So the full response function $\chi$ can be calculated from free response function $\chi_0$ and the kernel $K$, by iteration or inversion. Equivalently one can calculate the local potential directly by iteration of $\varphi_{\text{loc}} = \varphi_{\text{ext}} + K\chi_0\varphi_{\text{loc}}$ [8]. The problem is that the exchange-correlation part of the kernel

$$K_{XC}(\mathbf{r}t, \mathbf{r}'t') = \frac{\delta V_{XC}(\mathbf{r}t)}{\delta n(\mathbf{r}'t')} \quad (3.28)$$

is not known exactly. The adiabatic approximation consists in using a static exchange-correlation potential, which may be taken from standard time-independent DFT. In this case, $K_{XC} = [\delta V_{XC}(\mathbf{r})/\delta n(\mathbf{r}')]\delta(t-t')$ such that $\varphi_{\text{ind}}(t)$ changes instantaneously with $\delta n(t)$. As a result, $K(\mathbf{r}, \mathbf{r}', \omega)$ is frequency independent and dynamical screening is neglected. X-ray fields correspond to fast oscillations, so neglecting dynamical effects is questionable.

### 3.5.3 Absorption Spectra

The optical absorption coefficient is essentially the imaginary part of the response function $\chi$ as we shall show now. We consider an electromagnetic wave given by $E(\mathbf{r}, t) = eE_0 e^{i\mathbf{q}\cdot\mathbf{r}-i\omega t}$, where $e$ is the light polarization vector, not to be confused with the electric charge $e$. The induced electrical polarization is $P(\mathbf{r}, t) = -e\delta n(\mathbf{r}, t)\mathbf{r}$ and so the change in energy density is $-E \cdot P = eE \cdot \mathbf{r}\delta n$. In the dipole approximation, $E(\mathbf{r}, t) \approx eE_0 e^{-i\omega t}$ and the perturbation in (3.17) is given by

$$\varphi_{\text{ext}}(\mathbf{r}, \omega) = eE_0 e \cdot \mathbf{r}. \quad (3.29)$$

The total induced dipole moment is

$$\mu(\omega) = -e \int \mathbf{r}\delta n(\mathbf{r}, \omega) d\mathbf{r} = -e^2 E_0 \int \mathbf{r}\chi(\mathbf{r}, \mathbf{r}', \omega) e \cdot \mathbf{r}' d\mathbf{r} d\mathbf{r}', \quad (3.30)$$

and the absorbed energy is $\text{Re}\{E \cdot d\mu/dt\}$ or equivalently $\text{Im}\{\omega E \cdot \mu(\omega)\}$. The absorption coefficient $\sigma(\omega)$ is the absorbed energy divided by $E_0^2$, which yields

$$\sigma(\omega) = -4\pi\alpha\hbar\omega \int d\mathbf{r} d\mathbf{r}' e \cdot \mathbf{r} \operatorname{Im}\chi(\mathbf{r}, \mathbf{r}', \omega) e \cdot \mathbf{r}'. \quad (3.31)$$

This expression of the absorption coefficient is fully equivalent to (3.3). In the independent particle approximation, we put $\chi \to \chi_0$ and obtain from (3.23)

$$\sigma_0(\omega) = 4\pi^2\alpha\hbar\omega \sum_{hp} |\langle\phi_p|\mathbf{e}\cdot\mathbf{r}|\phi_h\rangle|^2 \delta(\hbar\omega + \epsilon_h - \epsilon_p) \quad (3.32)$$

in agreement with (3.9).
In summary, TDDFT with linear response provides a rigorous and efficient framework for calculating absorption spectra. Compared to the independent particle approximation, TDDFT takes the screening of the electromagnetic field into account by introducing the induced field $\varphi_{ind}(t)$ which is calculated self-consistently with the density change $\delta n(t)$. In practice, the problem is to find good approximations for the unknown exchange-correlation kernel $K_{XC}$. The Hartree part alone, i.e. putting $K_{XC} = 0$, yields the well-known random-phase approximation (RPA) [10]. Apart from the single particle-hole excitations included in $\chi_0$, the RPA can describe plasmon excitations, i.e. collective oscillations of electron gas, which can be observed, for example, as satellite peaks in core-level photoemission spectra. The RPA kernel also gives rise to a redistribution of spectral weight between different transitions. This may strongly change the peak intensity ratio, e.g. between the $L_2$ and $L_3$ white lines in transition elements [10]. In adiabatic TDDFT, $K_{XC}$ can be obtained from standard DFT [8], but such static approximations to $K_{XC}$ do not improve much over the RPA [10, 12]. It appears that complex configuration mixing such as multiplet excitations cannot be described by the common, adiabatic kernels. Going beyond the adiabatic approximation is difficult, but some non-adiabatic kernels have been proposed and applied to the X-ray absorption problem [13].

## 3.6 Photoemission Spectroscopy

PES is probably the most direct way of probing the electronic structure of materials. In a PES, light is shone on a surface and the kinetic energy, and possibly exit angle and spin, of the emitted electrons is measured. In core-level PES, electrons from the inner atomic shells are excited. As these levels are element specific, core-level photoemission is a powerful tool for chemical analysis.

Angle-resolved core-level photoemission from crystal surfaces is known as X-ray photoelectron diffraction [14]. The photoelectron wave spreads from the core hole site and is diffracted by the neighbouring atoms. Analysis of the diffraction pattern gives precise information about the local structure around the atoms of a given chemical species. X-ray photoelectron diffraction can be well modelled with real-space single or multiple scattering theory on a finite cluster of atoms.

### 3.6.1 Angle-Resolved Photoemission Spectroscopy

Angle-resolved photoemission spectroscopy (ARPES) is the major method for measuring energy band dispersion (the 'band structure') of crystals. An intuitive picture of ARPES is provided by the three-step model [15]. The three steps are as follows:

1. Photon absorption in the bulk of the material resulting in an inter-band transition $|m\mathbf{k}\rangle \rightarrow |n\mathbf{k}\rangle$, with $\epsilon(n, \mathbf{k}) = \epsilon(m, \mathbf{k}) + \hbar\omega$. Here $n, m$ are band indices and the
three-dimensional crystal momentum vector **k** is conserved up to a reciprocal lattice vector **G**.

2. Propagation of the excited wave to the surface, with damping due to inelastic scattering.

3. Transmission through the surface by matching the Bloch wave $|nk\rangle$ to a plane wave $\exp(ik' \cdot r)$. The matching conditions are dictated by conservation of energy and the surface parallel component of **k**, i.e. $k_{||}^2 = k_{\perp}^2$ and $k_{\perp}^2 = k_{\perp}^2 + 2mV_0/\hbar^2$, where $V_0$ represents the surface potential barrier.

The three-step model is very useful for relating the photoemission data to the three-dimensional band structure of the material. However, for an accurate calculation of ARPES intensities, the one-step model should be used, where the photoelectron final state is calculated in all space (bulk, surface and vacuum) as a single wave function with proper boundary conditions (so-called 'time-reversed low-energy electron diffraction' boundary conditions). A suitable computational scheme is the layered Korringa-Kohn-Rostoker method [16].

The hole left behind in the photoemission process is not an independent particle, but it interacts with the electrons and the lattice, giving rise to various many-body effects, which are conveniently described using quasiparticle theory.

## 3.7 Quasiparticle Theory

In a photoemission experiment, an electron is ejected from the system, which becomes ionized. Neglecting the interaction between the photoelectron and the hole left behind, i.e. applying the so-called sudden approximation, the photoemission excitation is a one-electron removal process from the *N*-particle ground state to a *N* − 1 particle excited state. In the same fashion, inverse photoemission probes the one-electron addition process from the *N*-particle ground state to a *N* + 1-particle excited state. The true excitations are called quasiparticles. In the limit of vanishing electron interaction, the quasiparticle wave functions are the spin-orbitals of the ground state Slater determinant and the quasiparticle energies are the one-electron levels. In the independent particle approximation, the quasiparticles are taken as HF or Kohn-Sham orbitals. This neglects electron correlation and the interaction of the electrons with the lattice vibrations. These effects change the quasiparticle energies and wave functions. The quasiparticles are said to be renormalized or 'dressed' by the interaction. In particular, due to inelastic scattering at collective excitations such as phonons and plasmons, the one-electron quasiparticles will decay after a characteristic lifetime. As a result, compared to the delta-function-like photoemission peaks corresponding to the independent particle approximation, the true photoemission peaks are energy shifted and lifetime broadened. Moreover, some spectral weight of the main peak is lost to extra ('satellite') peaks, corresponding to some inelastic process.
### 3.7.1 Green's Functions

Quasiparticles can be described using many-body Green's function techniques. We introduce the retarded one-electron Green's function

$$G(\mathbf{r}, \mathbf{r}', t-t') = -i\theta(t-t')\langle 0 | \{\Psi(\mathbf{r}t), \Psi^{+}(\mathbf{r}'t')\} | 0 \rangle, \quad (3.33)$$

where $\Psi^{+}(\mathbf{r}t)$ is a Heisenberg field operator which creates an electron at point $\mathbf{r}$ and time $t$, and $\Psi$ destroys one. $|0\rangle$ is the many-particle ground state and $\{A, B\} = AB + BA$ denotes the anti-commutator. Note that we have suppressed spin for convenience. This Green's function, or 'propagator', gives the probability amplitude for an electron to be found at $\mathbf{r}t$ if one was added at $\mathbf{r}'t'$. The one-electron removal and addition spectrum is given by the spectral function

$$A(\mathbf{k}, \omega) = -\frac{1}{\pi} \mathrm{Im} G(\mathbf{k}, \omega), \quad (3.34)$$

where $G(\mathbf{k}, \omega)$ is the space and time Fourier transform of (3.33). In the following, we focus on a perfect crystal and suppress the band index. The Hamiltonian of the non-interacting system is then given by $H_0 = \sum_{\mathbf{k}} \epsilon_{\mathbf{k}} \hat{n}_{\mathbf{k}}$, where $\mathbf{k}$ labels the Bloch eigenstates with energy $\epsilon_{\mathbf{k}}$ and $\hat{n}_{\mathbf{k}}$ is the corresponding occupation number operator. It is easy to see that in this non-interacting case, Green's and spectral functions are given by

$$G_0(\mathbf{k}, \omega) = (\omega - \epsilon_{\mathbf{k}} + i\eta)^{-1}, \quad A(\mathbf{k}, \omega) = \delta(\omega - \epsilon_{\mathbf{k}}). \quad (3.35)$$

