Superexchange, or Kramers–Anderson superexchange, is the strong (usually)
antiferromagnetic
In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usually related to the spins of electrons, align in a regular pattern with neighboring spins (on different sublattices) pointing in opposite directions. ...
coupling
A coupling is a device used to connect two shafts together at their ends for the purpose of transmitting power. The primary purpose of couplings is to join two pieces of rotating equipment while permitting some degree of misalignment or end mov ...
between two next-to-nearest neighbour
cation
An ion () is an atom or molecule with a net electrical charge.
The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by convent ...
s through a non-magnetic
anion
An ion () is an atom or molecule with a net electrical charge.
The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by convent ...
. In this way, it differs from direct exchange, in which there is coupling between nearest neighbor cations not involving an intermediary anion. Superexchange is a result of the electrons having come from the same donor atom and being coupled with the receiving ions' spins. If the two next-to-nearest neighbor positive ions are connected at 90 degrees to the bridging non-magnetic anion, then the interaction can be a
ferromagnetic
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 ...
interaction.
![MnO-superaustausch](https://upload.wikimedia.org/wikipedia/commons/1/1d/MnO-superaustausch.GIF)
Superexchange was proposed by
Hendrik Kramers
Hendrik Anthony "Hans" Kramers (17 December 1894 – 24 April 1952) was a Dutch physicist who worked with Niels Bohr to understand how electromagnetic waves interact with matter and made important contributions to quantum mechanics and statistical ...
in 1934, when he noticed that in crystals like MnO, there are Mn atoms that interact with one another despite having nonmagnetic oxygen atoms between them.
Phillip Anderson later refined Kramers' model in 1950.
A set of semi-empirical rules were developed by
John B. Goodenough and in the 1950s. These rules, now referred to as the Goodenough–Kanamori rules, have proven highly successful in rationalizing the magnetic properties of a wide range of materials on a qualitative level. They are based on the symmetry relations and electron occupancy of the overlapping atomic orbitals (assuming the localized
Heitler–London, or
valence-bond, model is more representative of the chemical bonding than is the delocalized, or
Hund–Mulliken–Bloch, model). Essentially, the Pauli exclusion principle dictates that between two magnetic ions with half-occupied orbitals, which couple through an intermediary non-magnetic ion (e.g. O
2−), the superexchange will be strongly anti-ferromagnetic while the coupling between an ion with a filled orbital and one with a half-filled orbital will be ferromagnetic. The coupling between an ion with either a half-filled or filled orbital and one with a vacant orbital can be either antiferromagnetic or ferromagnetic, but generally favors ferromagnetic. When multiple types of interactions are present simultaneously, the antiferromagnetic one is generally dominant, since it is independent of the intra-atomic exchange term.
For simple cases, the Goodenough–Kanamori rules readily allow the prediction of the net magnetic exchange expected for the coupling between ions. Complications begin to arise in various situations: 1) when
direct exchange and superexchange mechanisms compete with one another; 2) when the cation–anion–cation bond angle deviates away from 180°; 3) when the electron occupancy of the orbitals is non-static, or dynamical; and 4) when spin–orbit coupling becomes important.
Double exchange The double-exchange mechanism is a type of a magnetic exchange that may arise between ions in different oxidation states. First proposed by Clarence Zener, this theory predicts the relative ease with which an electron may be exchanged between two s ...
is a related magnetic coupling interaction proposed by
Clarence Zener
Clarence Melvin Zener (December 1, 1905 – July 2, 1993) was the American physicist who first (1934) described the property concerning the breakdown of electrical insulators. These findings were later exploited by Bell Labs in the development of ...
to account for electrical transport properties. It differs from superexchange in the following manner: in superexchange, the occupancy of the d-shell of the two metal ions is the same or differs by two, and the electrons are localized. For other occupations (double exchange), the electrons are itinerant (delocalized); this results in the material displaying magnetic exchange coupling, as well as metallic conductivity.
Manganese oxide
The ''p'' orbitals from oxygen and ''d'' orbitals from manganese can form a direct exchange.
There is antiferromagnetic order because the singlet state is energetically favoured. This configuration allows a delocalization of the involved electrons due to a lowering of the kinetic energy.
Quantum-mechanical
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
results in an antiferromagnetic interaction of the spins of neighboring Mn atoms with the energy operator (
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
)
:
where ''t''
Mn,O is the so-called hopping energy between a Mn 3''d'' and the oxygen ''p'' orbitals, while ''U'' is a so-called
Hubbard Hubbard may refer to:
Places Canada
*Hubbard, Saskatchewan
*Hubbards, Nova Scotia
Canada/United States
* Mount Hubbard, a mountain on the Alaska/Yukon border
*Hubbard Glacier, a large freshwater glacier in Alaska and Yukon
Greenland
*Hubbard Gla ...
energy for Mn. The expression
is the
scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
between the Mn spin-vector operators (
Heisenberg model).
References
External links
* {{cite book, author=Erik Koch, chapter-url=http://www.cond-mat.de/events/correl12/manuscripts/koch.pdf, chapter=Exchange Mechanisms, editor1= E. Pavarini , editor2=E. Koch , editor3=F. Anders , editor4=M. Jarrell , title= Correlated Electrons: From Models to Materials, publisher= Jülich , year=2012, isbn=978-3-89336-796-2
Condensed matter physics
Magnetic exchange interactions