# Goodenough-Kanamori rule

John B. Goodenough (2008), Scholarpedia, 3(10):7382. | doi:10.4249/scholarpedia.7382 | revision #91326 [link to/cite this article] |

The **Goodenough-Kanamori rule**, first formulated by Goodenough in 1955 (Goodenough 1958; Goodenough 1995) and subsequently provided more rigorous mathematical underpinning by Kanamori (1959), applies to interatomic spin-spin interactions between two atoms, each carrying a net spin, that are mediated by virtual electron transfers between the atoms **(superexchange)** and/or between a shared anion and the two atoms **(semicovalent exchange)**. A virtual electron transfer occurs between overlapping orbitals of electronic states that are separated by an energy ∆E; and a spin-spin interaction mediated by an electron transfer, virtual (or real as in ferromagnetic **double exchange**), is a **kinetic exchange**. Orthogonal orbitals do not overlap, so there is no electron transfer and the exchange interaction between spins in orthogonal orbitals is a **ferromagnetic potential exchange**; it is responsible for the Hund highest-spin rule for the free atom or ion.

The Goodenough-Kanamori rule states that superexchange interactions are antiferromagnetic where the virtual electron transfer is between overlapping orbitals that are each half-filled, but they are ferromagnetic where the virtual electron transfer is from a half-filled to an empty orbital or from a filled to a half-filled orbital. The Goodenough-Kanamori rule is the same for both superexchange and semicovalent exchange.

Where the two cation orbitals overlap the same p orbital of a shared anion as in a 180° cation-anion-cation bridge, it is customary to introduce the virtual electron transfer from the shared anion to the interacting cations first as the covalent component of the cation orbital. The net spin of the cation orbital is not changed by addition of a covalent component, but the covalent component extends the cation wavefunction out over the anions to give an orbital overlap for the superexchange electron transfer. However, a pure semicovalent antiferromagnetic exchange can occur between two empty orbitals provided each cation carries a net spin and the empty orbitals share the same anion p orbital.

## Contents |

## Theoretical basis

The theoretical basis for the Goodenough-Kanamori rule rests on four pillars:

- The spin angular momentum is conserved in an electron transfer, virtual or real.
- The Pauli exclusion principal restricts electron transfer from a half-filled to a half-filled orbital or two-electron transfer from the same anion-p orbital.
- The intraatomic spin-spin potential exchange interaction is ferromagnetic and is determinative where the Pauli exclusion principle is not restrictive.
- Since spins are only oriented up (\(\alpha\)) or down (\(\beta\)) with respect to their spin axis, the angular dependence for an electron-spin transfer between cation spins having an angle \(\theta\) between their spin axes is

- \[ \alpha_1 = \alpha_2 \cos (\theta/2) + \beta_2 \sin (\theta/2) \ :\]

\[\tag{1} \beta_1 = -\alpha_2 \sin (\theta/2) + \beta_2 \cos (\theta/2) \ :\]

Therefore, the expectation value for an electron transfer between atoms carrying a net spin becomes spin-dependent:

- \[\tag{2} t_{ij}^{\uparrow \uparrow} = b_{ij} \cos (\theta/2) \quad \mbox{and} \quad t_{ij}^{\uparrow \downarrow} = b_{ij} \sin (\theta/2) \ :\]

where the spin-independent expectation value for a charge transfer is

- \[\tag{3} b_{ij} = (\psi_i, H'\psi_j) = \epsilon_{ij}(\psi_i, \psi_j) \ :\]

in which \(\epsilon_{ij}\) is a one-electron energy, H′ is the perturbation of the potential of the electron in orbital \(\psi_j\) by the presence of the other cation with an overlapping orbital \(\psi_i\ ,\) and \((\psi_i, \psi_j)\) is the overlap integral for the two orbitals. Orthogonal orbitals have a \((\psi_i, \psi_j) = 0\ .\)

A virtual electron transfer between states of different energy \(\Delta \Epsilon\) can be treated in higher-order perturbation theory provided the ratio \(b_{ij}/\Delta \Epsilon\) is much less than unity. Where \(\Delta \Epsilon\) becomes too small for the perturbation approximation to be reliable, the mathematics breaks down, but the Goodenough-Kanamori rule remains valid.