Thus, the photoemission peaks are delta functions, meaning that Bloch states are exact excitations of energy $\epsilon_{\mathbf{k}}$ (band energy) and infinite lifetime. As mentioned above, due to electron interaction, the true photoemission peaks are shifted, broadened and may have satellite structures. In quasiparticle theory, these effects are described by the so-called self-energy $\Sigma$, which is essentially the difference between the inverses of the exact and the free Green's function. The self-energy is defined through the Dyson equation

$$G = G_0 + G_0 \Sigma G \Leftrightarrow G^{-1} = G_0^{-1} - \Sigma. \quad (3.36)$$

For a single band in a crystal, we have

$$G^{-1}(\mathbf{k}, \omega) = \omega - \epsilon_{\mathbf{k}} - \Sigma_{\mathbf{k}}(\omega). \quad (3.37)$$

It is clear from (3.35) and (3.37) that Re$\Sigma$ describes a shift of the eigenvalues $\epsilon_{\mathbf{k}}$ (band energy) and Im$\Sigma$ results in peak broadening, i.e. it reflects the finite lifetime $\tau = \hbar/\mathrm{Im}\Sigma$ of the quasiparticle. There are various methods to find (approximate) self-energies. For the electron correlation effect, two of the most popular methods are the so-called GW approximation and dynamical mean-field theory.
### 3.7.2 GW Approximation

The GW approximation was invented by Hedin in 1965 [17] and owes its name from the form of this self-energy, which is $\Sigma = iGW$, i.e. the product (or convolution) of Green's function ($G$) and the screened Coulomb interaction ($W$). The latter is given by [18]

$$W(\mathbf{r}, \mathbf{r}', \omega) = \frac{e^2}{4\pi\epsilon_0} \int d\mathbf{r}'' \frac{\epsilon^{-1}(\mathbf{r}, \mathbf{r}'', \omega)}{|\mathbf{r} - \mathbf{r}'|}, \quad (3.38)$$

where $\epsilon(\mathbf{r}, \mathbf{r}', \omega)^{-1}$ is the inverse dielectric function. This expression may be understood by analogy to the electrostatic energy between two electrons in a polarizable medium, which is given by $e^2/[4\pi\epsilon_0\epsilon_r|\mathbf{r} - \mathbf{r}'|]$, where $\epsilon_r$ is the relative dielectric permittivity. The dielectric function $\epsilon(\mathbf{r}, \mathbf{r}', \omega)$ generalizes $\epsilon_r$ to inhomogeneous media and dynamic screening effects. The GW approximation is most often used in a non-self-consistent way, i.e. as $\Sigma = iG_0W_0$ with the free Green's function $G_0$ instead of the full Green's function $G$. The GW approximation has been very successful for correcting band energies of weakly correlated systems. In particular, bandgaps of semi-conductors are very well reproduced in the GW approximation, while the values obtained in DFT (except for DFT-HF hybrid functionals) are systematically too small [19].

### 3.7.3 Bethe–Salpeter Equation

At this point, we briefly switch back to the problem of absorption spectroscopy. Since light absorption creates an electron–hole pair, absorption spectra are described with an electron–hole (i.e. a two-particle) Green’s function $G_{eh}$. If the excited electron and the hole do not interact, $G_{eh}$ is just the product of the one-particle removal (hole) Green’s function $G_h$ and the addition (electron) Green’s function $G_e$. Electron–hole interaction leads to coupling of these two Green’s functions, which may be expressed in a Dyson-type equation as [20]

$$G_{eh}(1, 2; 1', 2') = G_e(1, 1')G_h(2, 2') + \int G_e(1, 3)G_h(2, 4)K(3, 4; 5, 6)G_{eh}(5, 6; 1', 2')d3d4d5d6, \quad (3.39)$$

where 1 stands for all coordinates of particle 1 and K is the interaction kernel. In the Bethe–Salpeter equation (BSE) approach, (3.39) is solved with K given by the screened Coulomb interaction in (3.38) and the bare exchange interaction. The electron and hole Green’s functions, $G_e$ and $G_h$, are commonly computed in the GW approximation. The BSE approach is arguably the most accurate first-principles method for absorption spectroscopy in solids, but it is computationally very demanding. It was first applied to X-ray spectra by Shirley in 1998 [21]. It accounts well
for strong excitonic effects and features electron–hole multiplet coupling in L-edge spectra [22]. Let us note that the latter effect is also well described with multichannel multiple scattering theory [23, 24], where the electron–hole coupling is dealt by a CI calculation of the scattering matrix. However, at present, none of these particle–hole theories can fully account for the complex multiplet structure of L-edge spectra of open-shell transition metal compounds. These spectra are still best described with CI methods, either the semi-empirical ligand-field multiplet model (see Chap. 4) or the ab initio complete active space approach on small clusters [25].

### 3.7.4 Static and Dynamical Mean-Field Theory

In strongly correlated electron systems, e.g. 3d transition metal oxides and 5f elements, collective phenomena such as band magnetism, metal–insulator transition and high-$T_c$ superconductivity are observed. These are genuine many-body effects that cannot be explained in the independent particle picture. Itinerant magnetism and the metal–insulator transition are due to the competition between the kinetic energy, which leads to delocalized band states, and strong local Coulomb repulsion which favours electron localization and formation of magnetic moments. The most simple model to study these problems is the (one-band) Hubbard model [26], whose Hamiltonian is given by

$$H = \sum_{k\sigma} \epsilon_k n_{k\sigma} + U \sum_{i} n_{i\uparrow} n_{i\downarrow} = \sum_{ij\sigma} t_{ij} c_{i\sigma}^{\dagger} c_{j\sigma} + U \sum_{i} n_{i\uparrow} n_{i\downarrow},$$

where lattice sites are labelled by i and j, the crystal momentum by k and spin by $\sigma$. Further, $c_v^\dagger$ ($c_v$) creates (destroys) an electron in state $v$, and $n_v \equiv c_v^\dagger c_v$ counts them. $t_{ij}$ is the hopping (or ‘transfer’) integral between sites i and j, and U is the Coulomb energy between two electrons occupying the same site. The corresponding one-electron Green’s function is

$$G_{k\sigma}(\omega) = [\omega - \epsilon_k - \Sigma_{k\sigma}(\omega)]^{-1}.$$

Despite the apparent simplicity of the Hubbard model, the exact solution is unknown (except in one dimension) and the self-energy $\Sigma$ must be approximated. At the lowest level, there is the normal mean-field (i.e. HF) approximation, where $\Sigma$ is taken to be static, i.e. independent of frequency $\omega$. It is given by $\Sigma_{k\sigma} = U \langle n_{k-\sigma} \rangle$, where the occupation numbers $\langle n_{k-\sigma} \rangle$ must be calculated self-consistently. When the lowest energy solution corresponds to a different occupation between spin-up and spin-down bands $\langle n_{k\uparrow} \rangle \neq \langle n_{k\downarrow} \rangle$, the band energies $E_{k\sigma} = \epsilon_k + U \langle n_{k-\sigma} \rangle$ become exchange split, and the ground state is ferromagnetic. HF and LDA are such static mean-field theories and can account for certain static exchange effects, such as ferromagnetism. But they lack all dynamic correlation, which is crucial for the metal–insulator transition and
for various phenomena seen in photoemission spectra, such as band narrowing and satellite structures.

Dynamic correlation effects can, to some extend, be described by the dynamical mean-field theory (DMFT) [27], where the self-energy is taken to be frequency dependent but local, i.e. momentum independent, $\Sigma_{k\sigma}(\omega) \rightarrow \Sigma_{\sigma}(\omega)$. We note that self-energies from static mean-field theory (such as LDA) and dynamical mean-field theory can be, and often are, combined. Although the total self-energy is then both momentum and frequency dependent, it is still an approximation.

The basic idea of DMFT is to map the Hubbard model with correlation ($U \neq 0$) on all lattice sites onto the Anderson model, which describes one correlated atom (the 'impurity') coupled to an effective bath of band states. In the Anderson model, we have $U \neq 0$ only at the impurity site ($i=0$), and as a consequence, the self-energy is a frequency dependent, but local, quantity $\Sigma_{ij}(\omega) = \Sigma_0(\omega)\delta_{i0}\delta_{ij}$. In DMFT, this local self-energy is taken as the self-energy of the lattice problem (Hubbard model). The mapping, i.e. the definition of the effective bath, must be done in a self-consistent manner such that the on-site matrix elements of the lattice model Green's function $G_{ii}(\omega)$ coincide with those of the impurity model [27]. While the Anderson impurity model is simpler than the Hubbard model, it is nonetheless a complex many-body problem. Implementations of DMFT mostly differ in the approximations used for solving the impurity problem.

DMFT has been applied to photoemission spectroscopy of correlated systems [28] and results in improved spectra compared to independent particle approximation (HF or LDA). In transition metal systems, for example, photoemission spectra calculated in DMFT can account for finite temperature effects, correlation-driven band narrowing and satellite peaks [29].

