# 6 Itinerant magnetism

On the way to justifying the origin of different terms entering the spin Hamiltonian, we have computed the Landé and the anisotropy tensors for Mn\(^{3+}\) in octahedral environment, and we have listed some possible mechanisms that are responsible for inter-atomic exchange coupling in transition-metal oxides. These concepts provide the basis to understand the magnetic behavior of insulators. However, magnetism in conductors shows some peculiarities that cannot be explained within the framework developed so far. The crucial questions we are going to address are:

1. What happens to an atomic magnetic moment when it is embedded into sea of conduction electrons?

2. If magnetic moments “survive” in contact with the conduction electrons, which mechanism aligns them parallel (or antiparallel) to each other?

A definitive answer to these two questions was achieved only in the late 70s of the last century, with the outstanding article by Mary B. Stearns “Why is iron magnetic?”^{16}.

## 6.1 Local magnetic moments in metals

A very concrete experimental fact is related to the first question posed above.
As derived according to Hund’s rules, the magnetic moment of a free ion consists of an orbital contribution and a spin contribution. When the free ion is embedded in a solid we have seen that the orbital contribution becomes – partially or totally – *quenched*. Assuming total quenching of the orbital momentum in conductors, the spin contribution would yield a magnetic moment equal to 4, 3 and 2 \(\mu_B\) for Fe, Co and Ni, respectively. In other words, magnetic moments should be integer multiples of \(\mu_B\). In contrast to this expectation, the values measured experimentally in metallic bulk samples are \(2.216\) for Fe, \(1.715\) for Co and \(0.616\) for Ni, namely they are not multiples of the Bohr magneton but rather fractions of it. This indicates that – even with the inclusion of crystal-field effects – the description based on atomic levels severely overestimates the magnetic moments per atom as they appear in metallic lattices. The simplest model that captures the feature of non-integer magnetic moments of metal Fe, Co and Ni is the
Stoner-Wohlfarth-Slater model of metallic ferromagnetism. This is a band-like model, whose basic assumption is that the bonding interaction between the 3d electrons causes a smearing of their energy into a band. Before sketching the main ideas behind the Stoner criterion, it is convenient to refresh some useful concepts of band theory.

### Band theory in a nutshell

^{17}Band theory applies to the electronic structure of solids and allows getting insights into many physical properties of matter including magnetism. Band theory deals with an arrangement of a large number of atoms, as opposed to “free” atoms. As we have seen, the electrons in free atoms occupy defined discrete energy levels that are filled following the prescriptions of the Pauli exclusion principle. When atoms are brought closer together and form a solid, these discrete energy levels can be considered as continuums of allowed energies, called bands.
There is no electron cloud overlap for two atoms at large distance, and thus there can be two electrons with the same quantum numbers (same energy level, same state). However, as these two atoms are brought together (the inter-atomic distance \(d\) decreases), the electron clouds start overlapping and thus the energy levels have to split in two to comply with Pauli principle. Hence, when N atoms come together to form a solid, each level of the free atom for which there is an overlap of the electronic density must split into N levels, so that the Pauli principle is satisfied for the whole group of atoms.

Figure6.1 shows the splitting of the energy levels as the inter-atomic distance \(d\) decreases to the equilibrium \(d_0\) in the solid. The extent of the splitting depends on the electron cloud overlap for each level. In transition metals the outer electrons are the 3d and 4s and, therefore, these levels will be the first to split as the atoms are brought closer and closer. At the equilibrium distance, the 3d electrons are spread into a band extending from B to C, while the 4s levels are spread over a wider band, extending from A to D. This happens becasue 4s orbitals extend farther from the nucleus than 3d orbitals. As the number N of atoms in a solid is very large (typically 10\(^{19}\) for 1mg of Fe), the discrete energy levels become a band, a continuum of allowed energies, as N tends to \(\infty\).

Band theory has been developed for solids as an infinite periodic arrangement of atoms in a lattice, so that Bloch wavefunctions can be used to describe the electronic orbitals. Band structure of a material is thus calculated in momentum space where electrons occupy a series of bands composed of single-particle energy eigenstates, and usually represented as in figure 6.2a, with the dashed line showing the Fermi energy \(E_{\rm F}\). As it would be difficult to visualize the shape of a band as a function of wave-vector, since it requires a plot in four-dimensional space (Energy as a function of \(k_x\), \(k_y\) and \(k_z\)), it is common to see band structure plots which show the values of \(E(k)\) for values of \(k\) along straight lines connecting symmetry points. These points are generally labeled \(\Delta\), \(\Lambda\), \(\Sigma\), etc., or [100], [111], and [110], respectively.