A virtual electron transfer between half-filled orbitals is restricted by the Pauli exclusion principle to be antiferromagnetic, so a \(t^{\uparrow \downarrow}\) must be used in the second-order perturbation expression \[ \begin{array}{lcl} \Delta \epsilon & \approx & - |t^{\uparrow \downarrow}|^2 / \Delta \Epsilon = - (b^2/\Delta \Epsilon) \sin^2(\theta/2) \\ & = & - \mbox{constant} + (b^2/2\Delta \Epsilon) \cos \theta \end{array} \]

in which the subscript ij is dropped for a near-neighbor interaction. The spin-dependent part of this energy is \[\tag{4} \Delta \epsilon_{ex} = - J^{ex} S_i \cdot S_j \]

so that the exchange-energy parameter is
\[\tag{5}
J^{ex} \approx -2b^2 / (4S_i S_j \Delta \Epsilon)
\]

which is antiferromagnetic, i.e. \(J^{ex} < 0\) in Eq.(4), and a \(J^{ex} \approx 2b^2/\Delta \Epsilon\) for the interaction between two electrons with S = \(^1/_2\) is the same as originally derived by P.W. Anderson (1950).

A virtual electron transfer from a half-filled to an empty orbital is not restricted by the Pauli exclusion principle, but the ferromagnetic intraatomic potential exchange on the receiving cation favors transfer of a spin that is parallel to the pre-existing spin on the cation, so the interaction is ferromagnetic and a \(t^{\uparrow \uparrow} = b \cos (\theta/2)\) is used. However, this interaction is treated in third-order perturbation theory, which adds a factor \((\Delta_{ex}/\Delta \Epsilon)\) to give \[\tag{6} J^{ex} = +2b^2 \Delta_{ex} / [4S_i S_j (\Delta \Epsilon)^2] \]

where \(\Delta_{ex}\) is the intraatomic exchange energy and a \(J^{ex} > 0\) stabilizes ferromagnetic interactions according to Eq.(4).

To illustrate application of the Goodenough-Kanamori rule, (Goodenough,1963)(Goodenough and Zhou, 2001), consider first the perovskite LaFeO_{3}; the ideal perovskite structure is shown in Figure 1.
In this and other examples, formal charges on the cations designate the d-electron occupation not the charge at the cation site.
The octahedral-site cation d orbitals are split by the cubic crystalline field of the site into a threefold-degenerate set of t orbitals and a twofold-degenerate set of e orbitals.
The high-spin 3d-electron configuration t^{3}e^{2} at the Fe^{3+} ions undergo \((180^{\circ} - \phi)\) Fe – O – Fe interactions; the small angle \(\phi\) is introduced by a cooperative rotation of the octahedral sites that lowers the crystal symmetry from cubic to orthorhombic.
The energy \(\Delta \Epsilon\) in \(J^{ex}\) is the on-site electrostatic energy U to add an electron to the corresponding t or e orbitals of the localized-electron manifold, i.e. the separation of the d^{5} and d^{6} energies of the Fe^{4+}/Fe^{3+} and Fe^{3+}/Fe^{2+} redox couples.
For LaFeO_{3}, all five 3d orbitals are half-filled; they each give antiferromagnetic interactions that add to one another. The Fe – O – Fe interactions each consist of one σ and two π interactions.

