5.6 Fermi surface

The Fermi surface is a crucial concept in solid state physics, describing the boundary between occupied and unoccupied electron states in reciprocal space. It provides essential information about a material's electronic structure, influencing properties like conductivity and magnetism.

Understanding the Fermi surface is key to grasping how electrons behave in solids. Its shape, topology, and nesting characteristics determine various physical phenomena, from electrical conductivity to superconductivity, making it a fundamental tool for studying and engineering materials.

Fermi surface definition

• The Fermi surface is a fundamental concept in solid state physics that describes the distribution of electrons in a material at absolute zero temperature
• It represents the boundary between occupied and unoccupied electronic states in reciprocal space (k-space)

Fermi energy

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• The Fermi energy ($E_F$) is the highest occupied energy level in a material at absolute zero temperature
• Electrons with energies below $E_F$ are occupied, while those above $E_F$ are unoccupied
• The Fermi energy depends on the electronic structure and density of states of the material

Fermi wave vector

• The Fermi wave vector ($k_F$) is the wave vector corresponding to the Fermi energy
• It represents the radius of the Fermi surface in reciprocal space for an isotropic material
• The magnitude of $k_F$ is related to the electron density ($n$) by $k_F = (3\pi^2n)^{1/3}$

Fermi velocity

• The Fermi velocity ($v_F$) is the velocity of electrons at the Fermi surface
• It is given by $v_F = \hbar k_F / m^*$, where $\hbar$ is the reduced Planck's constant and $m^*$ is the effective mass of the electron
• The Fermi velocity is important for understanding transport properties (electrical conductivity) and optical properties (plasma frequency) of materials

Fermi surface properties

• The Fermi surface encapsulates essential information about the electronic structure and properties of a material
• Its shape, topology, and nesting characteristics influence various physical phenomena in solid state physics

Fermi surface shape

• The shape of the Fermi surface depends on the electronic band structure of the material
• In simple metals (free electron model), the Fermi surface is a sphere in reciprocal space
• In more complex materials, the Fermi surface can have intricate shapes (sheets, pockets, or tubes) due to the presence of multiple energy bands and their interactions

Fermi surface topology

• The topology of the Fermi surface refers to its connectivity and genus (number of holes)
• Fermi surfaces can be closed (electron pockets) or open (hole pockets)
• Topological changes in the Fermi surface (Lifshitz transitions) can occur due to variations in temperature, pressure, or chemical composition, leading to changes in the material's properties

Fermi surface nesting

• Fermi surface nesting occurs when segments of the Fermi surface are parallel to each other, separated by a specific wave vector ($q$)
• Nesting can lead to instabilities in the electronic system, such as charge density waves (CDWs) or spin density waves (SDWs)
• Materials with strong nesting (quasi-1D or quasi-2D systems) are more susceptible to these instabilities

Experimental determination of Fermi surfaces

• Various experimental techniques are used to probe the Fermi surface of materials, providing insights into their electronic structure and properties

De Haas-van Alphen effect

• The de Haas-van Alphen effect is a quantum oscillation phenomenon observed in the magnetization of a material as a function of the applied magnetic field
• It arises from the quantization of electron orbits in the presence of a magnetic field (Landau levels)
• The oscillation frequency is directly related to the extremal cross-sectional area of the Fermi surface perpendicular to the magnetic field direction

Shubnikov-de Haas effect

• The Shubnikov-de Haas effect is a quantum oscillation phenomenon observed in the electrical resistivity of a material as a function of the applied magnetic field
• It shares the same physical origin as the de Haas-van Alphen effect (Landau level quantization)
• The oscillation frequency provides information about the Fermi surface cross-sectional area

Angle-resolved photoemission spectroscopy (ARPES)

• ARPES is a powerful technique that directly measures the electronic band structure and Fermi surface of a material
• It involves the excitation of electrons from the sample using high-energy photons and the measurement of their kinetic energy and emission angle
• ARPES provides a direct visualization of the Fermi surface in reciprocal space and allows for the determination of the electronic dispersion relation $E(k)$

Fermi surface in metals

• In metals, the Fermi surface plays a crucial role in determining their electronic and transport properties
• Different models are used to describe the Fermi surface in metals, depending on the strength of the electron-ion interactions

