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 ↓


Fermi surface topology


In the field of condensed matter physics, understanding the properties of materials involves examining their electronic structure. One of the powerful concepts that has emerged in this field is the idea of the Fermi surface. The Fermi surface is a structure that represents the collection of energy states occupied by electrons at zero temperature. It is important for describing the electronic properties of metals and semiconductors.

What is the Fermi surface?

The Fermi surface arises from the study of band theory, where electrons can occupy only certain energy bands. The energy level at which the conduction band is half-filled at absolute zero temperature is known as the Fermi energy. The set of points in reciprocal space corresponding to this energy defines the Fermi surface.

Formally, if E(k) is the energy of the electronic state associated with the wave vector k, then the Fermi surface is defined by the equation:

E(k) = E_F

where E_F is the Fermi energy.

Visualization of the Fermi surface

To see this, consider a simple metal free electron model in three dimensions. In this simple case the Fermi surface is a sphere in reciprocal space, because the free electron energies are isotropic—they depend only on the magnitude of the wave vector.

E F

In this illustration, the circle represents the cross-section of a spherical Fermi surface in 3D reciprocal space. The radius of this sphere depends on the electron density and is proportional to the square root of the Fermi energy.

Complexity in real substances

Real materials exhibit much more complex Fermi surfaces, which often deviate from simple geometric shapes due to interactions within the crystal lattice and the presence of multiple electron bands. In such materials, Fermi surfaces can take complex shapes such as ellipsoids, toroids, and even more complex structures. These shapes are determined through experimental techniques such as angle-resolved photoemission spectroscopy (ARPES) or calculated theoretically using computational methods.

For example, the Fermi surface of copper, which crystallizes in a face-centered cubic (FCC) structure, is not spherical but instead bulges outward in certain directions within the Brillouin zone:

Cu Fermi surface

Why is the Fermi surface important?

The topology of the Fermi surface is deeply connected to the electronic properties of metals and semiconductors. It governs phenomena such as electrical conductivity, heat capacity, and electromagnetic response of materials. The geometry of the Fermi surface often dictates how electrons will scatter, which in turn affects the resistance and superconductivity properties of the material.

Transport theory and the Fermi surface

Transport properties of materials such as electrical and thermal conductivity can be better understood through the concept of the Fermi surface. Electrons near the Fermi surface are responsible for conducting electricity because they have the highest energy and lowest momentum states, allowing them to be easily excited or perturbed by external fields.

Boltzmann transport equation

In weak electric fields, the behavior of electrons can be analyzed using the Boltzmann transport equation, which describes the statistical behavior of a thermodynamic system out of equilibrium:

df/dt = (∂f/∂t)_coll + (∂f/∂t)_external

where f is the distribution function, (∂f/∂t)_coll is the change caused by collisions, and (∂f/∂t)_external is the change caused by external forces such as electric fields. The solution of this equation gives information about conductivity, the Seebeck effect, and other transport phenomena.

Mean free path and scattering

Electrons moving through the solid scatter due to imperfections such as lattice vibrations (phonons), impurities, and electron-electron interactions. The mean free path, which is the average distance an electron travels between collisions, is affected by the Fermi surface topology. A highly anisotropic Fermi surface means that the mean free path will vary considerably in different crystal directions, which affects transport properties.

Quantum oscillation

The De Haas–Van Alphen effect and the Shubnikov–De Haas effect are quantum oscillatory phenomena observed in the magnetization and electrical resistance of metals, respectively. These effects provide direct measurements of the Fermi surface. They arise due to the quantization of the Landau level of electron orbits in a magnetic field, leading to oscillations that depend on the cross-sectional area of the Fermi surface perpendicular to the field.

The frequency of these oscillations is related to the peak cross-sectional areas of the Fermi surface, allowing scientists to perform detailed mapping of the surface.

Topology and Fermi surface reconstruction

Topological aspects of Fermi surfaces are a major topic of interest. Materials with non-trivial Fermi surface topology, such as topological insulators and Weyl semimetals, host exotic electronic states on their surfaces while retaining insulating behavior in their bulk.

These materials have Fermi surfaces that reconstruct due to phase transitions or changes in temperature and pressure. The discovery of these reconstructions has led to new physics insights and potential technological applications in quantum computing and nano-electronics.

Conclusion

The topology of Fermi surfaces provides an important window into the electronic properties of materials. By investigating the size, shape, and topology of the Fermi surface, scientists not only gain a deeper understanding of the fundamental properties of materials, but also advance the discovery of new materials with tailored electronic and transport properties.

The study of Fermi surface topologies is a vibrant area of research, with significant implications for understanding the underlying principles of technology and materials science.


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Fermi surface topology

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