As a second example, consider the perovskite LaCrO_{3}, which contains Cr^{3+}:t^{3}e^{0} octahedral-site cations.
The three half-filled orbitals give antiferromagnetic π-bond superexchange interactions as in LaFeO_{3}; the two empty e orbitals can give no superexchange interaction, but they give antiferromagnetic \(\sigma\)-bond spin-spin interactions via a purely semicovalent-exchange interaction in which \(\Delta \Epsilon = 2\Delta\ ,\) where \(\Delta\) is the energy difference between the empty \(\sigma\)-bond cation e orbital and the \(O-2p_\sigma\) orbitals; the factor 2 arises because two electrons are virtually transferred from the same itinerant-electron \(2p_\sigma\) orbital.
Since the two \(2p_\sigma\) orbitals have antiparallel spins by the Pauli exclusion principle and intaatomic ferromagnetic exchange on the cations stabilizes transfer of a spin parallel to the preexisting cation spin, the virtual transfer of opposite spins from the same \(O-2p_\sigma\) orbital couples the two cation spins antiferromagnetically.

The ideally ordered La_{2}NiMnO_{6} double perovskite offers an example of ferromagnetic Ni^{2+}:t^{6}e^{2} – O – Mn^{4+} : t^{3}e^{0} interactions in which electron transfer of \(\sigma\)-bond e electrons and \(\pi\)-bond t electrons from the half-filled e^{2} and full t^{6} manifolds on Ni^{2+} are, respectively, to empty e^{0} and half-filled t^{3} orbitals on the Mn^{4+} ions. The \(\Delta \Epsilon\) for the e orbitals is small as it is the energy difference between Ni^{3+}/Ni^{2+} and Mn^{4+}/Mn^{3+} redox couples. Experimentally, the small \(\Delta \Epsilon\) makes it difficult to obtain ideal ordering of the Ni^{2+} and Mn^{4+} ions on the simple-cubic array of octahedral sites, but the Ni^{2+} – O – Mn^{4+} interactions have been shown to be ferromagnetic in accordance with the rule. By using layer-by-layer chemical-vapor deposition of thin films of the double perovskite La_{2}FeCrO_{6}, Ueda *et al* (1998) succeeded in ordering the Fe^{3+} and Cr^{3+} ions first into alternate (111) planes and then into alternate (001) planes. The former order gave ferromagnetism since the virtual electron transfer from a half-filled \(\sigma\)-bond e orbital on the iron to an empty e orbital on the chromium dominates the antiferromagnetic \(\pi\)-bonding t-electron transfer. On the other hand, ordering the Fe^{3+} and Cr^{3+} ions into alternate (001) planes gave antiferromagnetic Fe^{3+}-O-Fe^{3+} and Cr^{3+}-O-Cr^{3+} interactions within the (001) planes and ferromagnetic Fe^{3+}-O-Cr^{3+} interactions along the [001] axis.

As a fourth illustration, consider the perovskite LaMnO_{3} in which the twofold e-orbital degeneracy of the Mn^{3+} : t^{3}e^{1} manifold is removed by a cooperative site distortion that alternates long and short O – Mn – O bonds along the cubic [100] and [010] axes. The cooperative site distortions below a temperature T_{t} > T_{N} order half-filled \(\sigma\)-bond orbitals of e parentage in the long O – Mn – O bonds and empty \(\sigma\)-bond orbitals in the short O – Mn – O bonds of the (001) planes. As a result of the orbital ordering, half-filled \(\sigma\)-bond orbitals overlap empty \(\sigma\)-bond orbitals in the (001) planes to give ferromagnetic interactions whereas the half-filled \(\pi\)-bonding t orbitals give, according to the rule, antiferromagnetic interactions between all near neighbor Mn^{3+} ions as in LaFeO_{3} and LaCrO_{3}. Because the overlap of the \(\sigma\)-bond electrons is greater and their \(\Delta \Epsilon\) is smaller than those of the \(\pi\)-bonding electrons, the ferromagnetic interactions prove to be stronger in the (001) planes, but the t^{3} – O – t^{3} interactions dominate the [001] interactions to couple antiferromagnetically the ferromagnetic (001) planes below a T_{N} < T_{t}.