## 3.8 Conclusions

In this chapter, I have tried to give a brief introduction to the theory of X-ray absorption and photoemission spectroscopy. Along the way, it appeared useful to present succinctly the principles of several computational methods of electronic structure that are used in spectroscopic calculations. Given the vast nature of the subject, this account is necessarily very incomplete. But I hope that the reader got an idea of the physics underlying the different theoretical methods and that it aroused his/her curiosity to dwell deeper into the subject by reading some of the cited literature.

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# Chapter 4
## X-ray Dichroisms in Spherical Tensor and Green's Function Formalism

Hebatalla Elnaggar, Pieter Glatzel, Marius Retegan, Christian Brouder, and Amélie Juhin

**Abstract** In this book chapter, our goal is to provide experimentalists and theoreticians with an accessible approach to the measurement or calculation of X-ray dichroisms in X-ray absorption spectroscopy (XAS). We start by presenting the key ideas of different calculation methods such as density functional theory (DFT) and ligand-field multiplet (LFM) theory and discuss the pros and cons for each approach. The second part of the chapter is dedicated to the expansion of the XAS cross section using spherical tensors for electric dipole and quadrupole transitions. This expansion enables to identify a set of linearly independent spectra that represent the smallest number of measurements (or calculations) to be performed on a sample, in order to extract all spectroscopic information. Examples of the different dichroic effects which can be expected depending on the type of transitions and on the symmetry of the system are then given.

### 4.1 Introduction

#### 4.1.1 The X-ray Absorption Cross Section

The X-ray absorption cross section is obtained by dividing the transition rate by the flux of photons and summing over all possible final states. It is given in (4.1) where $\hbar\omega$ is the photon energy, and $\alpha$ the fine structure constant. $I(E_I)$ and $F(E_F)$ are the initial and final state wave functions (energies), and $T$ is the transition operator,

H. Elnaggar
Debye Institute for Nanomaterials Science, Utrecht University, 3584 CA Utrecht,
The Netherlands

P. Glatzel · M. Retegan
European Synchrotron Radiation Facility, 71 Rue des Martyrs, 38000 Grenoble, France

Ch. Brouder · A. Juhin (✉)
Institut de Minéralogie, Physique des Matériaux et Cosmochimie, CNRS-Sorbonne Université,
4 Place Jussieu, 75252 Paris Cedex 05, France
e-mail: amelie.juhin@sorbonne-universite.fr
$$ \sigma_{\omega} = 4\pi^2\alpha\hbar\omega \sum_F \langle I | T^{\dagger} | F \rangle \langle F | T | I \rangle \delta(E_I + \hbar\omega - E_F). \quad (4.1) $$

The transition operator describes the interaction of photons with the system. In the case of an electromagnetic plane wave, the transition operator writes $T \propto e^{ik \cdot r} [\hbar\epsilon \cdot \nabla - \frac{g}{2s} \mathbf{k} \times \mathbf{\epsilon}]$ where $\epsilon$ is the polarization vector of the incident photon, and $\mathbf{k}$ is the incident wave vector, $g$ the gyromagnetic ratio ($g \approx 2$ for the electron), and $s$ the electron spin [1]. The exponential in the transition operator can be expanded as a Taylor series

$$ e^{ik \cdot r} \approx 1 + i\mathbf{k} \cdot \mathbf{r} - \frac{(\mathbf{k} \cdot \mathbf{r})^2}{2!} + \dots \quad (4.2) $$

The first term in the expansion approximates the interaction of light with the atom as an electric dipole ($\langle F | T | I \rangle = \hbar \langle F | \epsilon \cdot \nabla | I \rangle = -\frac{m(E_F - E_I)}{\hbar} \langle F | \epsilon \cdot \mathbf{r} | I \rangle$). The second term gives rise to the electric quadrupole interaction ($-i \frac{m(E_F - E_I)}{\hbar} \langle F | \epsilon \cdot \mathbf{r} \mathbf{k} \cdot \mathbf{r} | I \rangle$) and to the (negligible) magnetic dipole one ($-\frac{1}{2} \langle F | (\epsilon \times \mathbf{k}) \cdot (\mathbf{L} + g\mathbf{s}) | I \rangle$) [1], the third term is the octupole transition ($\frac{m(E_F - E_I)}{6\hbar} \langle F | (\epsilon \cdot \mathbf{r})(\mathbf{k} \cdot \mathbf{r})^2 | I \rangle$) [2], and so on. In this chapter, we focus on electric dipole and quadrupole transitions.

The summation over final states in (4.1) implies that one has first to calculate the ground state, all possible final states, and then compute the transition matrix elements between the ground state and the final states. This is not always the most efficient way to numerically calculate XAS. Instead, Green's function can be used to replace the summation over final states by a propagator of the transition operator. Hence $\sum_F |F\rangle\langle F|\delta(E_I + \hbar\omega - E_F) \rightarrow \frac{-1}{2\pi i}(G^+ - G^-)$ with $G^\pm(E_I + \hbar\omega) = \frac{1}{\hbar\omega - H_F + E_I \pm \frac{1}{2}i\Gamma}$, where $H_F$ is the final state Hamiltonian. The “Fermi Golden Rule” can be expressed as in (4.3). Most modern codes calculating core level spectra use this expression

$$ \sigma_{\omega} = -4\pi\alpha\hbar\omega\mathrm{Im}[\langle I | T^{\dagger} G^{+} (E_I + \hbar\omega) T | I \rangle]. \quad (4.3) $$

Let us now discuss electric dipole transitions according to the first term of the expansion in (4.2). For electric dipole transitions we have $T = \epsilon \cdot \mathbf{r}$. One can see from the expression of the transition operator that the cross section will depend on the orientation of the polarization vector ($\epsilon$) with respect to the absorbing system ($\mathbf{r}$). The X-ray absorption spectrum measured on any sample is in fact the sum of several linearly independent spectra as will be discussed further in this chapter. They can be disentangled by macroscopically orienting the sample, e.g., by using a single crystal or orienting the magnetic moments. Consequently, one may wonder:

- How many independent spectra exist for a given system?

- What information do they give us about the absorbing system?
## 4.1.2 Definition of Dichroisms

*X-ray dichroism* can be defined as the difference in the X-ray absorption cross section measured for two orthogonal polarization states of the incident light. There exist different types of dichroism. Dichroism measurements can be classified according to the type of polarization used for the measurements into linear and circular. *Linear dichroism* (LD) is the difference measured with linearly polarized light, where in most cases the polarization vector is set parallel and perpendicular to an orientation axis, while *circular dichroism* (CD) is the difference measured with circularly polarized light (left handed and right handed).

Not all systems exhibit dichroism effects when the polarization of the light is changed. Certain symmetry conditions regarding the interaction operator between light and matter have to be satisfied for dichroism effects to occur, which brings us to the second classification of dichroism types. Two symmetry operations are essential for this classification:

- Time-reversal symmetry,

- Space inversion (also called parity).

*Natural dichroism* (ND) refers to dichroism effects that occur in non-magnetic systems where time-reversal symmetry is conserved (i.e., the system is even under time-reversal operation). Using linearly polarized light, one can measure *X-ray natural linear dichroism* (XNLD). On the other hand, using circularly polarized light, one can measure *X-ray natural circular dichroism* (XNCD) only for systems that do not have a centre of inversion (i.e., the system is of odd parity).

*Magnetic dichroism* (MD) relates to dichroism effects measured in magnetic (ferro, ferri, or antiferromagnetic) systems where time-reversal symmetry is broken either by spontaneous magnetic ordering in the sample or by the application of an external magnetic field. *X-ray magnetic linear dichroism* (XMLD) is parity-even and time-reversal even and *non-reciprocal linear dichroism* (NRLD) is parity-odd and time-reversal odd. Using circularly polarized light, *X-ray magnetic circular dichroism* (XMCD) and *X-ray magneto-optical dichroism* (XMχD) effects can be measured. The former is parity-even and time-reversal odd while the latter is parity-odd and time-reversal odd.

In magnetic materials, which will be discussed in this chapter, several cases are possible:

- In the case of centrosymmetric crystals with ferro- or ferrimagnetic properties, one can measure XMCD.

- In the case of centrosymmetric crystals with antiferromagnetic properties, XMLD can be measured.

- In the case of magnetized, non-centrosymmetric crystals, XMχD and NRLD can be measured.

XMCD and XMLD measurements give, respectively, access to the average value of $\langle M \rangle$ and $\langle M^2 \rangle$ of the local magnetization for the absorber. On the other hand,
XMχD and NRLD signals are related to moments that are more complex, the anapole orbital moment and other higher order moments.

### 4.1.3 The Many-Body Problem in Spectra Calculations

The calculation of an absorption spectrum is a formidable task: it requires the calculation of the ground state of the system, the excited states of the system, and the interaction of the system with the electromagnetic field (X-ray beam). This means that the theoretical approach required to calculate XAS has to be suitable for calculating the electronic structure in addition to properly considering the interaction with the electromagnetic field. Approximations have often to be made to calculate the absorption (or the scattering Kramers–Heisenberg) cross section and the spectroscopist therefore has to choose which theoretical approach is the most suitable for the problem at hand.

In principle, the ground and excited states can be determined by solving the Dirac equation which accounts for all relativistic effects and includes all possible interactions in the Hamiltonian. This full treatment provides a relativistic, many-body, extended description of the electronic states. Unfortunately, in practice, it is not possible to perform such a calculation as it is computationally very consuming. In most cases one solves instead the Schrödinger equation and introduces relativistic effects as perturbations (e.g., the spin-orbit interaction). Furthermore, one can make use of the Born–Oppenheimer approximation to separate the electronic properties of the system from the dynamics of the nuclei. In order to describe the electronic part of the wave function, various theoretical approaches can be used such as (i) the single-particle extended picture (DFT-based approaches), (ii) the many-body atomic picture (multiplet theory), and (iii) the many-body extended picture (beyond DFT methods).

