The Stoner criterion is a condition to be fulfilled for the
ferromagnetic order to arise in a simplified model of a solid. It is named after
Edmund Clifton Stoner
Edmund Clifton Stoner FRS (2 October 1899 – 27 December 1968) was a British theoretical physicist. He is principally known for his work on the origin and nature of itinerant ferromagnetism (the type of ferromagnetic behaviour associated with ...
.
Stoner model of ferromagnetism
Ferromagnetism
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
ultimately stems from Pauli exclusion. The simplified model of a solid which is nowadays usually called the Stoner model, can be formulated in terms of dispersion relations for
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
up and spin down electrons,
:
where the second term accounts for the
exchange energy
In chemistry and physics, the exchange interaction (with an exchange energy and exchange term) is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classica ...
,
is the Stoner parameter,
(
) is the dimensionless density
[Having a lattice model in mind, is the number of lattice sites and is the number of spin-up electrons in the whole system. The density of states has the units of inverse energy. On a finite lattice, is replaced by discrete levels and then .] of spin up (down) electrons and
is the
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the d ...
of spinless electrons where the electron-electron interaction is disregarded. If
is fixed,
can be used to calculate the total energy of the system as a function of its polarization
. If the lowest total energy is found for
, the system prefers to remain
paramagnetic
Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, d ...
but for larger values of
, polarized
ground states occur. It can be shown that for
:
the
state will spontaneously pass into a polarized one. This is the Stoner criterion, expressed in terms of the
density of states
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
at the
Fermi energy
The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature.
In a Fermi ga ...
.
A non-zero
state may be favoured over
even before the Stoner criterion is fulfilled.
Relationship to the Hubbard model
The Stoner model can be obtained from the
Hubbard model
The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems.
It is particularly useful in solid-state physics. The model is named for John Hubbard.
The Hubbard model states that each el ...
by applying the
mean-field approximation
In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random ( stochastic) models by studying a simpler model that approximates the original by averaging over degrees o ...
. The particle density operators are written as their mean value
plus fluctuation
and the product of spin-up and spin-down fluctuations is neglected. We obtain
:
With the third term included, which was omitted in the definition above, we arrive at the better-known form of the Stoner criterion
:
Notes
References
*
Stephen Blundell Stephen John Blundell (born 1967) is a professor of physics at the University of Oxford. He was previously head of Condensed Matter Physics at Oxford, and is also a professorial fellow of Mansfield College, Oxford. His research is concerned with usi ...
, Magnetism in Condensed Matter (Oxford Master Series in Physics).
*
* {{cite journal, last=Stoner, first=Edmund Clifton, title=Collective electron ferronmagnetism, journal=Proc. R. Soc. Lond. A, date=April 1938, volume=165, issue=922, pages=372–414, doi=10.1098/rspa.1938.0066, bibcode=1938RSPSA.165..372S, doi-access=free
Ferromagnetism