A significant check of this analysis has been obtained by substituting R^{3+} rare-earth ions for the La^{3+} ion (Zhou and Goodenough, 2006). In LaMnO_{3}, the \(\Delta \Epsilon\) of the \(\sigma\)-bond superexchange proves to be kT_{t}, and T_{t} increases as the size of the R^{3+} decreases. As \(\Delta \Epsilon\) increases, the strength of the \(\sigma\)-bond ferromagnetic interaction decreases relative to the antiferromagnetic \(\pi\)-bond interactions. As a result, T_{N} decreases with increasing atomic weight of the R^{3+} ion until the \(\pi\)-bond and \(\sigma\)-bond interactions within the (001) planes become of comparable magnitude, at which point an exchange-density wave appears in which ferromagnetic Mn – O – Mn interactions alternate with antiferromagnetic Mn – O – Mn interactions within the (001) planes.

Comparison of the two spinels Zn[Cr_{2}]O_{4} and Cd[Cr_{2}]S_{4} provides another example of competing interactions. In these spinels, the Cr^{3+}: t^{3}e^{0} ions occupy octahedral sites that share common edges, and only \(90^\circ\) Cr – anion – Cr interactions are associated with a shared anion. In this case, the two competing interactions are the antiferromagnetic t^{3}-t^{3} superexchange interactions across a shared octahedral-site edge and ferromagnetic \(90^\circ\) Cr^{3+} – anion – Cr^{3+} interactions. In the \(90^\circ\) interactions, a half-filled \(\pi\)-bonding t orbital shares a common anion-p orbital with an empty \(\sigma\)-bonding e orbital on the neighboring cation. The shorter Cr – Cr separation and weaker covalent component in Zn[Cr_{2}]O_{4} compared to Cd[Cr_{2}]S_{4} makes the antiferromagnetic Cr – Cr interactions dominant in Zn[Cr_{2}]O_{4} whereas a dominant \(90^\circ\) Cr – S – Cr interaction makes Cd[Cr_{2}]S_{4} ferromagnetic.

The spinel Ge[Ni_{2}]O_{4} has filled t^{6} orbitals on the Ni^{2+} : t^{6}e^{2} ions, so there are no Ni – Ni interactions. The \(90^\circ\) Ni – O – Ni interactions are ferromagnetic because the e orbitals on neighboring cations overlap different anion-\(p_\sigma\) orbitals and are, therefore, orthogonal to one another. In addition, shared anion-p orbitals couple a filled t^{6} configuration with a half-filled e^{2} configuration. However, weak antiferromagnetic Ni-O-Ge-O-Ni interactions between next near neighbors via overlapping half-filled e^{2} orbitals are strongs enough to induce formation of a ferromagnetic helical-spin configuration.

A final example is ferromagnetic EuO, which has the rock-salt structure. The seven 4f orbitals on the Eu^{2+} ions are all half-filled, so the rule states that the 4f – 4f interactions should be antiferromagnetic. However, the orbitals of the 4f^{7} configuration have a small overlap integral and a \(\Delta \Epsilon = U \geqslant 10\) eV separating the 4f^{7} and 4f^{8} energies. Therefore, the 4f – 4f interactions are weak and the Eu – Eu interactions are dominated by a virtual electron transfer from a half-filled 4f^{7} configuration on one cation to an empty 5d orbital on a near neighbor, which has a larger orbital overlap and only requires a \(\Delta \Epsilon \approx 1\) eV. This dominant interaction is ferromagnetic.

In all the examples cited, any orbital angular momentum was quenched by the crystalline electric field at the octahedral sites. Where there is a t-orbital degeneracy, the orbital angular momentum is not fully quenched and it is necessary to consider whether cooperative site distortions below a T_{t }> T_{N} remove the orbital degeneracy to give anisotopic interactions as in LaMnO_{3} or spin-spin coupling first establishes a magnetic order that, through spin-orbit coupling, then provides the needed cooperativity between site distortions to remove the t-orbital degeneracy. Long-range collinear-spin order establishes a cooperativity between even dilute site distortions resulting from spin-orbit coupling. This latter case is, therefore, responsible for a single-ion **magnetostriction** that is to be distinguished from **exchange striction**, which reflects the bonding stabilization due to long-range magnetic order. Both of these distortions are stabilized below the long-range magnetic-ordering temperature.