Free electron model

• The free electron model assumes that electrons in a metal behave as a non-interacting gas, moving freely through the crystal lattice
• In this model, the Fermi surface is a sphere in reciprocal space, with a radius determined by the electron density
• The free electron model provides a good approximation for simple metals (alkali metals) but fails to capture the complexity of the Fermi surface in more complex materials

Nearly free electron model

• The nearly free electron model takes into account the weak periodic potential of the crystal lattice
• It leads to the formation of energy gaps at the Brillouin zone boundaries and the distortion of the Fermi surface from a perfect sphere
• The nearly free electron model explains the formation of electron and hole pockets in the Fermi surface and the presence of Fermi surface nesting

Tight-binding model

• The tight-binding model considers the strong localization of electrons around the atomic sites in a crystal
• It describes the electronic structure in terms of the overlap of atomic orbitals and the hopping of electrons between neighboring sites
• The tight-binding model is particularly useful for describing the Fermi surface in materials with strong electron correlations (transition metals, rare earth elements)

Fermi surface in semiconductors

• In semiconductors, the Fermi surface plays a crucial role in determining their electronic and optical properties
• The position of the Fermi level relative to the conduction and valence bands influences the carrier concentration and transport characteristics

Intrinsic vs extrinsic semiconductors

• Intrinsic semiconductors are pure materials with an equal number of electrons and holes at thermal equilibrium
• Extrinsic semiconductors are doped with impurities to introduce excess electrons (n-type) or holes (p-type)
• The position of the Fermi level in extrinsic semiconductors depends on the type and concentration of dopants

Fermi level in semiconductors

• In intrinsic semiconductors, the Fermi level lies approximately in the middle of the band gap
• In n-type semiconductors, the Fermi level is closer to the conduction band, while in p-type semiconductors, it is closer to the valence band
• The position of the Fermi level determines the carrier concentration and the electrical conductivity of the semiconductor

Band gap effects on Fermi surface

• The presence of a band gap in semiconductors leads to a Fermi surface that is not continuous, unlike in metals
• The Fermi surface in semiconductors consists of separate electron and hole pockets, depending on the position of the Fermi level relative to the conduction and valence bands
• The size and shape of these pockets are determined by the band structure and the effective mass of the carriers

Fermi surface in superconductors

• In superconductors, the Fermi surface undergoes significant modifications due to the formation of Cooper pairs and the opening of a superconducting gap

BCS theory

• The Bardeen-Cooper-Schrieffer (BCS) theory is a microscopic theory that explains the origin of superconductivity
• It describes the formation of Cooper pairs, which are bound states of two electrons with opposite momenta and spins, mediated by electron-phonon interactions
• The BCS theory predicts the existence of a superconducting gap in the electronic density of states around the Fermi level

Cooper pairs

• Cooper pairs are the fundamental building blocks of superconductivity
• They consist of two electrons with opposite momenta and spins, bound together by an attractive interaction mediated by phonons
• The formation of Cooper pairs leads to a lowering of the electronic energy and the establishment of a coherent superconducting state

Superconducting gap

• The superconducting gap ($\Delta$) is the energy gap that opens up in the electronic density of states around the Fermi level in a superconductor
• It represents the minimum energy required to break a Cooper pair and create excitations (quasiparticles) in the superconducting state
• The magnitude of the superconducting gap is related to the critical temperature ($T_c$) of the superconductor by $\Delta \approx 1.76 k_B T_c$, where $k_B$ is the Boltzmann constant

Fermi surface applications

• The Fermi surface plays a crucial role in determining various physical properties of materials, including their electrical, thermal, and thermoelectric characteristics

Electrical conductivity

• The electrical conductivity of a material is directly related to the Fermi surface and the scattering of electrons near the Fermi level
• The shape and size of the Fermi surface, as well as the presence of scattering centers (impurities, phonons), influence the electron mobility and conductivity
• Materials with a large Fermi surface area and low scattering rates exhibit high electrical conductivity

Thermal conductivity

• The thermal conductivity of a material is determined by the contributions from both electrons and phonons
• The electronic contribution to thermal conductivity is related to the Fermi surface and the electron mean free path
• Materials with a large Fermi surface area and long electron mean free path have higher electronic thermal conductivity

