2.5 Fermi surface
7 min read•august 21, 2024
The 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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Top images from around the web for Definition and significance
solid state physics - The Fermi Surface of the Free Electron Model for T>0K - Physics Stack Exchange View original
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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
- represents highest occupied energy state at absolute zero temperature
- 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 and electron distribution in solids
Brillouin zone relationship
- Fermi surface typically represented within first
- 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 and
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
- 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
- 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
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
- 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
- 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 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 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
- Important for understanding stability of Fermi surface in many-body systems
- Violations of indicate exotic phases of matter
- Connects Fermi surface physics to fundamental principles of many-body quantum mechanics
Key Terms to Review (28)
Angle-resolved photoemission spectroscopy: Angle-resolved photoemission spectroscopy (ARPES) is a powerful experimental technique used to study the electronic structure of materials by measuring the energy and momentum of electrons emitted from a sample when it is illuminated by ultraviolet or X-ray radiation. This method provides critical information about the band structure, Fermi surface, and other electronic properties of solids, which are essential for understanding phenomena like superconductivity, surface states, and quantum phase transitions.
Band Structure: Band structure describes the range of energy levels that electrons can occupy in a solid and the gaps between these energy levels. This concept is crucial for understanding the electrical, thermal, and optical properties of materials, as it explains how electrons behave in different conditions and influences phenomena such as conductivity and magnetism.
Berry Phase: Berry phase is a geometric phase acquired over the course of a cycle when a quantum system is subjected to adiabatic, cyclic changes in its parameters. This concept is crucial in understanding phenomena in condensed matter physics, as it connects to the geometric properties of the wavefunctions, which can influence observable physical effects such as the behavior of electrons in various materials, including topological insulators and systems experiencing the quantum Hall effect.
Brillouin zone: A Brillouin zone is a uniquely defined region in reciprocal space that contains all the distinct wave vectors for a periodic lattice. It plays a crucial role in understanding the electronic properties of solids, particularly in defining energy bands and the behavior of electrons under periodic potentials.
Closed Fermi Surface: A closed Fermi surface refers to a specific type of Fermi surface in the momentum space that is entirely enclosed and does not extend infinitely. This surface is critical in understanding the electronic properties of materials, particularly in metals and semiconductors, where it helps determine how electrons occupy energy states and influences conduction properties. The shape and topology of the closed Fermi surface can give insight into the material's electronic behavior, revealing whether it behaves as a metal, insulator, or semiconductor.
De Haas-van Alphen effect: The de Haas-van Alphen effect is a phenomenon observed in metals and some other materials where the magnetic susceptibility exhibits oscillatory behavior as a function of an applied magnetic field. This effect arises from the quantization of electron orbits in the presence of a magnetic field, providing valuable insights into the Fermi surface properties and electronic structure of materials. It helps in understanding the geometrical and topological aspects of the Fermi surface, making it crucial for studying electron dynamics in solids.
Electron-electron interactions: Electron-electron interactions refer to the forces and effects that arise when electrons interact with each other due to their electric charge. These interactions are essential in determining the electronic properties of materials, as they influence phenomena such as screening, correlation effects, and the overall behavior of electrons in solids. Understanding these interactions is crucial for modeling systems like the Fermi surface and analyzing the energy levels in tight-binding models.
Electron-phonon coupling: Electron-phonon coupling refers to the interaction between electrons and phonons, which are quantized modes of vibrations in a crystal lattice. This coupling plays a crucial role in various physical properties of materials, such as electrical conductivity, thermal conductivity, and superconductivity. Understanding this interaction helps to explain phenomena like the formation of Cooper pairs in superconductors and how the Fermi surface can be affected by lattice vibrations.
F. Wilczek: F. Wilczek is a theoretical physicist known for his groundbreaking work in the field of quantum physics, particularly for contributions to the understanding of the strong force and quantum chromodynamics (QCD). His work has significant implications in condensed matter physics, especially in the context of Fermi surfaces, where understanding particle interactions at different energy levels is crucial.
Fermi Energy: Fermi energy is the highest energy level that electrons occupy at absolute zero temperature in a solid. It plays a crucial role in understanding how electrons fill available energy states in a material, determining the electronic properties and behaviors of metals and semiconductors. The Fermi energy is also central to concepts like the density of states and the shape of the Fermi surface, which are important for analyzing how electrons contribute to conduction and other properties.
Fermi Level: The Fermi level is the highest energy level occupied by electrons in a solid at absolute zero temperature. It plays a crucial role in determining the electrical properties of materials, as it helps to distinguish between conductors, insulators, and semiconductors, and influences how energy bands are filled with electrons.
Fermi Liquid Theory: Fermi Liquid Theory is a theoretical framework that describes the behavior of interacting fermions at low temperatures, positing that they can be treated as a collection of quasi-particles that behave like non-interacting particles. This theory is crucial for understanding the properties of metals and liquid helium, where the interactions between particles do not disrupt the overall Fermi surface, which defines the momentum distribution of electrons in a solid.
Fermi surface: The Fermi surface is the boundary in momentum space that separates occupied from unoccupied electron states at absolute zero temperature. It plays a crucial role in determining the electronic properties of materials, as it defines how electrons behave in solids and influences phenomena like electrical conductivity and heat capacity.
