Spin density wave
Encyclopedia
Spin-density wave and charge-density wave (CDW) 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 
. Other low-temperature ground state
s that occur in such materials are superconductivity
, ferromagnetism
and antiferromagnetism
. The transition to the ordered states is driven by the condensation energy which is approximately
where 
is the magnitude of the energy gap opened by the transition. Note that SDWs are distinct from spin wave
s, which are an excitation mode of ferromagnets and antiferromagnets.
Fundamentally SDWs and CDWs involve the development of a superstructure in the form of a periodic modulation in the density of the electronic spins
and charges with a characteristic spatial frequency
that does not transform according to the symmetry group that describes the ionic positions.
The new periodicity associated with CDWs can easily be observed using scanning tunneling microscopy or electron diffraction
while the more elusive SDWs are typically observed via neutron diffraction
or susceptibility
measurements. If the new periodicity is a rational fraction or multiple of the lattice constant
, the density wave is said to be commensurate; otherwise the density wave is termed incommensurate.
Some solids with a high
form density waves while others choose a superconducting or magnetic ground state at low temperatures, because of the existence of nesting vectors in the materials' Fermi surface
s. The concept of a nesting vector is illustrated in the Figure for the famous case of Cr, which transitions from a paramagnetic to SDW state at a Néel temperature of 311 K. Cr is a body-centered cubic metal whose Fermi surface features many parallel boundaries between electron pockets centered at
and hole pockets at H. These large parallel regions can be spanned by the nesting wavevector 
shown in red. The real-space periodicity of the resulting spin-density wave is given by 
. The formation of an SDW with a corresponding spatial frequency causes the opening of an energy gap that lowers the system's energy. The existence of the SDW in Cr was first posited in 1960 by Albert Overhauser
of Purdue. Cliff Shull of MIT won the Nobel Prize in Physics
in 1994 for his experimental observation of the Cr SDW. The theory of CDWs was first put forth by Rudolf Peierls
of Oxford University, who was trying to explain superconductivity.
Many low-dimensional solids have anisotropic Fermi surfaces that have prominent nesting vectors. Well-known examples include layered materials like
, 
and 
(a Chevrel phase) and quasi-1D organic conductors like TMTSF or TTF-TCNQ. CDWs are also common at the surface of solids where they are more commonly called surface reconstruction
s or even dimerization. Surface so often support CDWs because they can be described by two-dimensional Fermi surfaces like those of layered materials.
The most intriguing properties of density waves are their dynamics. Under an appropriate electric field or magnetic field, a density wave will "slide" in the direction indicated by the field due to the electrostatic or magnetostatic force. Typically the sliding will not begin until a "depinning" threshold field is exceeded where the wave can escape from a potential well caused by a defect. The hysteretic
motion of density waves is therefore not unlike that of dislocations or magnetic domains. The current-voltage curve of a CDW solid therefore shows a very high electrical resistance up to the depinning voltage, above which it shows a nearly ohmic
behavior. Under the depinning voltage (which depends on the purity of the material), the crystal is an insulator.
. Other low-temperature ground stateGround state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...
s that occur in such materials are superconductivity
Superconductivity
Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
, ferromagnetism
Ferromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...
and antiferromagnetism
Antiferromagnetism
In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usuallyrelated to the spins of electrons, align in a regular pattern with neighboring spins pointing in opposite directions. This is, like ferromagnetism and ferrimagnetism, a manifestation of ordered magnetism...
. The transition to the ordered states is driven by the condensation energy which is approximately
where is the magnitude of the energy gap opened by the transition. Note that SDWs are distinct from spin waveSpin wave
Spin waves are propagating disturbances in the ordering of magnetic materials. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as magnons, which are boson modes of the spin lattice...
s, which are an excitation mode of ferromagnets and antiferromagnets.