#### 4.1.3.1 The Single-Particle Extended Picture of Electronic States

DFT-based methods can be used to describe the electronic states using a single-particle extended picture. Although DFT methods should formally only apply to ground state calculations, they are often used for the calculation of excited states probed in core level spectroscopies. DFT methods simplify the ground state wave function of *N* electrons by replacing them with a fictitious, non-interacting system of independent electrons, that have the same electronic density as the real system. The correct charge density minimizes the total energy of the system. The Schrödinger equation is transformed into a system of equations (called the “Kohn–Sham equations”) with an effective Hamiltonian and wave functions, which are functions of only one space variable. This implies that DFT is essentially a single-particle approach, although some many-particle (many-body) interactions are contained in the exchange and correlation term of the electronic effective potential. The exact
analytical expression of this term is not known. This means that DFT can only be applied in an approximate form for example using the local density approximation (LDA) or the generalized gradient approximation (GGA)].

One can determine the ground state wave function by solving the Kohn–Sham equations which are constructed from the single-particle wave functions. The ground state wave function is therefore formed by a single Slater determinant. This is an important point because it limits the ability to treat a many-body response of the system described through a linear combination of Slater determinants. To illustrate this let us take an example with two electrons. Coupling $s = 1/2$ to $s = 1/2$ yields $S = 1$ or $0$. This corresponds to four $|S, Ms\rangle$ wave functions: $|S = 1, Ms = 1\rangle$, $|S = 1, Ms = -1\rangle$, $|S = 1, Ms = 0\rangle$, and $|S = 0, Ms = 0\rangle$. These functions need to meet the property of being anti-symmetric under particle exchange and it can be shown that only $|S = 1, Ms = 1\rangle$, $|S = 1, Ms = -1\rangle$ can each be expressed as a single Slater determinant. However, the $|S = 1, Ms = 0\rangle$ and $|S = 0, Ms = 0\rangle$ states can only be expressed as the combination of two Slater determinants and thus cannot be calculated in DFT.

One can group the various DFT-based methods according to their characteristics:

- Cluster or periodic: The Kohn–Sham equations can be solved either for a cluster centred around the absorbing atom (direct or real space methods) or starting from a unit cell of the crystal (or a multiple unit cell, which is called supercell) in order to take advantage of the 3D periodicity (reciprocal space methods).

- Self-consistency or not: The Kohn–Sham equations can be solved without or (preferably) with self-consistency, i.e., using an iterative cycle where two successive steps are mixed until a convergence criterion is reached to determine the charge density.

- Type of basis functions used to expand the orbital solutions of the Kohn–Sham equations: either localized functions [linear combination of atomic orbitals (LCAO), linear muffin-tin orbitals (LMTO)], or delocalized functions [plane waves (PW), full-potential linearized augmented plane waves (FLAPW)].

- Approximation made on the shape of the electronic potential: For example in LMTO or multiple scattering theory, the potential is approximated to be spherically symmetric in the atoms, and constant between them (muffin-tin). In full-potential methods [FLAPW, or projector augmented wave (PAW)-pseudopotentials], no approximation is made, which is generally preferable, even though it makes the calculations more consuming.

### 4.1.3.2 The Many-Body Atomic Picture of Electronic States

#### A Simple Introduction to the Many-Body Atomic Picture

Let us consider as an example a $Cr^{3+}$ ion in an octahedral ($O_h$) environment. Here the solid is reduced to an atom embedded in a mean field known as the crystal field (CF) that mimics the effect of the inter-atomic interactions. The atomic electronic configuration is $1s^22s^22p^63s^23p^63d^3$. The degeneracy of the Cr 3d levels is lifted
due to the CF and the 3d orbitals are split into two groups: the $e_g$ orbitals pointing towards the ligands and the $t_{2g}$ orbitals pointing between the ligands. The number of possible electronic states is given by the number of allowed arrangements of three electrons into ten spin orbitals, i.e., $C_{10}^3 = 120$ microstates. The energy separation between these states arises due to the combined effect of: (i) CF splitting, (ii) electronic repulsions, and (iii) spin-orbit coupling, i.e., the multiplet effects. Electrons occupying closed shells do not actually contribute to the energy splitting of the electronic levels; there is only one way to completely fill a shell giving a single average energy of the configuration.

All these multiplet states can be further grouped in so-called term symbols (or spectroscopic terms) according to their energy, spin and orbital moments. The relative energy positions of these spectroscopic terms for a $3d^n$ transition metal ion in $O_h$ CF (and neglecting 3d spin-orbit coupling) were calculated by Tanabe and Sugano and are available in several references (e.g., [3, 4]). Similar diagrams are available in [5] for symmetries lower than $O_h$, such as trigonal or tetragonal. The relative energies of the electronic states depend on the CF parameters as well as the Racah parameters that relate to the electronic repulsions. The determination of the spectroscopic terms becomes very complex when the spin-orbit coupling and Zeeman terms are included in the Hamiltonian and/or if lower symmetries are considered. LFM theory takes these effects into account and has been realized in several computer codes.

## Key Ideas of Ligand-Field Multiplet Theory

Atomic multiplet theory, crystal field multiplet theory, and LFM theory (sometimes collectively referred to as the multiplet theory) are based on concepts that were developed in atomic physics and make use of group theory. One has to solve the Schrödinger equation for the ion with its $N$ electrons in a given configuration

$$ \hat{H}|g\rangle = E|g\rangle, \qquad (4.4) $$

where $\hat{H}$ is the Hamiltonian of the system for the chosen configuration, $E$ and $|g\rangle$ are the eigenvalue and eigenstate, respectively. The different eigenstates are functions of $N$ electrons, hence they are called many-body (or multi-electronic) states. The Hamiltonian is expressed as

$$ \hat{H} = \hat{T} + \hat{V} + \hat{V}_{ee} + \hat{H}_{SO} + \hat{H}_{CF}, \qquad (4.5) $$

where $\hat{T}$ is the kinetic energy of the electrons, $\hat{V}$ the Coulomb attraction between electrons and the nucleus, $\hat{V}_{ee}$ the electron-electron Coulomb interaction, $\hat{H}_{SO}$ the spin-orbit coupling interaction, and $\hat{H}_{CF}$ the CF Hamiltonian, which takes into account the local environment of the absorbing atom.

These interactions will now be expressed in second quantization formalism. In this notation, any operator can be expressed in terms of creation ($c_\tau^\dagger$) and annihilation ($c_\tau$) operators. The operator $c_\tau^\dagger$ creates a state characterized by the quantum numbers $\tau$ (for example, if we choose to express the states as spin-orbitals, $\tau$ will be the set
of quantum numbers $n, l, m, \sigma$ that give the principal quantum number, the orbital momentum, the projected orbital momentum, and the projected spin momentum that uniquely identify this state) when it acts on the vacuum state $|0\rangle$. The operator $c_{\tau}$ is the annihilation operator of the state $\tau$. In this formalism, the (spherical) atomic interactions write

$$ \hat{T} = \sum_{\tau_1, \tau_2} \langle \tau_1 | \frac{\hat{p}^2}{2m} | \tau_2 \rangle c_{\tau_1}^{\dagger} c_{\tau_2}, \quad (4.6) $$

$$ \hat{V} = \sum_{\tau_1, \tau_2} \langle \tau_1 | -\frac{Ze^2}{\hat{r}} | \tau_2 \rangle c_{\tau_1}^{\dagger} c_{\tau_2}, \quad (4.7) $$

$$ \hat{V}_{ee} = \frac{1}{2} \sum_{\tau_1, \tau_2, \tau_3, \tau_4} \langle \tau_1 \tau_2 | \frac{e^2}{|\hat{r} - \hat{r}'|} | \tau_3 \tau_4 \rangle c_{\tau_2}^{\dagger} c_{\tau_1}^{\dagger} c_{\tau_3} c_{\tau_4}, \quad (4.8) $$

$$ \hat{V}_{SO} = \sum_{\tau_1, \tau_2} \langle \tau_1 | \xi \hat{l} \cdot \hat{s} | \tau_2 \rangle c_{\tau_1}^{\dagger} c_{\tau_2}. \quad (4.9) $$

Here $\hat{p}$ is the linear momentum operator, $m$ is the electron mass, $e$ is the electron charge, $\hat{r}$ is the position operator, $\hat{l}$ and $\hat{s}$ are the orbital and spin momenta operators, and $\xi$ is an atom dependent constant that is a function of the gradient of the atomic potential ($\xi \propto \frac{1}{r} \frac{dV}{dr}$). The kinetic energy of the electrons and the Coulomb interaction of the electrons with the nucleus are fixed for a given atomic configuration and they contribute only to the average energy of the configuration; hence $\hat{T}$ and $\hat{V}$ do not contribute to the multiplet splitting, and will not be further discussed. As a matter of fact, they are typically not evaluated in standard multiplet calculation programs. However, we are left with the task of simplifying the terms $\hat{V}_{SO}$ and $\hat{V}_{ee}$. Let us start with $\hat{V}_{SO}$ and assume that $z$ is the quantization axis.

