In
condensed matter physicsCondensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter. In particular, it is concerned with the "condensed" phases that appear whenever the number of constituents in a system is extremely large and the interactions between the...
, the
Fermi surface is an abstract boundary useful for predicting the thermal, electrical, magnetic, and optical properties of
metalA metal is a chemical element that is a good conductor of both electricity and heat, forms cations and ionic bonds with non-metals. In chemistry, a metal is an element, compound, or alloy characterized by high electrical conductivity. In a metal, atoms readily lose electrons to form positive ions...
s,
semimetalA semimetal is a material with a small overlap in the energy of the conduction band and valence bands.However, the bottom of the conduction band is typically situated in a different part of momentum space than the top of the valence band...
s, and doped
semiconductorA semiconductor is a material that has an electrical resistivity between that of a conductor and an insulator, that is, generally in the range 10
3 Siemens/cm to 10
−8 S/cm. Devices made from semiconductor materials are the foundation of modern electronics, including radio,...
s. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline lattice and from the occupation of
electronic energy bandsIn solid-state physics, the electronic band structure of a solid describes ranges of energy that an electron is "forbidden" or "allowed" to have. It is due to the diffraction of the quantum mechanical electron waves in the periodic crystal lattice...
. The existence of a Fermi surface is a direct consequence of the
Pauli exclusion principleThe Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. It states that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement of this principle is that, for two identical fermions, the total wave function...
, which allows a maximum of one electron per quantum state.
Consider a spinless ideal
Fermi gasFermi gas is a physical model assuming a collection of non-interacting fermions. It is the quantum mechanical version of an ideal gas, for the case of fermionic particles. The behavior of Electrons in metals and semiconductors and neutrons in a neutron star can be approximated by treating them as...
of particles.
In
condensed matter physicsCondensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter. In particular, it is concerned with the "condensed" phases that appear whenever the number of constituents in a system is extremely large and the interactions between the...
, the
Fermi surface is an abstract boundary useful for predicting the thermal, electrical, magnetic, and optical properties of
metalA metal is a chemical element that is a good conductor of both electricity and heat, forms cations and ionic bonds with non-metals. In chemistry, a metal is an element, compound, or alloy characterized by high electrical conductivity. In a metal, atoms readily lose electrons to form positive ions...
s,
semimetalA semimetal is a material with a small overlap in the energy of the conduction band and valence bands.However, the bottom of the conduction band is typically situated in a different part of momentum space than the top of the valence band...
s, and doped
semiconductorA semiconductor is a material that has an electrical resistivity between that of a conductor and an insulator, that is, generally in the range 10
3 Siemens/cm to 10
−8 S/cm. Devices made from semiconductor materials are the foundation of modern electronics, including radio,...
s. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline lattice and from the occupation of
electronic energy bandsIn solid-state physics, the electronic band structure of a solid describes ranges of energy that an electron is "forbidden" or "allowed" to have. It is due to the diffraction of the quantum mechanical electron waves in the periodic crystal lattice...
. The existence of a Fermi surface is a direct consequence of the
Pauli exclusion principleThe Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. It states that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement of this principle is that, for two identical fermions, the total wave function...
, which allows a maximum of one electron per quantum state.
Theory
Consider a spinless ideal
Fermi gasFermi gas is a physical model assuming a collection of non-interacting fermions. It is the quantum mechanical version of an ideal gas, for the case of fermionic particles. The behavior of Electrons in metals and semiconductors and neutrons in a neutron star can be approximated by treating them as...
of particles. According to
Fermi-Dirac statisticsFermi-Dirac statistics is a part of the science of physics that describes the energies of single particles in a system comprising many identical particles that obey the Pauli Exclusion Principle...
, the mean occupation number of a state with energy is given by
where,
- is the mean occupation number
- is the energy of the state
- is the fugacity
Fugacity is a measure of a chemical potential in the form of 'adjusted pressure'. It reflects the tendency of a substance to prefer one phase over another, and can be literally defined as “the tendency to flee or escape”. At a fixed temperature and pressure, a homogeneous substance will have a...
(which at low temperatures is called the Fermi energyThe Fermi energy is a concept in quantum mechanics usually referring to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature...