While the levels can be considered as a continuum of allowed energies due to the large number of atoms generally involved, the density of levels inside a band becomes a crucial quantity. This quantity is called the density of state \({\rm DOS}(E)\), and depends on the energy. The product of the density \({\rm DOS}(E)\) and any given energy range gives the number of levels in that range; thus \({\rm DOS}(E)dE\) is the number of levels lying between the energies \(E\) and \(E + dE\), and \(1/{\rm DOS}(E)\) is the average energy separation of adjacent levels in that range. The DOS as a function of energy, shown in figure 6.2b for manganese, can be obtained from the band structure in momentum space. As a rule of thumb, a peaked density of state (flat band) corresponds to a more localized character of the electron, while a smeared out density of state (parabolic band) corresponds to electrons with more itinerant character.

### Stoner criterion

Let us go back to the explanation of metallic ferromagnetism. We sketch in the following the main ideas behind the Stoner-Wohlfarth-Slater model, which is an extension of band theory to include the effect of an intra-atomic exchange interaction. These two ingredients allow establishing a condition for the existence of an imbalance between majority and minority spins in itinerant electrons. Defining the polarization \(P\) with respect to some spin quantization axis^{18} as the difference between the number of majority- and minority-spin electrons \(P= N^{\rm maj}-N^{\rm min}\), one can express the exchange contribution to the electron-gas energy as
\[\begin{equation}
\label{Exch-SWS}
\tag{6.1}
E_{\rm exch} = -J_{\rm intra} P\,.
\end{equation}\]
In the equation above \(J_{\rm intra}\) can be identified with the intra-atomic exchange interaction responsible for the first Hund’s rule in atoms.

Let \(D_0(E)\) be the density of states per atom common to both spin channels in the absence of spin imbalance. Then we apply an opposite shift to the energy of electrons with majority and minority spin to account for the exchange energy given in Eq.(6.1) (see Fig.6.3):
\[\begin{equation}
\begin{split}
D^{\rm maj}(E) = D_0\left(E + \frac{1}{2}J_{\rm intra}P\right) \nonumber\\
D^{\rm min}(E) = D_0\left(E - \frac{1}{2}J_{\rm intra}P\right) \,.
\end{split}
\end{equation}\]
At this point the problem is defined by the following pair of equations
\[\begin{equation}
\label{Implicit-SWS}
\tag{6.2}
\begin{split}
N = \int_0^{E_F(P)}\left[ D_0\left(E + \frac{1}{2}J_{\rm intra}P\right) + D_0\left(E - \frac{1}{2}J_{\rm intra}P\right) \right] dE\\
P =\int_0^{E_F(P)}\left[ D_0\left(E + \frac{1}{2}J_{\rm intra}P\right) - D_0\left(E - \frac{1}{2}J_{\rm intra}P\right) \right] dE\,,
\end{split}
\end{equation}\]
in which the unknowns are the spin polarization \(P\) and the Fermi energy \(E_F(P)\). We will come back to the dependence of \(E_F\) on \(P\) later on. Practically, the core levels are not relevant in this problem because the electrons occupying those levels are all paired in spin singlets and, thus, do not contribute to the spin imbalance.
Therefore, \(N=N^{\rm maj}+N^{\rm min}\) can be identified with the number of electrons in the outer shell of a magnetic atom.
Ideally, one can imagine solving the first equation to obtain \(E_F(P)\) and insert this quantity in the second one.
In this way, one would end up dealing with a self-consistent equation for the unknown \(P\) qualitatively similar to the equation of state of a ferromagnet obtained with the mean-field approximation: \(P = F(P)\). The function \(F(P)\) has the following important properties:

\(\bullet\) \(F(0) = 0\)

\(\bullet\) \(F(-P) = -F(P)\) , i.e. \(E_F (-P) = E_F (P)\)

\(\bullet\) \(F(\pm \infty) =\pm P_{\rm Hund}\) and \(-P_{\rm Hund} <F(P)< + P_{\rm Hund}\)

\(\bullet\) \(F'(0)=\left(d F/dP\right)_{P=0} \geq 0\).