## Influence of Spin-Orbit Coupling Interaction

The spin-orbit interaction is given as follows:

$$ \begin{aligned} \hat{V}_{SO} &= \sum_{\tau_1, \tau_2} \langle \tau_1 | \xi (l_x s_x + l_y s_y + l_z s_z) | \tau_2 \rangle c_{\tau_1}^{\dagger} c_{\tau_2} \\ &= \sum_{\tau_1, \tau_2} \langle \tau_1 | \xi l_z s_z | \tau_2 \rangle c_{\tau_1}^{\dagger} c_{\tau_2} \\ &\quad + \sum_{\tau_1, \tau_2} \langle \tau_1 | \xi (l_x s_x + l_y s_y) | \tau_2 \rangle c_{\tau_1}^{\dagger} c_{\tau_2}. \end{aligned} \quad (4.10) $$

We shall use a set of atomic spin-orbitals as basis functions, $\psi_i = R_{n_i, l_i}(r) Y_{l_i, m_i}(\theta, \phi) \sigma_i$ where $Y_{l_i, m_i}(\theta, \phi)$ is the spherical harmonic, and $R_{n_i, l_i}(r)$ is the radial part. Given that the potential in $\xi$ has a spherical form one can separate the radial and angular parts of the Hamiltonian. The angular part of the first term gives
$$ \sum_{(l_1,m_1),(l_2,m_2)} \langle Y_{l_1,m_1} \sigma_1 | l_z s_z | Y_{l_2,m_2} \sigma_2 \rangle c_{l_1,m_1,\sigma_1}^{\dagger} c_{l_2,m_2,\sigma_2} \\ = \frac{1}{2} \sum_{m=-l}^{l} m (c_{l,m,\uparrow}^{\dagger} c_{l,m,\uparrow} - c_{l,m,\downarrow}^{\dagger} c_{l,m,\downarrow}) . \qquad (4.11) $$

The second term gives

$$ \sum_{(l_1,m_1),(l_2,m_2)} \langle Y_{l_1,m_1} \sigma_1 | (l_x s_x + l_y s_y) | Y_{l_2,m_2} \sigma_2 \rangle c_{l_1,m_1,\sigma_1}^{\dagger} c_{l_2,m_2,\sigma_2} \\ = \sum_{(l_1,m_1),(l_2,m_2)} \langle Y_{l_1,m_1} \sigma_1 | \frac{1}{2} (l^+ s^- + l^- s^+) | Y_{l_2,m_2} \sigma_2 \rangle c_{l_1,m_1,\sigma_1}^{\dagger} c_{l_2,m_2,\sigma_2} \quad (4.12) \\ = \frac{1}{2} \sum_{m=-l}^{l-1} \sqrt{(l-m)(l+m+1)} (c_{l,m+1,\downarrow}^{\dagger} c_{l,m,\uparrow} + c_{l,m,\uparrow}^{\dagger} c_{l,m+1,\downarrow}) . $$

Hence the angular part of the spin-orbit Hamiltonian finally writes

$$ \hat{H}_{SO} = \frac{1}{2} \sum_{m=-l}^{l} m (c_{l,m,\uparrow}^{\dagger} c_{l,m,\uparrow} - c_{l,m,\downarrow}^{\dagger} c_{l,m,\downarrow}) \\ + \frac{1}{2} \sum_{m=-l}^{l-1} \sqrt{(l-m)(l+m+1)} (c_{l,m+1,\downarrow}^{\dagger} c_{l,m,\uparrow} + c_{l,m,\uparrow}^{\dagger} c_{l,m+1,\downarrow}) . \quad (4.13) $$

It is clear from (4.13) that the spin-orbit interaction mixes states with different projected orbital and spin momenta.

## The Electron–Electron Coulomb Interaction

Now we undertake the simplification of the electron–electron Coulomb interaction. This is more involved than the simplification of the spin-orbit coupling Hamiltonian. Cowan nicely explains the details of the derivation in his book [6]. We will rely on a combination of the derivations by Cowan [6] and Haverkort [7] in this section. This is not a thorough derivation; it is only meant to qualitatively explain the origin of multiplet splittings.

The first step to simplify this Hamiltonian is to perform a multipole expansion of the term $\frac{1}{\hat{r} - \hat{r}'} = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} Y_{k,m}^{*}(\theta', \phi') \frac{4\pi}{2k+1} \frac{r_{l}^{k}}{r_{l+1}^{k+1}} Y_{k,m}(\theta, \phi)$, where $r_{<}^{k}$ and $r_{>}^{k}$ are, respectively, the lesser and greater of the distances $r$ and $r'$. Now we are in a position to separate the radial and angular terms of the expression and separate the angular variables of each electron. Using the atomic spin-orbital basis to express the matrix elements of the angular part of the Hamiltonian one finds that
$$ 
\begin{aligned}
& \sum_{\tau_1, \tau_2, \tau_3, \tau_4} \langle Y_{l_1, m_1}(\theta, \phi) \sigma_1 Y_{l_2, m_2}(\theta', \phi') \sigma_2 | \\
& \qquad \sum_{k=0}^{\infty} \sum_{m=-k}^{k} Y_{k,m}^*(\theta', \phi') Y_{k,m}(\theta, \phi) | \\
& \qquad Y_{l_3, m_3}(\theta, \phi) \sigma_3 Y_{l_4, m_4}(\theta', \phi') \sigma_4 \rangle \\
& = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} \sum_{\tau_1, \tau_2, \tau_3, \tau_4} \langle Y_{l_1, m_1}(\theta, \phi) \sigma_1 | Y_{k, m}(\theta, \phi) | Y_{l_3, m_3}(\theta, \phi) \sigma_3 \rangle \\
& \qquad \langle Y_{l_2, m_2}(\theta', \phi') \sigma_2 | Y_{k, m}^*(\theta', \phi') | Y_{l_4, m_4}(\theta', \phi') \sigma_4 \rangle .
\end{aligned}
\quad (4.14) $$

We have in (4.14) integrals involving three spherical harmonics which are given by the Gaunt coefficients. This can be used to restrict the values of the summation over $k$ and $m$. The Gaunt coefficients are different from zero in the first integral only for $m = m_1 - m_3$. Similarly, the second integral is different from zero for $m = m_4 - m_2$. Hence, in combination, one concludes that the total $M_z$ is conserved ($m_1 + m_2 = m_3 + m_4$) for the integrals. Furthermore, the values of $k$ are restricted to values of $k \le \min(|l_1 + l_4|, |l_2 + l_3|)$. This simplifies the angular part.

Let us now investigate the radial part. The general expression for a scattering event involving four different shells is expressed in (4.15)

$$ \sum_{\tau_1, \tau_2, \tau_3, \tau_4} \langle R_{n_1, l_1}(r) R_{n_2, l_2}(r') | \sum_{k=0}^{\min(|l_1+l_4|, |l_2+l_3|)} \frac{r_k^k}{r^{k+1}} | R_{n_3, l_3}(r) R_{n_4, l_4}(r') \rangle . \quad (4.15) $$

In the case of Coulomb interaction in a single shell, $n_1 = n_2 = n_3 = n_4$ and $l_1 = l_2 = l_3 = l_4$ and hence one obtains the expression

$$ F^{(k)} = \sum_{k=0}^{2l} \langle R_{n,l}(r) R_{n,l}(r') | \frac{r_{<}^{k}}{r_{>}^{k+1}} | R_{n,l}(r) R_{n,l}(r') \rangle . \quad (4.16) $$

For a $3d^n$ configuration, $2l = 4$ yielding values of $k = 0, 2, 4$ and one has to evaluate three Slater integrals $F^{(k)}$ for such a configuration. For an excited state with a core hole, like the excited state of a $L$ absorption edge with a configuration $2p^53d^{n+1}$, it is necessary to take into consideration the Coulomb interaction between the $2p$ and the $3d$ shells. There are two possible cases for these scattering events (see also Chap. 2):

1. Direct interaction with $n_1 = n_3$ and $n_2 = n_4$. This means that each electron scatters in its shell. The matrix elements read

$$ F^{(k)} = \sum_{k=0}^{\min(2l_1, 2l_2)} \langle R_{n_1, l_1}(r) R_{n_2, l_2}(r') | \frac{r_{<}^{k}}{r_{>}^{k+1}} | R_{n_1, l_1}(r) R_{n_2, l_2}(r') \rangle . \quad (4.17) $$

2. Exchange interaction with $n_1 = n_4$ and $n_2 = n_3$. This means that electrons exchange shells. The matrix elements read
$$G^{(k)} = \sum_{k=0}^{\min(|2l_1, 2l_2|)} \langle R_{n_1, l_1}(r) R_{n_2, l_2}(r') | \frac{r_{<}^k}{r_{>}^{k+1}} | R_{n_2, l_2}(r) R_{n_1, l_1}(r') \rangle . \quad (4.18)$$

The $F^{(k)}$ and $G^{(k)}$ are called Slater integrals. The magnitude of the multiplet splitting depends on the magnitude of the Slater integrals. For an atomic calculation (corresponding to the case of a free ion in spherical symmetry) radial integrals are calculated self-consistently using a Hartree-Fock model, with values typically of the order of a few eV (not more than tens of eV). In the point group symmetry, where the absorber is considered with its environment, these values are reduced empirically in order to take into account the effect of the chemical bond which delocalizes the electrons. This reduction factor is an adjustable parameter (typically, 60-80% for a iono-covalent bond, 100% being the ionic limit case of a free ion).