Fundamentally SDWs and CDWs involve the development of a superstructure in the form of a periodic modulation in the density of the electronic spins
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
and charges with a characteristic spatial frequency
that does not transform according to the symmetry group that describes the ionic positions.The new periodicity associated with CDWs can easily be observed using scanning tunneling microscopy or electron diffraction
Electron diffraction
Electron diffraction refers to the wave nature of electrons. However, from a technical or practical point of view, it may be regarded as a technique used to study matter by firing electrons at a sample and observing the resulting interference pattern...
while the more elusive SDWs are typically observed via neutron diffraction
Neutron diffraction
Neutron diffraction or elastic neutron scattering is the application of neutron scattering to the determination of the atomic and/or magnetic structure of a material: A sample to be examined is placed in a beam of thermal or cold neutrons to obtain a diffraction pattern that provides information of...
or susceptibility
Susceptibility
*In physics, the susceptibility of a material or substance describes its response to an applied field. There are many kinds of susceptibilities, for example:These two susceptibilities are particular examples of a linear response function;...
measurements. If the new periodicity is a rational fraction or multiple of the lattice constant
Lattice constant
The 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...
, the density wave is said to be commensurate; otherwise the density wave is termed incommensurate.
Some solids with a high
form density waves while others choose a superconducting or magnetic ground state at low temperatures, because of the existence of nesting vectors in the materials' Fermi surfaceFermi surface
In condensed matter physics, the Fermi surface is an abstract boundary useful for predicting the thermal, electrical, magnetic, and optical properties of metals, semimetals, and doped semiconductors. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline...
s. The concept of a nesting vector is illustrated in the Figure for the famous case of Cr, which transitions from a paramagnetic to SDW state at a Néel temperature of 311 K. Cr is a body-centered cubic metal whose Fermi surface features many parallel boundaries between electron pockets centered at
and hole pockets at H. These large parallel regions can be spanned by the nesting wavevector shown in red. The real-space periodicity of the resulting spin-density wave is given by . The formation of an SDW with a corresponding spatial frequency causes the opening of an energy gap that lowers the system's energy. The existence of the SDW in Cr was first posited in 1960 by Albert OverhauserAlbert Overhauser
Albert W. Overhauser is an American physicist and member of the National Academy of Sciences. He is best known for his theory of dynamic nuclear polarization, also known as the Overhauser Effect....
of Purdue. Cliff Shull of MIT won the Nobel Prize in Physics
Nobel Prize in Physics
The Nobel Prize in Physics is awarded once a year by the Royal Swedish Academy of Sciences. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895 and awarded since 1901; the others are the Nobel Prize in Chemistry, Nobel Prize in Literature, Nobel Peace Prize, and...
in 1994 for his experimental observation of the Cr SDW. The theory of CDWs was first put forth by Rudolf Peierls
Rudolf Peierls
Sir Rudolf Ernst Peierls, CBE was a German-born British physicist. Rudolf Peierls had a major role in Britain's nuclear program, but he also had a role in many modern sciences...
of Oxford University, who was trying to explain superconductivity.
Many low-dimensional solids have anisotropic Fermi surfaces that have prominent nesting vectors. Well-known examples include layered materials like
, and (a Chevrel phase) and quasi-1D organic conductors like TMTSF or TTF-TCNQ. CDWs are also common at the surface of solids where they are more commonly called surface reconstructionSurface reconstruction
Surface reconstruction refers to the process by which atoms at the surface of a crystal assume a different structure than that of the bulk. Surface reconstructions are important in that they help in the understanding of surface chemistry for various materials, especially in the case where another...
s or even dimerization. Surface so often support CDWs because they can be described by two-dimensional Fermi surfaces like those of layered materials.
The most intriguing properties of density waves are their dynamics. Under an appropriate electric field or magnetic field, a density wave will "slide" in the direction indicated by the field due to the electrostatic or magnetostatic force. Typically the sliding will not begin until a "depinning" threshold field is exceeded where the wave can escape from a potential well caused by a defect. The hysteretic
Hysteresis
Hysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...
motion of density waves is therefore not unlike that of dislocations or magnetic domains. The current-voltage curve of a CDW solid therefore shows a very high electrical resistance up to the depinning voltage, above which it shows a nearly ohmic
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...
behavior. Under the depinning voltage (which depends on the purity of the material), the crystal is an insulator.