2.5 Fermi surface

The Fermi surface is a crucial concept in condensed matter physics, describing the boundary between occupied and unoccupied electron states in momentum space. It's key to understanding electronic properties of materials, including conductivity, optical behavior, and magnetism.

Fermi surface characteristics vary widely across materials, from simple spheres in free electron gases to complex shapes in real solids. These features determine material properties and play a vital role in designing new materials with specific electronic behaviors.

Concept of Fermi surface

• Fundamental concept in condensed matter physics describes the surface of constant energy in momentum space
• Crucial for understanding electronic properties of materials and their behavior under various conditions
• Provides insights into conductivity, optical properties, and magnetic behavior of solids

Definition and significance

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• Surface in reciprocal space separating occupied from unoccupied electron states at absolute zero temperature
• Determines electronic, thermal, and magnetic properties of materials
• Shape and size of Fermi surface influence conductivity and other material characteristics
• Allows prediction of material behavior under external fields (electric, magnetic)

Fermi energy vs Fermi level

• Fermi energy represents highest occupied energy state at absolute zero temperature
• Fermi level denotes chemical potential of electrons at finite temperatures
• Fermi energy remains constant while Fermi level shifts with temperature changes
• Difference between Fermi energy and Fermi level affects thermal and electrical properties
• Impacts band structure and electron distribution in solids

Brillouin zone relationship

• Fermi surface typically represented within first Brillouin zone
• Periodic nature of crystal lattice reflected in Fermi surface symmetry
• Brillouin zone boundaries often correspond to energy gaps or band crossings
• Understanding this relationship crucial for interpreting experimental data (ARPES)
• Helps in predicting and explaining phenomena like electron-phonon coupling and superconductivity

Fermi surface characteristics

• Key features of Fermi surfaces determine material properties and behavior
• Understanding these characteristics essential for predicting and manipulating material properties
• Plays crucial role in designing new materials with specific electronic properties

Shape and topology

• Varies widely depending on material composition and crystal structure
• Spherical for free electron gases, complex for real materials
• Can include multiple sheets, pockets, or disconnected regions
• Topology influences electronic transport properties and quantum oscillations
• Fermi surface features (necks, pockets) correspond to specific electronic states

Electron vs hole surfaces

• Electron surfaces enclose filled states, hole surfaces enclose empty states
• Distinction important for understanding carrier type and mobility
• Electron surfaces typically convex, hole surfaces typically concave
• Coexistence of electron and hole pockets leads to semimetallic behavior
• Affects Hall effect measurements and thermoelectric properties

Fermi surface nesting

• Occurs when portions of Fermi surface can be mapped onto each other by a translation vector
• Important for understanding charge density waves and spin density waves
• Enhances electron-phonon coupling and can lead to lattice instabilities
• Plays role in superconductivity of certain materials (iron-based superconductors)
• Can be manipulated through doping or applying pressure to alter material properties

Experimental techniques

• Various methods employed to probe and map Fermi surfaces in real materials
• Combination of techniques provides comprehensive understanding of electronic structure
• Advances in experimental techniques have revolutionized our understanding of complex materials

de Haas-van Alphen effect

• Oscillations in magnetic susceptibility of materials in strong magnetic fields
• Frequency of oscillations directly related to extremal cross-sectional areas of Fermi surface
• Allows mapping of Fermi surface topology and effective mass determination
• Requires low temperatures and high magnetic fields for observation
• Particularly useful for studying metals and semimetals

Angle-resolved photoemission spectroscopy

• Direct measurement of electronic band structure and Fermi surface
• Uses photoelectric effect to eject electrons and analyze their energy and momentum
• Provides detailed information about surface electronic structure
• Can resolve spin polarization of electronic states (spin-ARPES)
• Crucial for studying topological materials and high-temperature superconductors

Positron annihilation

• Probes electron momentum distribution in materials
• Positrons annihilate with electrons, producing gamma rays
• Angular correlation of annihilation radiation reveals Fermi surface information
• Particularly useful for studying buried interfaces and defects in materials
• Complements other techniques by providing bulk electronic structure information

Theoretical models

• Various approaches to calculate and predict Fermi surface properties
• Each model has strengths and limitations for different material classes
• Combination of models often necessary for accurate description of complex systems

Nearly free electron model

• Treats electrons as almost free particles with weak periodic potential from ions
• Accurately describes simple metals (alkali metals, noble metals)
• Predicts parabolic band structure and spherical Fermi surfaces
• Explains formation of energy gaps at Brillouin zone boundaries
• Serves as starting point for more sophisticated models

Tight-binding approximation

• Assumes electrons tightly bound to atoms with limited hopping between sites
• Well-suited for describing transition metals and strongly correlated systems
• Produces band structures with characteristic widths and shapes
• Can incorporate multiple orbitals and spin-orbit coupling
• Useful for studying magnetism and Mott insulator behavior