## The Crystal Field Hamiltonian

Let us now extend our theoretical framework to include the effect of the CF potential on the absorbing ion. This is done by considering the N nearest neighbours as point charges ($Z_i e$) at positions $R_i$. The electrostatic potential due to these point charges at position $\mathbf{r}$, $V_{CF}(\mathbf{r})$, is expressed as

$$V_{CF}(\mathbf{r}) = \sum_{i=1}^{N} \frac{Z_i e^2}{|\mathbf{r} - \mathbf{R}_i|} . \quad (4.19)$$

A multipole expansion of the potential can be used to expand (4.19), leading to the expression

$$V_{CF}(r, \theta, \phi) = e^2 \sum_{k=0}^{\infty} r^k \sum_{m=-k}^{k} C_{k,m}(\theta, \phi) Q_{k,m}, \quad (4.20)$$

with $Q_{k,m} = \left[ \frac{4\pi}{2k+1} \right]^{1/2} \sum_{i=1}^{N} Z_i (\frac{1}{R_i})^{k+1} Y_{k,m}^*(\theta_i, \phi_i)$. $Y_{k,m}$ is the spherical harmonic, $C_{k,m}$ is the renormalized spherical harmonic, and * is the complex conjugate. Note here that one assumes that the radial extent of the 3d orbitals is smaller than the distance between the absorbing ion and its first neighbours (i.e., $r \ll R_i$). The CF Hamiltonian can be developed using the single-particle basis (atomic spin-orbitals as discussed before for the Coulomb interaction). Separating the radial and angular parts leads to

$$\hat{H}_{CF} = \sum_{\tau_1, \tau_2} \sum_{k,m} A_{k,m} \langle Y_{\ell_1, m_1} \sigma_1 | C_{k,m} | Y_{\ell_2, m_2} \sigma_2 \rangle c_{\tau_1}^{\dagger} c_{\tau_2} . \quad (4.21)$$

Here, the $A_{k,m}$ combine all the radial parts of (4.20). They are related to the usual CF parameters (10Dq, Ds, Dt, ...) which are usually not known precisely. The CF parameters (or $A_{k,m}$) are either fitted parameters or taken from experiments (optical absorption, electron paramagnetic resonance, ...). It is important to warn the reader against the temptation to fit the calculation with an unreasonable number of CF
parameters: with adjustable parameters, one runs the risk of producing some spectra in good agreement with experiments not for a theoretically justifiable reason but rather due to some lucky cancellation between inappropriate choices of the parameters and inaccurate theoretical approximations employed in solving the model. It is therefore important that one uses some sensible limits for these parameters and critically examines the values used in the model.

Equation 4.21 is a useful expression because the matrix elements are integrals again over three spherical harmonics and are given by the Gaunt coefficients. Furthermore, the series can be truncated according to the triangular condition. For 3d orbitals (like in the case of an Fe ion), $ℓ_1 = ℓ_2 = 2$ so the maximum value of m possible is $m = 4$ with $k ≤ 4$. The actual form of the matrix elements depends on the symmetry of the CF potential.

We will present the example of an octahedral ($O_h$) cluster to illustrate the procedure of calculating the CF matrix elements. In the case of an $O_h$ cluster, six neighbours are positioned at equal distances from the central ion as shown in Fig. 4.1. We will first calculate the multipole terms possible for this configuration according to (4.20). This can be easily evaluated and many softwares are available such as the “multipoles” Python package [8]. One finds that the multipole expansion of the octahedral potential reduces to the following terms with coefficients listed below $C_{0,0} → 6$,
$$C_{4,0} → \sqrt{\frac{49}{4}}, C_{4,4} → \sqrt{\frac{35}{8}}, C_{4,-4} → \sqrt{\frac{35}{8}}.$$

Now we can evaluate the matrix elements of (4.21) using the Gaunt coefficients. One finds the following matrix for $O_h$ crystal field

$$\sum_{k,m} \langle Y_{2,m_1} | C_{k,m} | Y_{2,m_2} \rangle \propto \begin{bmatrix} 1 & 0 & 0 & 0 & 5 \\ 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 6 & 0 & 0 \\ 0 & 0 & 0 & -4 & 0 \\ 5 & 0 & 0 & 0 & 1 \end{bmatrix}. \quad (4.22)$$

Diagonalizing this matrix gives the following eigenvalues $E = 6, -4, -4, 6, -4$ for the eigenvectors

$$\begin{bmatrix} \frac{1}{\sqrt{2}} \\ 0 \\ 0 \\ 0 \\ \frac{1}{\sqrt{2}} \end{bmatrix}, \begin{bmatrix} -\frac{1}{\sqrt{2}} \\ 0 \\ 0 \\ 0 \\ \frac{1}{\sqrt{2}} \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}. \quad (4.23)$$

This illustrates the splitting of the 3d one electron orbitals in an $O_h$ crystal field as shown in Fig. 4.1. The energy difference between the $t_{2g}$ and $e_g$ orbitals is referred to as $10Dq$ and its magnitude depends on the radial part $A_{k,m}$. The five degenerate orbitals split into two types of orbital:

1. Three orbitals of energies $-4Dq$ referred to as the $t_{2g}$ orbitals and which are $\frac{1}{\sqrt{2}}(Y_{2,-2} - Y_{2,2})$, $Y_{2,1}$, and $Y_{2,-1}$.
<figure><img src="image_44.png" /><figcaption>**Fig. 4.1** Splitting of the five degenerate 3d orbitals (left) into $t_{2g}$ and $e_g$ orbitals in an octahedral crystal field (right)</figcaption></figure>

2. Two orbitals of energies $6Dq$ referred to as the $e_g$ orbitals and which are $\frac{1}{\sqrt{2}}(Y_{2,-2} + Y_{2,2})$, and $Y_{2,0}$.

The same procedure can be used to determine the eigenstates and energies of the 3d (or any other) orbitals embedded in a certain symmetry.

#### 4.1.3.3 The Many-Body Extended Picture of Electronic States

The most recently developed approaches aim at extending both DFT and LFM theories to provide a comprehensive description of many-body interactions. An example for a rather manageable improvement to DFT is to add a static Hubbard parameter $U$ to account for on-site electronic repulsion. In addition, there are other approaches that go well beyond DFT. They can be grouped into two main types: the quantum chemistry approaches (mainly, wave function-based methods) and the Green's function methods.

Quantum chemistry approaches (DFT-CI, multi-configurational self consistent field [configuration interaction (CI)], coupled cluster, quantum Monte Carlo) are many-body extended approaches. In DFT-CI, for example, a combination of Slater determinants is used to describe the wave function of the system. The number of configurations that can be considered is limited by computing power and one has to decide which configurations to include. These approaches can only be applied for small clusters and molecules due to computational demand. Green's functions based methods, such as GW and dynamical mean-field theory (DMFT) provide an alternative approach to calculate the electronic structure of strongly correlated materials. In a GW method, a screened Coulomb interaction ($W$) is calculated following a DFT calculation of the charge density. A nonlocal energy dependent self-energy operator is required. Furthermore, DMFT can be utilized to map the full lattice problem onto
a single site quantum impurity problem, using a local screened Coulomb interaction (U). In both GW and DMFT approaches, correlations (many-body effects) are significantly better described than in standard DFT.

#### 4.1.3.4 Which Approach Works Best for Core Level Spectroscopy Calculations?

Unfortunately this is a very complex question: it depends on the chemical system (ionic, covalent, strongly correlated, ...), the edge (K-, L-edges, ...) and the type of spectroscopy (absorption, photoemission, ...). Consequently, in the following we try to provide some guiding considerations.

A good starting criterion for the choice of the method of calculation is the localization of the final state wave function. When electrons are excited into highly delocalized orbitals, they interact much less with other electrons meaning that the intra-atomic interactions are expected to be small. In this case single-particle approaches often give satisfactory results as the multi-electronic effects are small. This may be the case for the *K* edge of 3d elements and the *L*_{2,3} edges of heavy elements (e.g., rare earths, 5d transition metals), as well as the *K* edges of ligands (such as C, N, O, S).

On the other hand, when electrons are excited into localized orbitals, they will strongly interact with each other and the core hole. In this case, the multi-electronic effects become significant, and description considering only the absorber atom may be more successful such as LFM theory. This is typically the case for *L*_{2,3} edges of 3d elements and *M*_{4,5} edges of 4f elements.

In intermediate cases where both intra- and inter-atomic effects are relevant, like the *K* pre-edge of 3d elements, single-particle and many-body approaches may work or fail. It may also happen that different energy ranges of a spectrum are best described by different theoretical approaches. This is often the case when a weak pre-edge feature is observed before the strong main edge. The pre-edges often arise from excitations into localized orbitals while the main edge has a more delocalized character. Whether the final state is localized or not may be derived from the shape of the absorption edge. When the edge is dominated by a step function, the final state can be assumed to be delocalized. If it exhibits distinct peaks that decrease sharply, this may be due (but not necessarily) to a localized final state.

### 4.1.4 Codes for Ligand-Field Multiplet Calculations

Regarding the practical aspects of calculating core level spectra, one faces yet another level of complexity as the plethora of theoretical methods mentioned above are implemented in a comparably large number of computational packages. Instead of trying to provide an overview of all the available computational tools, we will focus on introducing Quany, one of the currently available software packages that can
be used to perform ligand-field multiplet calculations, and Crispy a graphical user interface that uses Quanty as a computational engine.

Quanty [9–12] is developed by Maurits Haverkort and his collaborators at the Institute of Theoretical Physics at Heidelberg University. It is a computational library that can be used to write quantum mechanical programs in the second quantization formalism. While the library can be used to describe a wide range of problems, it is specifically aimed at calculating different spectroscopies, including core level spectroscopy. Briefly, the user starts by constructing the Hamiltonian for the system of interest, diagonalize it, selects several eigenstates, and then calculates the spectrum corresponding to these eigenstates. In Quanty the Hamiltonian is expressed in a basis of one particle modes, which can be both fermionic and bosonic. The fermionic modes are usually spin-orbitals. In semi-empirical multiplet calculations, the interactions between the spin-orbitals are parametrized using values calculated for isolated atoms, which are afterwards scaled to account for the effect of the surrounding atoms. Alternatively, the parameters can be calculated directly by using for example DFT-based methods [9]. After the diagonalization of the Hamiltonian and the selection of the lowest eigenstates, using, for example, Boltzmann statistics, Quanty can be used to calculate the spectrum. As mentioned previously, this is done using a Green’s function approach, thereby avoiding the sum-over-states calculation, which can lead to an important reduction in computational time.