)
Suppose we consider the limit . Then we have,
By the
Pauli exclusion principleThe Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. It states that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement of this principle is that, for two identical fermions, the total wave function...
, no two particles can be in the same state. Therefore, in the state of lowest energy, the particles fill up all energy levels till , which is equivalent to saying that is the energy level below which there are exactly states.
In momentum space, these particles fill up a sphere of radius , the surface of which is called the
Fermi surface
The linear response of a metal to an electric, magnetic or thermal gradient is determined by the shape of the Fermi surface, because currents are due to changes in the occupancy of states near the Fermi energy. Free-electron Fermi surfaces are spheres of radius determined by the valence electron concentration where is the reduced Planck's constant. A material whose Fermi level falls in a gap between bands is an
insulatorAn insulator, also called a dielectric, is a material that resists the flow of electric current. An insulating material has atoms with tightly bonded valence electrons. These materials are used in parts of electrical equipment, also called insulators or insulation, intended to support or separate...
or semiconductor depending on the size of the bandgap. When a material's Fermi level falls in a bandgap, there is no Fermi surface.
Materials with complex crystal structures can have quite intricate Fermi surfaces. The figure illustrates the anisotropic Fermi surface of graphite, which has both electron and hole pockets in its Fermi surface due to multiple bands crossing the Fermi energy along the direction. Often in a metal the Fermi surface radius is larger than the size of the first
Brillouin zoneIn mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell of the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice...
which results in a portion of the Fermi surface lying in the second (or higher) zones. As with the band structure itself, the Fermi surface can be displayed in an extended-zone scheme where is allowed to have arbitrarily large values or a reduced-zone scheme where wavevectors are shown
moduloIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus...
(in the 1-dimensional case) where a is the
lattice constantThe Lattice Constant [or lattice parameter] refers to the constant distance between unit cells in a crystal lattice. Lattices in three dimensions generally have three lattice constants, referred to as a, b, and c. However, in the special case of cubic crystal structures, all of the constants are...
. In the three dimensional case the reduced zone scheme means that from any wavevector there is an appropriate number of reciprocal lattice vectors subtracted that the new now is closer to the origin in -space than to any . Solids with a large density of states at the Fermi level become unstable at low temperatures and tend to form ground states where the condensation energy comes from opening a gap at the Fermi surface. Examples of such ground states are superconductors, ferromagnets,
Jahn-Teller distortionsThe Jahn–Teller effect, sometimes also known as Jahn–Teller distortion, or the Jahn–Teller theorem, describes the geometrical distortion of non-linear molecules under certain situations. This electronic effect is named after Hermann Arthur Jahn and Edward Teller, who proved, using group theory,...
and
spin density waveSpin-density wave and charge-density wave are names for two similar low-energy ordered states of solids. Both these states occur at low temperature in anisotropic, low-dimensional materials or in metals that have high densities of states at the Fermi level...
s.
The state occupancy of
fermionIn particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle. Thus, if more than one...
s like electrons is governed by
Fermi-Dirac statisticsFermi-Dirac statistics is a part of the science of physics that describes the energies of single particles in a system comprising many identical particles that obey the Pauli Exclusion Principle...
so at finite temperatures the Fermi surface is accordingly broadened. In principle all fermion energy level populations are bound by a Fermi surface although the term is not generally used outside of condensed-matter physics.
Experimental determination
Electronic Fermi surfaces have been measured through observation of the oscillation of transport properties in magnetic fields , for example the
de Haas-van Alphen effectThe de Haas–van Alphen effect, often abbreviated to dHvA, was discovered in 1930 by Wander Johannes de Haas and PM van Alphen.The dHvA effect is a quantum mechanical effect in which certain physical quantities of a pure metal crystal oscillate as the intensity of an applied magnetic field is...
(dHvA) and the
Shubnikov-De Haas effectAn oscillation in the conductivity of a material that occurs at low temperatures in the presence of very intense, time varying magnetic fields, the Shubnikov–de Haas effect is a macroscopic manifestation of the inherent quantum mechanical nature of matter...