\(P_{\rm Hund}\) is the largest attainable polarization of the electron gas and corresponds to the magnetic moment obtained applying the first Hund’s rule to the considered metal^{19}. Under these conditions, the graphical solution of the implicit equation for \(P\) evidences two possible scenarios. For \(F'(0)<1\), the equation \(P=F(P)\) has only the solution \(P=0\) (curve a in Fig.6.4). For \(F'(0)>1\), the equation has three solutions: \(P=0\) and two solutions with finite \(P\) of opposite sign (marked with open circles on the curves b and c in Fig.6.4). The last ones generally provide non-integer values of \(P\). In this case it can be shown that the \(P=0\) solution maximizes the total energy, while the two solutions with opposite sign are the minima of the energy associated with a finite spin imbalance in the ground state.

In particular, the atomic magnetic moment is related to the spin imbalance by the equation
\[\begin{equation}
\mu = \frac{1}{2} g \mu_B P\,,
\end{equation}\]
where \(P\) is a solution of the equation \(P=F(P)\). Various solutions of this equation are qualitatively summarized in Fig.6.4, where
they are indicated with \(\bar P\) to help distinguish between different cases. A finite polarization is obtained only when \(F'(0)>1\) and this condition defines the Stoner criterion for metallic ferromagnetism.
Taking the first derivative with respect to \(P\) of the lower row of Eq.(6.2), it is straightforward to show that
\[\begin{equation}
\label{Slope-of-F-Stoner}
\tag{6.3}
\left(\frac{d F}{dP} \right)_{P=0} = J_{\rm intra}\, D_0(E_F)\,.
\end{equation}\]
Therefore, the Stoner criterion reduces to
\[\begin{equation}
\label{Stoner-criterion}
\tag{6.4}
J_{\rm intra}\, D_0(E_F) >1\,.
\end{equation}\]
\(J_{\rm intra}\) is essentially an atomic quantity of the order of \(0.7\) eV for 3d atoms. The tendency to ferromagnetism, thus, requires a high density of states
\(D_0(E_F)\) of the spin unpolarized band structure at the Fermi level. This can only be achieved when the states close to the Fermi level are sufficiently localized, i.e. their bandwidth is small enough, as for metals with partially filled d shells.
At the same time this simple model shows that the observation of magnetic moments that are not integer multiples of \(\mu_B\) is related to *some* itinerant character of the very same d electrons.
As the bandwidth gets smaller and \(D_0(E_F)\) accordingly more peaked, the polarization approaches \(P_{\rm Hund}\).

Qualitatively, one can understand the non-integer magnetic moments as the result of a competition between the kinetic energy and the intra-atomic exchange interaction. While in the unpolarized ground state with \(P=0\) all the levels up to \(E_F\) can be filled with two electrons (paired in a spin singlet), in the state with finite spin polarization the minority and majority bands are split (see Fig.6.3). This configuration lowers the exchange energy given in Eq.(6.1) and this is the driving force to the formation of magnetic moments. However, the radius of the Fermi sphere must be increased to host all the electrons, as the double occupancy of each level is no longer possible when majority and minority bands are split. This fact produces an increase of the total kinetic energy, i.e. \(E_F(P)\), that goes against the formation of a finite spin polarization, namely of magnetic moments. For this reason, the formation of magnetic moments in metallic samples is the result of a delicate energy balance and is subject to strong restrictions.

## 6.2 The RKKY interaction

In 1954 Ruderman and Kittel computed an explicit form for an effective interaction between nuclear magnetic moments of different atoms mediated by the polarization of itinerant s-like electrons.

This coupling turned out to oscillate in sign (ferro- or antiferromagnetic) and decay with a power of the distance between nuclei. Later, Zener proposed that a mechanism analogous to this interaction between nuclear magnetic moments could occur between localized electronic magnetic moments of transition metals and, therefore, be at the origin of ferromagnetism in metals. Kasuya and Yosida developed this idea in more detail. To acknowledge all these developments, the indirect coupling between localized magnetic moments mediated by conduction electrons is today referred to as the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction.

The explicit form of the RKKY interaction can be deduced following different approaches, but all require relatively long calculations. Here we sketch the derivation presented in the book by White ^{20}, assuming the knowledge of the susceptibility of a free-electron gas.

