        σ = ne²τ/m

Here, n is the charge carrier density, e is the electron charge, τ is the average free time between collisions, and m is the electron mass.

Quantum transport

The quantum perspective considers wave nature and statistical mechanics. It takes into account factors such as electron-electron interactions and dislocations. This is important for understanding behaviour in nanostructures and thin films.

Quantum conductivity

At the quantum level, conductivity is quantized and occurs in discrete steps. Conductivity G can be expressed as:

        G = (2e²/h)T

where e is the electron charge, h is Planck's constant, and T is the probability of transmission of the electron through the conductor.

Visual example: Quantum tunneling

barrier Tunneling

This simplified illustration shows a quantum tunneling phenomenon in which an electron passes through a barrier, and highlights its difference from classical predictions, where such passage is forbidden if the electron does not have sufficient energy.

Electron–phonon interaction

Phonons, or quantized lattice vibrations, play an important role in determining the transport properties of materials. The interaction between electrons and phonons can lead to phenomena such as superconductivity.

In an ideal lattice, electrons would flow without scattering. However, lattice vibrations (phonons) disrupt the periodic potential, causing electron scattering and affecting conductivity.

Mathematical description

Understanding these interactions requires complex mathematical equations that often go beyond basic levels. However, a simplified approach can start with the electron scattering rate due to phonons:

        τ⁻¹ ≈ g²(2π/h)(n_phonon + 1)D(E_F)

This formula suggests the interaction between the electron and the phonon, where g is the electron-phonon coupling constant, n_phonon is the phonon number, and D(E_F) is the density of states at the Fermi energy.

Applications and implications

Understanding band structure and transport theory has direct technological implications. This has largely influenced the development of semiconductors, leading to the development of modern electronics, including computers and smartphones. Today's advanced understanding of graphene and other 2D materials is deeply rooted in these fundamental principles.

Future prospects

The theories of band structure and quantum transport are at the heart of ongoing research into quantum computing, new materials, and nanotechnology. These areas hold the potential for breakthroughs that could redefine technology and physics.

As research continues, new theoretical models and simulations are advancing understanding of material properties, paving the way for innovations in areas such as energy efficiency, superconductivity and beyond.

Conclusion

By understanding band structures and transport theory, physicists can predict how materials will conduct electricity and heat, design materials with specific properties, and ultimately take advantage of these properties in practical applications. This field remains a vibrant area of research with important implications for future technological advancements.


Graduate → 6.1


U
username
0%
completed in Graduate

← Prev (6)
Condensed matter physics
Next (6.1.1) →
Block theorem and the Kronig–Penney model

Comments 



Band structure and transport theory

https://www.physicsflow.com/g/6.1?pfal=en