Fermi surface nesting: Fermi surface nesting is a phenomenon where the shape of the Fermi surface in a solid allows for certain wave vectors to connect points on the surface, promoting enhanced electronic interactions. This feature can lead to instabilities in the electronic structure of materials, often resulting in phase transitions such as charge density waves or magnetism. Understanding Fermi surface nesting is crucial for interpreting the electronic properties and behaviors of many condensed matter systems.
High-temperature superconductors: High-temperature superconductors are materials that exhibit superconductivity at temperatures significantly higher than traditional superconductors, typically above 77 K. These materials often have complex crystal structures and are primarily composed of copper oxide compounds, which play a critical role in their unique electronic properties and the underlying mechanisms of superconductivity.
K-space: K-space, or reciprocal space, is a conceptual framework used in physics to represent wave vectors associated with particles, particularly in the context of solids. It provides a way to understand the periodic structure of crystals by translating real-space coordinates into momentum space, allowing for the analysis of electronic band structures and scattering processes. K-space is closely tied to important concepts such as Brillouin zones, Fermi surfaces, and models of electron behavior in solids.
L. D. Landau: L. D. Landau was a prominent Soviet physicist known for his contributions to condensed matter physics, particularly in the development of the theory of Fermi liquids and phase transitions. His work fundamentally changed the understanding of electron interactions in metals and superconductors, linking microscopic behaviors to macroscopic properties through concepts like the Fermi surface and quasiparticles.
Landau Theory: Landau Theory is a theoretical framework used to describe phase transitions and critical phenomena in physical systems, particularly focusing on the concept of order parameters. It connects thermodynamic properties to microscopic interactions and offers a mathematical approach to understand various states of matter, such as solid, liquid, and gas, as well as magnetic and superconducting states.
Lifshitz Transition: A Lifshitz transition is a quantum phase transition that occurs when the topology of the Fermi surface changes as a function of external parameters, such as pressure, temperature, or doping. This change can lead to significant alterations in the electronic properties of materials, connecting deeply with the concept of Fermi surfaces and providing insights into the behavior of topological semimetals.
Luttinger's Theorem: Luttinger's Theorem states that the number of particles in a Fermi liquid can be determined solely from the topology of the Fermi surface, regardless of interactions between particles. This theorem is crucial for understanding the properties of many-body systems, particularly in how the shape and volume of the Fermi surface relate to the system's behavior at low temperatures.
Metallicity: Metallicity refers to the abundance of elements heavier than helium in a material, typically quantified as the ratio of the number of such elements to the total number of nucleons. This concept is crucial for understanding the electronic properties and conductivity of materials, especially metals, where higher metallicity indicates a greater presence of free electrons available for conduction. The variation in metallicity influences various physical behaviors, including thermal and electrical conductivity, as well as magnetic properties.
Non-Fermi Liquids: Non-Fermi liquids are a class of materials where the electronic properties deviate from the standard behavior predicted by Fermi liquid theory, particularly at low temperatures. In these systems, the concept of quasiparticles, which are a cornerstone of Fermi liquid theory, breaks down, leading to unusual temperature dependencies and collective behaviors that cannot be explained by traditional theories. This behavior is often associated with strongly correlated electron systems and can significantly impact their Fermi surface characteristics.
Open Fermi Surface: An open Fermi surface refers to a type of Fermi surface in a solid where the energy states at the Fermi level do not form a closed surface in momentum space. Instead, it extends infinitely in one or more directions, which can lead to unique electronic properties and behaviors. This kind of surface is often associated with materials that have low dimensionality or certain types of electron interactions, influencing conductivity and other physical properties.
Quantum Hall Effect: The quantum Hall effect is a quantum phenomenon observed in two-dimensional electron systems under low temperatures and strong magnetic fields, where the Hall conductivity becomes quantized in integer or fractional values. This effect is crucial for understanding electron behavior in low-dimensional systems and has deep connections to topological phases of matter and various advanced materials.
Quantum Oscillations: Quantum oscillations are phenomena observed in solid-state physics that arise due to the quantum mechanical behavior of electrons in a material, particularly under the influence of a magnetic field. These oscillations reflect the periodic nature of the energy levels and are directly related to the shape and characteristics of the Fermi surface, as well as revealing insights into topological features of materials.
Semimetallic behavior: Semimetallic behavior refers to the electronic properties of certain materials that exhibit characteristics of both metals and semiconductors. These materials possess a small overlap between the conduction band and valence band, allowing them to conduct electricity under specific conditions while still displaying some insulating properties. This unique blend makes semimetals interesting for applications in electronics, where their conductivity can change with temperature or doping.
Superconductivity: Superconductivity is a phenomenon where a material can conduct electricity without any resistance when cooled below a certain critical temperature. This unique property allows superconductors to carry electric current with zero energy loss, which has implications for various advanced technologies and is deeply connected to the behavior of electrons in materials, crystal structures, and quantum mechanics.
Topological Insulators: Topological insulators are materials that behave as insulators in their bulk while supporting conducting states on their surfaces or edges. This unique property arises from the topological order of the electronic band structure, which distinguishes them from ordinary insulators, allowing for robust surface states that are protected against scattering by impurities or defects.