This distinction, first pointed out by Kanamori (1957, 1960), is illustrated by the distortions that occur below the antiferromagnetic ordering temperature T_{N} in MnO and CoO, each of which have the rock-salt structure and the same magnetic order below T_{N}. The high-spin Mn^{2+} : t^{3}e^{2} ions have all their 3d-orbitals half-filled, so both the Mn-Mn interactions across shared octahedral-site edges and the \(180^\circ\) Mn – O – Mn interactions across shared octahedral-site corners are antiferromagnetic. Stronger Mn – O – Mn interactions create ferromagnetic (111) planes coupled antiferromagnetically to one another. The six Mn-Mn interactions between (111) planes are stabilized by antiferromagnetic coupling whereas the six Mn-Mn interactions within the (111) planes are destabilized by being forced to be ferromagnetically coupled. Therefore, below the Néel temperature T_{N}, the cubic symmetry of the lattice is distorted to rhombohedral \(\alpha > 60^\circ\) by a contraction along the [111] axis perpendicular to the ferromagnetic (111) sheets. This contraction represents an exchange striction, and the character of the distortion can be predicted by the Goodenough-Kanamori rule. On the other hand, the lattice-symmetry change below T_{N} of CoO exhibits a strong tetragonal (c/a < 1) component to the distortion even though the half-filled e orbitals of the Co^{2+} : t^{5}e^{2} ions give the same dominant antiferromagnetic Co – O – Co interactions as found in MnO. In CoO, the Co^{2+} ions have a t-orbital degeneracy and, therefore, the orbital angular momentum is not quenched. Since there is no bias of the site distortions in the paramagnetic, cubic phase and the site distortions due to the t-orbital ordering are smaller than those due to e-orbital ordering, cooperative site distortions are not stabilized above T_{N}. However, once long-range spin order is established, a cooperative orbital ordering by spin-orbit coupling is possible. The cooperative distortion that results has a sign that maximizes the orbital angular momentum. For the Co^{2+} : t^{5}e^{2} ion, the empty down-spin t orbital is therefore ordered into the yz ± izx orbital where z is the direction of the spin; an imaginary component in the wave function is required to maximize the orbital angular momentum since the z component of the orbital-angular-momentum operator is \(L_Z = -i \hbar \partial /\partial \phi\ ,\) where \(\phi\) is the angle in the x-y plane. Note that the magnetostriction reflects the axis of the cation spins whereas exchange striction reflects the symmetry of the magnetic order. In the orthorhombic perovskite YVO_{3}, on the other hand, an intrinsic distortion within the VO_{6/2} sites to orthorhombic symmetry as a result of cooperative site rotations and a weaker \(\pi\)-bond spin-spin interaction introduce orbital ordering below a \(T_\infty > T_N\) that suppresses the orbital angular momentum.

Finally, competing spin-spin interactions can give rise to a **frustration** that results in a non-collinear ordering of the spins. For example, antiferromagnetic interactions between spins of a close-packed plane may resolve their frustration by ordering the spin axes at 120° with respect to one another. The Goodenough-Kanamori rule provides the signs of the competitive interactions that are responsible for non-collinear spin ordering.

## References

- Anderson, P W (1950).
*Phys. Rev.*79: 350. - Kanamori, J (1957).
*Prog. Theoret. Phys.*17: 177-196, 197-222. - Goodenough, J B (1958).
*J. Phys. Chem. Solids*6: 287. - Kanamori, J (1959).
*J. Phys. Chem. Solids*10: 87. - Kanamori, J (1960).
*J. Appl. Phys. Suppl.*31: 145. - Goodenough, J B (1963). Magnetism and the Chemical Bond. Interscience-Wiley, New York.
- Goodenough, J B (1995).
*Phys. Rev.*100: 564. - Ueda, K; Tabata, M and Kawai, T (1998).
*Science*280: 1064. - Goodenough, J B and Zhou, J-S (2001).
*Structure and Bonding*98: 17-114. - Zhou, J-S and Goodenough, J B (2006).
*Phys. Rev. Lett.*96: 247202.