Let us model the interaction between the total spin of a magnetic atom \(\hat{\mathbf S}_l\) (the subscript standing for *localized*) located at some lattice site \(\underline{R}\) and the spin \(\hat{\mathbf s}_i\) (the subscript standing for *itinerant*) of a conduction electron as
\[\begin{equation}
\label{RKKY-1}
\tag{6.5}
\mathcal{H}_{} = -J_{\rm intra} \,\hat{\mathbf S}_l \cdot \hat{\mathbf s}_i \,\delta(\underline{r}-\underline{R})\,,
\end{equation}\]
where \(J_{\rm intra}\) describes the intra-atomic exchange constant and the Dirac-delta function
limits the interaction to be of contact form. First, we focus on the polarizing effect of a single magnetic impurity in the conduction-electron sea. We set this local magnetic impurity at \(R=0\), for simplicity. The interaction in Eq.(6.5) can be interpreted as an effective field acting on the spin \(\hat{\mathbf s}_i\) of a conduction electron:
\[\begin{equation}
\label{RKKY-2}
\tag{6.6}
{\vec B}_{\rm eff} =-\frac{J_{\rm intra}}{g \mu_B } \,\hat{\mathbf S}_l \,\delta(\underline{r})\,.
\end{equation}\]
In a rigorous approach, this field should be an operator. However, our goal is to deduce an effective coupling between spins \(\hat{\mathbf S}_l\) lying at different lattice sites. For this reason, we think of “freezing” their dependence as operators into a parametric dependence and integrate out the degrees of freedom of conduction electrons (note the similarity with the assumption made in the calculation to estimate the magnetic anisotropy of Mn\(^{3+}\)). Under this hypothesis, the magnetic response of the conduction-electron gas is just the response to a \(\delta\)-like excitation, which in the framework of magnetism is described by the susceptibility dependent on the wave vector \(\chi(\underline{q})\):
\[\begin{equation}
\label{RKKY-3}
\tag{6.7}
\hat{\mathbf s}_i({\underline r}) \propto J_{\rm intra} \, \hat{\mathbf S}_l \int \chi(\underline{q}) \, e^{-i {\underline q}\cdot{\underline r}}
\, d^{\rm d}q \,.
\end{equation}\]

The equation above relates the spatial dependence of the spin polarization of the electron gas surrounding a magnetic atom (“impurity”) to the susceptibility \(\chi(\underline{q})\) of the gas itself.
In order to provide an explicit dependence of \(\hat{\mathbf s}_i\) on \({\underline r}\), one needs to know the expression of the susceptibility.
Apart from constant prefactors – in which we are not interested – this is given by
\[\begin{equation}
\label{RKKY-4}
\tag{6.8}
\chi(\underline{q}) \propto \frac{1}{2} + \frac{k_F}{2q} \left(1- \frac{q^2}{4k_F^2}\right)
{\rm ln} \left| \frac{2k_F+q}{2k_F-q}\right| \,,
\end{equation}\]
where \(k_F\) is the Fermi wave vector. Using a mathematical equivalence^{21},
from Eq.(6.8) one obtains that
\[\begin{equation}
\label{RKKY-4a}
\int \chi(\underline{q}) \, e^{-i {\underline q}\cdot{\underline r}} \, d^{\rm d}q
\propto \frac{\sin(2k_F r)- 2k_F r\cos(2k_F r)}{(k_F r)^4} \,.
\end{equation}\]
Combined with Eq.(6.7), the last expression tells us that when a localized magnetic moment is introduced into a metal,
the spins of conduction electrons develop an oscillating polarization in the vicinity of the local moment^{22}.