Density functional theory

• First-principles approach based on electron density rather than many-body wavefunction
• Provides accurate predictions of Fermi surface topology for many materials
• Can handle complex crystal structures and chemical compositions
• Incorporates exchange-correlation effects through various approximations
• Widely used in materials design and prediction of new phases

Fermi surface in materials

• Fermi surface characteristics vary dramatically across different material classes
• Understanding these differences crucial for explaining and predicting material properties
• Provides insights into designing materials with specific electronic functionalities

Metals vs semiconductors

• Metals have partially filled bands with Fermi surface crossing energy bands
• Semiconductors have fully occupied valence bands and empty conduction bands at T=0K
• Metallic Fermi surfaces can be complex with multiple sheets and pockets
• Doped semiconductors develop small Fermi pockets near band edges
• Semimetals have small electron and hole pockets with unique transport properties

Superconductors and Fermi surface

• Cooper pair formation occurs near Fermi surface in conventional superconductors
• Fermi surface nesting can enhance electron-phonon coupling and superconductivity
• Unconventional superconductors may have non-trivial Fermi surface topology
• Pseudogap in high-temperature superconductors alters Fermi surface structure
• Understanding Fermi surface evolution crucial for explaining superconducting mechanisms

Topological materials

• Exhibit unique Fermi surface features due to non-trivial band topology
• Dirac and Weyl semimetals have point-like Fermi surfaces with linear dispersion
• Topological insulators have conducting surface states with spin-momentum locking
• Nodal line semimetals have line-like Fermi surfaces with high mobility
• Fermi surface characteristics determine topological response functions (anomalous Hall effect)

Quantum oscillations

• Periodic variations in material properties as function of inverse magnetic field
• Provide valuable information about Fermi surface topology and electronic structure
• Crucial tool for studying complex materials and novel quantum phases

Shubnikov-de Haas effect

• Oscillations in electrical resistivity under strong magnetic fields
• Frequency of oscillations related to extremal orbits on Fermi surface
• Allows determination of effective mass and scattering time of charge carriers
• Particularly useful for studying two-dimensional electron systems
• Can reveal information about Berry phase and topological properties

Quantum Hall effect

• Quantization of Hall conductance in two-dimensional electron systems
• Occurs when Fermi level lies between Landau levels
• Integer and fractional quantum Hall effects reveal fundamental aspects of many-body physics
• Provides platform for studying topological states of matter
• Has led to development of new concepts (composite fermions, anyons)

Berry phase and topology

• Quantum oscillations can reveal Berry phase accumulated by electrons orbiting Fermi surface
• Non-trivial Berry phase indicates topological nature of electronic states
• Manifests as phase shift in quantum oscillation patterns
• Important for identifying and characterizing topological materials
• Connects Fermi surface properties to broader concepts in topological band theory

Applications and implications

• Understanding Fermi surface crucial for explaining and predicting various material properties
• Enables design and optimization of materials for specific technological applications
• Bridges fundamental physics concepts with practical device engineering

Transport properties

• Fermi surface shape and topology directly influence electrical conductivity
• Determines carrier type, density, and mobility in materials
• Explains anisotropic transport behavior in layered and low-dimensional materials
• Impacts thermoelectric properties and potential for energy harvesting applications
• Crucial for understanding and optimizing performance of electronic devices

Optical properties

• Intraband and interband transitions near Fermi surface determine optical response
• Explains color and reflectivity of metals
• Influences plasmon resonances and plasmonic applications
• Affects transparency and absorption in semiconductors and insulators
• Important for designing materials for photovoltaics and optoelectronics

Magnetic susceptibility

• Fermi surface topology affects Pauli paramagnetism and Landau diamagnetism
• Explains magnetic behavior of conduction electrons in metals
• Influences spin-orbit coupling effects and magnetic anisotropy
• Important for understanding and designing magnetic materials
• Plays role in spintronics and quantum information applications

Advanced concepts

• Cutting-edge theories and phenomena related to Fermi surface physics
• Addresses limitations of conventional models and explains exotic material behavior
• Connects Fermi surface concepts to broader themes in condensed matter physics

Fermi liquid theory

• Describes low-energy excitations in interacting fermion systems
• Explains why many materials behave similarly to non-interacting electron gas
• Introduces concept of quasiparticles with renormalized mass and interactions
• Predicts temperature dependence of thermodynamic and transport properties
• Breaks down in certain systems, leading to non-Fermi liquid behavior

Non-Fermi liquids

• Systems where Fermi liquid theory fails to describe low-energy excitations
• Occurs in strongly correlated electron systems and low-dimensional materials
• Examples include strange metals in high-temperature superconductors
• Exhibit unusual power-law dependences in physical properties
• Challenges conventional understanding of metallic states

Luttinger's theorem

• Relates volume enclosed by Fermi surface to total number of electrons
• Holds even in presence of strong electron-electron interactions
• Important for understanding stability of Fermi surface in many-body systems
• Violations of Luttinger's theorem indicate exotic phases of matter
• Connects Fermi surface physics to fundamental principles of many-body quantum mechanics