The core of the library is written using the C/C++ programming language for maximum efficiency. The users do not interact directly with this part of the code, but rather with the Lua-based layer that wraps it. To run calculations the users are required to write small programs using the functions defined in Quanty. Doing this in a scripting language such as Lua has the advantage of providing an ideal environment for experimentation, circumventing the limitations of compiled programming languages such as C or C++. While this is indeed very helpful, it is not uncommon for such programs to reach more than a few tens of lines of code, which in itself can be intimidating for the majority of new users. Also, as it is the case for many scientific libraries, because of the flexibility given to the users when writing these programs, it is impossible to check for all the things that might be incorrect. This leads to errors that are difficult to trace even for experienced users.

To help users to more easily perform Quanty calculations, one of us (M. Rete gan) has developed a friendly user interface that exposes the library’s capabilities for a large part of core level spectroscopies. Crispy [13] was developed using the Python programming language and relies on additional packages from the Python ecosystem. The main window of the application is shown in Fig. 4.2. Using Crispy, the users can quickly adjust the parameters of the calculation, run it, and plot the resulting spectrum without the need of writing any programs. The approach has many advantages for novice users, but even experienced ones can use Crispy to generate a starting program that will become the basis of their calculation. Crispy is a free and open-source program that can be installed on any operating system that has an up-to-date Python distribution. For Windows® and macOS® the program comes in easy to install packages that can be downloaded from the official website, http:// www.esrf.eu/computing/scientific/crispy.
<figure><img src="image_45.png" /><figcaption>**Fig. 4.2** Crispy's main window showing a calculated XAS $L_{2,3}$ spectrum for a Co$^{2+}$ ion in octahedral symmetry</figcaption></figure>

## 4.2 Spherical Tensor Expansion of the XAS Cross Section

A spherical tensor is a set of components that transform into each other under arbitrary rotations. Another way to state this is to say that the components of a spherical tensor generate a vector space which is invariant under rotation. A spherical tensor is irreducible if this vector space cannot be written as the sum of two invariant (non-zero) subspaces. For an irreducible tensor of rank *j*, the dimension of the corresponding vector space is $2j + 1$. For example, spherical harmonics $Y_{l,m}$ are the spherical tensor components of a spherical tensor of rank *l*. Note that while a $3 \times 3$ matrix is an irreducible Cartesian tensor, it is a reducible spherical tensor which is the sum of $j = 0, j = 1$, and $j = 2$ irreducible spherical tensors. It is evident that such an expansion would provide us with deeper insights by identifying groups of spectra that obey certain symmetry transformation rules which one could easily relate back to the system symmetry [14].

Spherical tensor analysis has been used with great success for the X-ray photoelectron of localized magnetic systems [15–19] and in XAS [14, 20–22], including XNLD [23]. The underlying idea is to determine a finite set of fundamental spectra in terms of which all possible experimental spectra can be expressed. More precisely, the XAS spectrum obtained for a given polarization vector ($\epsilon$) and wave vector ($k$) of the incident beam is written as a sum of terms which are fundamental spectra [21, 22] (depending only on the sample properties) multiplied by an angular coefficient
depending only on the experimental conditions (k, ε). Such a geometric and fully decoupled expression is useful: (i) to disentangle the properties of the sample from those of the measurement; (ii) to determine specific experimental arrangements aiming at the observation of specific sample properties; (iii) to provide the most convenient starting point to investigate the reduction of the number of fundamental spectra due to crystal symmetries.

### 4.2.1 The Case of Electric Dipole Transitions

The first step is to build rank one spherical tensors from the vectors appearing in the transition operator. The polarization vector $\epsilon = [\epsilon_x, \epsilon_y, \epsilon_z]$ can be written as a spherical tensor $\epsilon^1$ with components $\epsilon_{-1}^1 = \frac{\epsilon_x - i\epsilon_y}{\sqrt{2}}$, $\epsilon_1^1 = -\frac{\epsilon_x + i\epsilon_y}{\sqrt{2}}$, and $\epsilon_0^1 = \epsilon_z$. Similarly, the position spherical tensor, $\mathbf{r}^1$, can be constructed. In the following we shall use the following notation for the coupling of spherical tensors $\mathbf{P}^a$ and $\mathbf{Q}^b$ of ranks $a$ and $b$ into a spherical tensor of rank $c$

$$ \{\mathbf{P}^a \otimes \mathbf{Q}^b\}_\gamma^c = \sum_{\alpha=-a}^a \sum_{\alpha=-b}^b (a\alpha b\beta | c\gamma) P_\alpha^a Q_\beta^b, \quad (4.24) $$

with $(a\alpha b\beta | c\gamma)$ being the Clebsch–Gordan coefficients. Therefore,

$$ \mathbf{P}^a \cdot \mathbf{Q}^a = \sum_{\alpha=-a}^{a} (-1)^a P_{-\alpha}^a Q_{\alpha}^a = (-1)^a \sqrt{2a+1} \{\mathbf{P}^a \otimes \mathbf{Q}^a\}^0. \quad (4.25) $$

One has now to compute the scalar product of both tensors which is given by (4.25). The dipole transition operator can be written as in (4.26) taking into consideration that $\mathbf{r}$ is real while $\epsilon$ is in general complex

$$ \begin{aligned} T &= -\sqrt{3}\{\epsilon^1 \otimes \mathbf{r}^1\}^0 \\ T^\dagger &= -\sqrt{3}\{\epsilon^{1*} \otimes \mathbf{r}^1\}^0. \end{aligned} \quad (4.26) $$

We can recouple the cross section such that polarization tensors are coupled to each other and position tensors are coupled to each other. This means that the expression will have a part that depends only on the experimental geometry (polarization vector) and a part that depends only on the sample properties. This recoupling can be done using the identity

$$ \{\mathbf{P}^g \otimes \mathbf{Q}^g\}^0 \cdot \{\mathbf{R}^d \otimes \mathbf{S}^d\}^0 = \sum_a (-1)^a \frac{\{\mathbf{P}^g \otimes \mathbf{R}^d\}^a \cdot \{\mathbf{Q}^g \otimes \mathbf{S}^d\}^a}{\sqrt{(2g+1)(2d+1)}}. \quad (4.27) $$
Here $a$ is constrained to $|g-d| \le a \le g+d$. Hence, for the dipole transition, $g=d=1$ and $0 \le a \le 2$. The recoupled XAS cross section is finally expressed as follows:

$$ \sigma_{\omega} = -4\pi\alpha\hbar\omega \mathrm{Im} \left[ \sum_{a=0}^{2} (-1)^a \{\epsilon^{1*} \otimes \epsilon^1\}^a \cdot \langle I | \mathbf{r}^1 G^+ \mathbf{r}^1 | I \rangle^a \right] . \quad (4.28) $$

Quantity can calculate the energy dependent tensors $R^{(a)} = \langle I | \mathbf{r}^1 G^+ \mathbf{r}^1 | I \rangle^a$ that depend only on the properties of the sample. We will refer to these elements as the fundamental spectra. Note that these fundamental spectra are sometimes referred to as $\sigma^{(a)}$. This could be confused with the total cross section $\sigma_{\omega}$ so we shall not use this notation here. The experimental geometry tensor is $E^a = \{\epsilon^{1*} \otimes \epsilon^1\}^a$.

### 4.2.1.1 Term $a=0$

The first term can be found by substituting $a=0$ in (4.28). This is the zero rank of the tensor, given in (4.29).

$$ \sigma(0, 0) = 4\pi\alpha\hbar\omega \times \mathrm{Im} \left[ \frac{1}{3} \left( \langle I | r C_{1,0}^* G^+ r C_{1,0} | I \rangle + \langle I | r C_{1,-1}^* G^+ r C_{1,-1} | I \rangle + \langle I | r C_{1,1}^* G^+ r C_{1,1} | I \rangle \right) \right]. \quad (4.29) $$

Here $C_{lm} = C_m^l = \sqrt{\frac{4\pi}{2l+1}} Y_l^m$. The term $\sigma(0, 0)$ is independent of the incident polarization vector and as such is rotation invariant. It gives the isotropic contribution of the XAS cross section.

### 4.2.1.2 Term $a=1$

The term $a=1$ consists of three components, namely, $\sigma(1, 0)$, $\sigma(1, 1)$, and $\sigma(1, -1)$:

$$ \sigma(1, 0) = -4\pi\alpha\hbar\omega \times \mathrm{Im} \left[ \frac{1}{2} (i\epsilon_x^* \epsilon_y - i\epsilon_x \epsilon_y^*) \times \left( \langle I | r C_{1,1}^* G^+ r C_{1,1} | I \rangle - \langle I | r C_{1,-1}^* G^+ r C_{1,-1} | I \rangle \right) \right], \quad (4.30) $$
$$ \sigma(1, 1) = -4\pi\alpha\hbar\omega \times \mathrm{Im}\left[\frac{-1}{2\sqrt{2}}(\epsilon_x^*\epsilon_z - \epsilon_x\epsilon_z^* + i\epsilon_y\epsilon_z^* - i\epsilon_y^*\epsilon_z)\right] \quad (4.31) $$

$$ \left(\langle I|rC_{1,0}^*G^+rC_{1,1}|I\rangle + \langle I|rC_{1,-1}^*G^+rC_{1,0}|I\rangle\right), \quad (4.32) $$

$$ \sigma(1, -1) = -4\pi\alpha\hbar\omega \times \mathrm{Im}\left[\frac{1}{2\sqrt{2}}(\epsilon_x^*\epsilon_z - \epsilon_x\epsilon_z^* + i\epsilon_y^*\epsilon_z - i\epsilon_y\epsilon_z^*) \times (\langle I|rC_{1,0}^*G^+rC_{1,-1}|I\rangle + \langle I|rC_{1,1}^*G^+rC_{1,0}|I\rangle)\right]. \quad (4.33) $$

One notices from (4.30), (4.32) and (4.33) that the spectra of $a = 1$ are not active if any of these two cases are satisfied:

1. If linearly polarized light is used for the measurement.

2. If all off-diagonal elements are zero and the diagonal elements are equal.

Another conclusion that can be drawn from the recoupling is the necessity to perform XAS measurements using both linearly and circularly polarized light to probe these fundamental spectra.