The polarized conduction electrons, in turn, shall produce an effective field on the magnetic moment localized at another lattice site, by virtue of Eq.(6.5) itself. This eventually leads to an effective exchange coupling between the spins \(\hat{\mathbf S}_{l1}\) and \(\hat{\mathbf S}_{l2}\) associated with two localized magnetic moments in the metal \[\begin{equation} \label{RKKY-5} \tag{6.9} \mathcal{H}_{\rm RKKY} =- J_{\rm RKKY} \,\hat{\mathbf S}_{l1}\cdot\hat{\mathbf S}_{l2} \end{equation}\] with \[\begin{equation} \label{RKKY-6} \tag{6.10} J_{\rm RKKY} = J_0 \,\frac{\sin(2k_F R) -2k_F R\cos(2k_F R)}{(k_F R)^4} \,. \end{equation}\] It is important to remark that behind the expression of the susceptibility \(\chi(\underline{q})\) given in Eq.(6.8) a scattering calculation is hidden, which involves the Fermi-Dirac distribution of electrons in the metal. As usual, scattering happens only among states around the Fermi level and this is the reason why \(k_F\) enters the expression of the electron spin polarization. In particular, \(k_F\) sets the spatial period of modulation of the RKKY interaction in Eq.(6.10). Depending upon the separation between the localized magnetic moments \(\hat{\mathbf S}_{l1}\) and \(\hat{\mathbf S}_{l2}\), the RKKY exchange coupling can be either ferromagnetic or antiferromagnetic. Like the super exchange coupling, it is an indirect mechanism. Differently from the super and double exchange coupling described some weeks ago, it decays with a power of the separation between magnetic moments \(R\), meaning that it may couple localized moments over relatively large distances. The coupling constant \(J_0\) in Eq.(6.10) is proportional to the square of the intra-atomic exchange constant \(J_{\rm intra}\). However, due to other terms that we have not considered explicitly in our derivation, \(J_{\rm RKKY}\) for neighboring atoms turns out to be 1 or 2 orders of magnitude smaller than \(J_{\rm intra}\).

Since the RKKY exchange interaction around magnetic impurities is non-directional, the same mechanism applies to two magnetic layers (labelled with \(a\) and \(b\)) separated by a non-magnetic spacer of thickness \(d\). In this case, the oscillatory coupling is in the direction perpendicular to the layers and takes the form \[\begin{equation} \label{RKKY-7} \tag{6.11} J_{ab} = J_0 \,\frac{d^2}{z^2} \, \sin(2k_F z) \,, \end{equation}\] where \(z\) is the distance from one of the layers and is assumed significantly larger than a lattice unit. If a non-magnetic spacer is interposed between two ferromagnetic films, Eq.(6.11) prescribes that the exchange coupling between the two magnetic films should oscillate by varying the thickness of the spacer. This prediction was directly verified using a Cu wedge as a spacer between two Co films, with the experimental geometry sketched in Fig.6.5. The fact that the corresponding \(J_{ab}\) was observed to oscillate periodically with a period matching the Fermi wave length of Cu provided a strong argument supporting the RKKY mechanism as the origin of itinerant ferromagnetism.

The reader is encouraged to read the famous article by Mary B. Stearns, suggested at the beginning of this chapter, that provides an interesting historical review of the long quest to understand why metallic Fe is magnetic. Therein, he/she will be able to appreciate the role of the Stoner criterion and the RKKY interaction as well as their subtle interplay. In fact, in spite of the fact that – apart from Zn – all 3d transition metals have a finite magnetic moment as free ions, when these atoms are in the metallic phase few of them show ferromagnetism.

Apart from transition metals, the RKKY interaction is also at the origin of the exchange coupling between the localized moments in rare-earth metals. In those materials, its oscillatory nature leads to the formation of non-collinear spin textures, such as spin helices.

Physics Today 31, 4, 34 (1978); https://doi.org/10.1063/1.2994993↩︎

This section was contributed by Dr. Diane Lançon taking inspiration from the book

*Introduction to magnetic materials*by B.D. Cullity.↩︎To help visualization the spin quantization direction has been defined in Fig.6.3 along the direction of an external magnetic field.↩︎

To be precise, by \(P_{\rm Hund}\) we mean the polarization that the atom in a solid would have without taking into account the itinerant character of conduction electrons. Therefore, \(P_{\rm Hund}\) shall generally be smaller than the free-ion value because of the quenching of the orbital momentum operated by the crystal field. ↩︎

R.~M. White,

*Quantum Theory of Magnetism*Springer-Verlag Berlin Heidelberg, 2007 (chapter available also in Moodle).↩︎The equivalence of interest is \[\begin{equation} \label{RKKY-trick} \tag{14.20} {\rm ln} \left| \frac{2k_F+q}{2k_F-q}\right| = 2 \int_0^\infty \frac{\sin(2k_Fx) \sin(qx)}{x} dx \,. \end{equation}\]↩︎

These spin-density oscillations have the same form as the Friedel charge-density oscillations that result when an electron gas screens out a charge impurity.↩︎