(SdH). The former is an oscillation in
magnetic susceptibilityIn electromagnetism the magnetic susceptibility is the degree of magnetization of a material in response to an applied magnetic field....
and the latter in
resistivityElectrical resistivity is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electrical charge...
. The oscillations are periodic versus and occur because of the quantization of energy levels in the plane perpendicular to a magnetic field, a phenomenon first predicted by
Lev LandauLev Davidovich Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics...
. The new states are called Landau levels and are separated by an energy where is called the
cyclotron frequencyElectron cyclotron resonance is a phenomenon observed both in plasma physics and condensed matter physics. An electron in a static and uniform magnetic field will move in a circle due to the Lorentz force...
, is the electronic charge, is the electron
effective massIn solid state physics, a particle's effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. It can be shown that electrons and holes in a crystal respond to electric and magnetic fields almost as if they were particles with a mass dependent upon their...
and is the
speed of lightIn physics, the speed of light is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space . Its value is 299,792,458 metres per second...
. In a famous result,
Lars OnsagerLars Onsager was a Norwegian–American physical chemist and theoretical physicist, winner of the 1968 Nobel Prize in Chemistry.He had the Gibbs Professorship of Theoretical Chemistry at Yale University....
proved that the period of oscillation is related to the cross-section of the Fermi surface (typically given in ) perpendicular to the magnetic field direction by the equation . Thus the determination of the periods of oscillation for various applied field directions allows mapping of the Fermi surface.
Observation of the dHvA and SdH oscillations requires magnetic fields large enough that the circumference of the cyclotron orbit is smaller than a
mean free pathIn physics the mean free path of a particle is the average distance covered by a particle between successive impacts.. Alternatively, it is the distance at which the intensity of particles drops by 1/e.-Derivation:...
. Therefore dHvA and SdH experiments are usually performed at high-field facilities like the
High Field Magnet Laboratory in Netherlands,
Grenoble High Magnetic Field Laboratory in France, the
Tsukuba Magnet Laboratory in Japan or the
National High Magnetic Field Laboratory in the United States.
The most direct experimental technique to resolve the electronic structure of crystals in the momentum-energy space (see
reciprocal latticeIn crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such thatfor all lattice point position vectors R...
), and, consequently, the Fermi surface, is the angle resolved photoemission spectroscopy (
ARPESAngle resolved photoemission spectroscopy , also known as ARUPS , is a direct experimental technique to observe the distribution of the electrons in the reciprocal space of solids...
). An example of the
Fermi surface of superconducting cupratesThe electronic structure of superconducting cuprates, also called high temperature superconductors , is highly anisotropic. First, the cuprates are almost two-dimensional : the conducting electrons are mainly localized in the CuO2 layers, the common building blocks of all HTSC compounds...
measured by
ARPESAngle resolved photoemission spectroscopy , also known as ARUPS , is a direct experimental technique to observe the distribution of the electrons in the reciprocal space of solids...
is shown in figure.
With positron annihilation the two photons carry the momentum of the electron away; as the momentum of a thermalized positron is negligible, in this way also information about the momentum distribution can be obtained. Because the positron can be polarized, also the momentum distribution for the two
spinIn particle physics and quantum mechanics, spin is a fundamental characteristic property of elementary particles including the force carriers , composite particles , and atomic nuclei....
states in magnetized materials can be obtained. Another advantage with De Haas-Van Alphen-effect is that the technique can be applied to non-dilute alloys. In this way the first determination of a smeared Fermi surface in a 30% alloy was obtained in 1978.
See also
- Fermi energy
The Fermi energy is a concept in quantum mechanics usually referring to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature...
- Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell of the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice...
- Fermi surface of superconducting cuprates
The electronic structure of superconducting cuprates, also called high temperature superconductors , is highly anisotropic. First, the cuprates are almost two-dimensional : the conducting electrons are mainly localized in the CuO2 layers, the common building blocks of all HTSC compounds...
- Kelvin probe force microscope
Kelvin probe force microscopy , also known as surface potential microscopy, is a noncontact variant of atomic force microscopy that was invented in 1991. With KPFM, the work function of surfaces can be observed at atomic or molecular scales...