Thermoelectric effects

• Thermoelectric effects, such as the Seebeck effect and the Peltier effect, are governed by the Fermi surface and the asymmetry of the electronic density of states near the Fermi level
• The Seebeck coefficient, which measures the voltage generated by a temperature gradient, is sensitive to the shape and topology of the Fermi surface
• Materials with a high Seebeck coefficient and low thermal conductivity are promising for thermoelectric applications (energy harvesting, cooling)

Fermi surface in reciprocal space

• The Fermi surface is defined in reciprocal space (k-space), which is the Fourier transform of the real space lattice
• Reciprocal space provides a convenient framework for describing the electronic structure and properties of materials

Brillouin zones

• The Brillouin zone is the primitive unit cell in reciprocal space, analogous to the Wigner-Seitz cell in real space
• It is defined as the Wigner-Seitz cell of the reciprocal lattice and contains all the unique wave vectors (k-points) that characterize the electronic states
• The Fermi surface is often depicted within the first Brillouin zone, but it can extend to higher Brillouin zones in some materials

Fermi surface mapping

• Fermi surface mapping refers to the experimental or computational determination of the shape and topology of the Fermi surface in reciprocal space
• Techniques such as angle-resolved photoemission spectroscopy (ARPES) and quantum oscillation measurements (de Haas-van Alphen effect, Shubnikov-de Haas effect) are used to map the Fermi surface
• Fermi surface mapping provides valuable insights into the electronic structure and properties of materials

Fermi surface reconstruction

• Fermi surface reconstruction refers to the changes in the shape and topology of the Fermi surface due to external perturbations or interactions
• It can occur as a result of charge or spin ordering, structural phase transitions, or the application of external fields (magnetic field, pressure)
• Fermi surface reconstruction can lead to significant changes in the electronic and transport properties of materials

Fermi surface instabilities

• Fermi surface instabilities are electronic instabilities that occur when the Fermi surface is susceptible to certain types of interactions or perturbations
• These instabilities can lead to the formation of new ordered states or phase transitions in materials

Charge density waves (CDWs)

• Charge density waves are periodic modulations of the electron density in a material, accompanied by a periodic lattice distortion
• CDWs can occur when the Fermi surface exhibits strong nesting, i.e., when large portions of the Fermi surface are parallel to each other and separated by a specific wave vector ($q$)
• The formation of CDWs opens up a gap in the electronic density of states and can lead to changes in the electrical and optical properties of the material

Spin density waves (SDWs)

• Spin density waves are periodic modulations of the electron spin density in a material, often accompanied by a periodic lattice distortion
• SDWs can occur when the Fermi surface exhibits strong nesting and there is a strong exchange interaction between the electrons
• The formation of SDWs leads to a splitting of the electronic bands and the opening of a gap in the electronic density of states

Peierls instability

• The Peierls instability is a type of CDW instability that occurs in one-dimensional (1D) or quasi-1D systems
• It arises from the perfect nesting of the Fermi surface in 1D systems, where the entire Fermi surface can be mapped onto itself by a single wave vector ($2k_F$)
• The Peierls instability leads to a periodic lattice distortion and the opening of a gap at the Fermi level, resulting in a metal-to-insulator transition

Fermi surface engineering

• Fermi surface engineering involves the manipulation and control of the Fermi surface properties to achieve desired electronic and transport characteristics in materials

Doping effects on Fermi surface

• Doping a material with impurities can significantly alter its Fermi surface properties
• In semiconductors, doping can shift the Fermi level towards the conduction band (n-type) or valence band (p-type), modifying the carrier concentration and the size of the electron or hole pockets
• In metals, doping can introduce additional electronic states near the Fermi level, leading to changes in the shape and topology of the Fermi surface

Strain effects on Fermi surface

• Applying strain to a material can modify its electronic band structure and Fermi surface properties
• Tensile strain can increase the overlap between atomic orbitals, leading to a broadening of the electronic bands and a change in the Fermi surface shape
• Compressive strain can have the opposite effect, reducing the orbital overlap and modifying the Fermi surface accordingly

Fermi surface tuning in 2D materials

• Two-dimensional (2D) materials, such as graphene and transition metal dichalcogenides (TMDs), offer unique opportunities for Fermi surface engineering
• The electronic properties of 2D materials are highly sensitive to external perturbations, such as electric fields, substrate interactions, and chemical functionalization
• By controlling these external factors, it is possible to tune the Fermi level, modify the Fermi surface topology, and induce phase transitions in 2D materials