44
S. J. Blundell
Fig. 2.2 The two different
conventions used for the
definition of J . In this
chapter (and in [2]), we are
using the 2J convention (so
that 2J is the energy
associated with a single
pairwise interaction between
two spins). The various
alternative expressions that
one can use for the
Heisenberg interaction are
shown under the heading
‘many spins’, as well as an
expression for a
one-dimensional (1D) chain
of spins
“J convention”
“2J convention”
J
2J
2 spins:
2 spins:
−JS 1 · S 2
−2JS 1 · S 2
many spins:
many spins:
− i>j
J ij S i · S j
−2 i>j
J ij S i · S j
− 1
2 ij
J ij S i · S j
− ij
J ij S i · S j
− J
2 ij
S i · S j
−J ij
S i · S j
−J i
S i · S i+1 (1D)
−2J i
S i · S i+1 (1D)
2.2.2 Indirect Exchange
If the electrons on neighbouring magnetic atoms interact via an exchange interaction,
this is known as direct exchange. This is because the exchange interaction proceeds
directly without the need for an intermediary, and this was considered in the previous
section.
Very often direct exchange cannot be an important mechanism in controlling the
magnetic properties because there is insufficient direct overlap between neighbouring
magnetic orbitals. For example, in rare earths the 4 f electrons are strongly localized
and lie very close to the nucleus, with little probability density extending significantly
further than about a tenth of the interatomic spacing. This means that the direct
exchange interaction is unlikely to be very effective in rare earths. Even in transition
metals, such as Fe, Co and Ni, where the 3d orbitals extend further from the nucleus,
it is extremely difficult to justify why direct exchange should lead to the observed
magnetic properties. These materials are metals which means that the role of the
conduction electrons should not be neglected, and a correct description needs to be
taken into account of both the localized and band character of the electrons.
In metals the exchange interaction between magnetic ions can be mediated by the
conduction electrons. A localized magnetic moment spin-polarizes the conduction
electrons and this polarization in turn couples to a neighbouring localized magnetic
moment a distance r away. The exchange interaction is thus indirect because it does
not involve direct coupling between magnetic moments. It is known as the RKKY
interaction (or also as itinerant exchange). The name RKKY is used because of the
initial letters of the surnames of the discoverers of the effect, Ruderman, Kittel,
Kasuya and Yosida [6–8]. The coupling takes the form of an r -dependent exchange
interaction J RKKY (r ) given by
S. J. Blundell
Fig. 2.2 The two different
conventions used for the
definition of J . In this
chapter (and in [2]), we are
using the 2J convention (so
that 2J is the energy
associated with a single
pairwise interaction between
two spins). The various
alternative expressions that
one can use for the
Heisenberg interaction are
shown under the heading
‘many spins’, as well as an
expression for a
one-dimensional (1D) chain
of spins
“J convention”
“2J convention”
J
2J
2 spins:
2 spins:
−JS 1 · S 2
−2JS 1 · S 2
many spins:
many spins:
− i>j
J ij S i · S j
−2 i>j
J ij S i · S j
− 1
2 ij
J ij S i · S j
− ij
J ij S i · S j
− J
2 ij
S i · S j
−J ij
S i · S j
−J i
S i · S i+1 (1D)
−2J i
S i · S i+1 (1D)
2.2.2 Indirect Exchange
If the electrons on neighbouring magnetic atoms interact via an exchange interaction,
this is known as direct exchange. This is because the exchange interaction proceeds
directly without the need for an intermediary, and this was considered in the previous
section.
Very often direct exchange cannot be an important mechanism in controlling the
magnetic properties because there is insufficient direct overlap between neighbouring
magnetic orbitals. For example, in rare earths the 4 f electrons are strongly localized
and lie very close to the nucleus, with little probability density extending significantly
further than about a tenth of the interatomic spacing. This means that the direct
exchange interaction is unlikely to be very effective in rare earths. Even in transition
metals, such as Fe, Co and Ni, where the 3d orbitals extend further from the nucleus,
it is extremely difficult to justify why direct exchange should lead to the observed
magnetic properties. These materials are metals which means that the role of the
conduction electrons should not be neglected, and a correct description needs to be
taken into account of both the localized and band character of the electrons.
In metals the exchange interaction between magnetic ions can be mediated by the
conduction electrons. A localized magnetic moment spin-polarizes the conduction
electrons and this polarization in turn couples to a neighbouring localized magnetic
moment a distance r away. The exchange interaction is thus indirect because it does
not involve direct coupling between magnetic moments. It is known as the RKKY
interaction (or also as itinerant exchange). The name RKKY is used because of the
initial letters of the surnames of the discoverers of the effect, Ruderman, Kittel,
Kasuya and Yosida [6–8]. The coupling takes the form of an r -dependent exchange
interaction J RKKY (r ) given by