### 4.2.1.3 Term $a = 2$

The term $a = 2$ consists of five components, namely, $\sigma(2, 0)$, $\sigma(2, 1)$, $\sigma(2, -1)$, $\sigma(2, 2)$, and $\sigma(2, -2)$. These are given below as follows:

$$ \sigma(2, 0) = -4\pi\alpha\hbar\omega \times \mathrm{Im}\left[\frac{1}{6}(2|\epsilon_z|^2 - |\epsilon_x|^2 - |\epsilon_y|^2)(2\langle I|rC_{1,0}^*G^+rC_{1,0}|I\rangle - \langle I|rC_{1,-1}^*G^+rC_{1,-1}|I\rangle - \langle I|rC_{1,1}^*G^+rC_{1,1}|I\rangle)\right], \quad (4.34) $$

$$ \sigma(2, 1) = -4\pi\alpha\hbar\omega \times \mathrm{Im}\left[\frac{1}{2\sqrt{2}}(\epsilon_x\epsilon_z^* + \epsilon_x^*\epsilon_z - i\epsilon_y\epsilon_z^* - i\epsilon_y^*\epsilon_z) \times (\langle I|rC_{1,-1}^*G^+rC_{1,0}|I\rangle - \langle I|rC_{1,0}^*G^+rC_{1,-1}|I\rangle)\right], \quad (4.35) $$

$$ \sigma(2, -1) = -4\pi\alpha\hbar\omega \times \mathrm{Im}\left[\frac{1}{2\sqrt{2}}(\epsilon_x\epsilon_z^* + \epsilon_x^*\epsilon_z + i\epsilon_y\epsilon_z^* + i\epsilon_y\epsilon_z) \times (\langle I|rC_{1,0}^*G^+rC_{1,-1}|I\rangle - \langle I|rC_{1,1}^*G^+rC_{1,0}|I\rangle)\right], \quad (4.36) $$
$$ \sigma(2, 2) = -4\pi\alpha\hbar\omega \times \mathrm{Im} \left[ \frac{-1}{2} \left( (\epsilon_x - i\epsilon_y)(\epsilon_x^* - i\epsilon_y^*) \right) \times \left( \langle I | rC_{1,-1}^* G^+ rC_{1,1} | I \rangle \right) \right], \quad (4.37) $$

$$ \sigma(2, -2) = -4\pi\alpha\hbar\omega \times \mathrm{Im} \left[ \frac{-1}{2} \left( (\epsilon_x + i\epsilon_y)(\epsilon_x^* + i\epsilon_y^*) \right) \times \left( \langle I | rC_{1,1}^* G^+ rC_{1,-1} | I \rangle \right) \right]. \quad (4.38) $$

It can be noted from (4.34), (4.35), (4.36), (4.37), and (4.38) that the $a = 2$ spectra are active for linearly polarized light and hence these spectra are responsible for the angular dependence observed with linear light. On the contrary, no difference can be observed between right and left circularly polarized light. Another feature of these terms is that they are not active if the following two conditions are satisfied:

1. The diagonal matrix elements are equal.

2. The off-diagonal matrix elements are zero.

#### 4.2.1.4 General Dipole Expression

The general dipole expression is given in (4.39). From this equation, the dipole XAS cross-section for an arbitrary polarization ($\epsilon$) can be constructed from the nine fundamental spectra derived above.

$$ \sigma_{\omega}^{\text{Dipole}}(\epsilon) = -4\pi\alpha\hbar\omega \times \mathrm{Im} \left[ \frac{1}{3} R(0, 0) + \frac{1}{2} (i\epsilon_x^*\epsilon_y - i\epsilon_x\epsilon_y^*) R(1, 0) \right. \\ \left. - \frac{1}{2\sqrt{2}} (\epsilon_x^*\epsilon_z - \epsilon_x\epsilon_z^* + i\epsilon_y\epsilon_z^* - i\epsilon_y\epsilon_z^*) R(1, 1) \right. \\ \left. + \frac{1}{2\sqrt{2}} (\epsilon_x^*\epsilon_z - \epsilon_x\epsilon_z^* + i\epsilon_y^*\epsilon_z - i\epsilon_y\epsilon_z^*) R(1, -1) \right. \\ \left. + \frac{1}{6} (2|\epsilon_z|^2 - |\epsilon_x|^2 - |\epsilon_y|^2) R(2, 0) \right. \\ \left. + \frac{1}{2\sqrt{2}} (\epsilon_x\epsilon_z^* + \epsilon_x^*\epsilon_z - i\epsilon_y\epsilon_z^* - i\epsilon_y\epsilon_z^*) R(2, 1) \right. \\ \left. + \frac{1}{2\sqrt{2}} (\epsilon_x\epsilon_z^* + \epsilon_x^*\epsilon_z + i\epsilon_y\epsilon_z^* + i\epsilon_y\epsilon_z^*) R(2, -1) \right. \\ \left. - \frac{1}{2} ((\epsilon_x - i\epsilon_y)(\epsilon_x^* - i\epsilon_y^*)) R(2, 2) \right. \\ \left. - \frac{1}{2} ((\epsilon_x + i\epsilon_y)(\epsilon_x^* + i\epsilon_y^*)) R(2, -2) \right], \quad (4.39) $$
where the *R* are the fundamental spectra and are defined as

$$R(0, 0) = \langle I|rC_{1,0}^*G^+rC_{1,0}|I\rangle + \langle I|rC_{1,-1}^*G^+rC_{1,-1}|I\rangle \\ +\langle I|rC_{1,1}^{1*}G^+rC_{1,1}|I\rangle, \quad (4.40)$$

$$R(1, 0) = \langle I|rC_{1,1}^*G^+rC_{1,1}|I\rangle - \langle I|rC_{1,-1}^*G^+rC_{1,-1}|I\rangle, \quad (4.41)$$

$$R(1, 1) = \langle I|rC_{1,0}^*G^+rC_{1,1}|I\rangle + \langle I|rC_{1,-1}^*G^+rC_{1,0}|I\rangle, \quad (4.42)$$

$$R(1, -1) = \langle I|rC_{1,0}^*G^+rC_{1,-1}|I\rangle + \langle I|rC_{1,1}^*G^+rC_{1,0}|I\rangle, \quad (4.43)$$

$$R(2, 0) = 2\langle I|rC_{1,0}^*G^+rC_{1,0}|I\rangle - \langle I|rC_{1,-1}^*G^+rC_{1,-1}|I\rangle \\ -\langle I|rC_{1,1}^*G^+rC_{1,1}|I\rangle, \quad (4.44)$$

$$R(2, 1) = \langle I|rC_{1,-1}^*G^+rC_{1,0}|I\rangle - \langle I|rC_{1,0}^*G^+rC_{1,1}|I\rangle, \quad (4.45)$$

$$R(2, -1) = \langle I|rC_{1,0}^*G^+rC_{1,-1}|I\rangle - \langle I|rC_{1,1}^*G^+rC_{1,0}|I\rangle, \quad (4.46)$$

$$R(2, 2) = \langle I|rC_{1,-1}^*G^+rC_{1,1}|I\rangle, \quad (4.47)$$

$$R(2, -2) = \langle I|rC_{1,1}^*G^+rC_{1,-1}|I\rangle. \quad (4.48)$$

### 4.2.1.5 Case Study of a $d^9$ ion

#### Octahedral Crystal Field

Equation (4.39) can be simplified when the symmetry of the absorbing system is taken into account. As a demonstration, we shall study a $d^9$ ion in octahedral ($O_h$) symmetry. Figure 4.3 (top) shows the matrix elements for such a system. These matrix elements are the direct output of Quantity and will be referred to as the conductivity tensor. One finds that all the off-diagonal matrix elements are equal to zero and all the diagonal matrix elements are equal. This leaves only the $R(0, 0)$ term of (4.39) not equal to zero. Hence one can conclude that the cross section of a dipole transition is isotropic in an $O_h$ system.

#### Tetragonal Crystal Field

Let us now consider a tetragonal distortion such that the octahedron is compressed along the z-axis. The ground state in this case has a hole in the $d_{z^2}$ orbital (neglecting spin-orbit coupling) and the z-axis is now different from the x- and y-axes. The conductivity tensor for such a system is shown in Fig. 4.3 (bottom). As could be intuitively expected, the middle panel corresponding to $C_0^1G^+C_0^1$ is different from the other two diagonal elements. This implies that the following terms come into play (see Fig. 4.4):

* $R(0, 0)$ which gives the isotropic cross section.

* $R(2, 0)$ which has a polarization dependence of the form $\frac{1}{6}(2|\epsilon_z|^2 - |\epsilon_x|^2 - |\epsilon_y|^2)$.

It is interesting in this case to investigate what types of dichroism effect could be observed. Consider rotating the incident linear polarization vector in the x-y-plane.
<figure><img src="image_46.png" /><figcaption>**Fig. 4.3** Conductivity tensor calculated for a *d*⁹ ion in an octahedral crystal field (top) and in a tetragonal crystal field (bottom). Calculations are done with a crystal field parameter $10D_q = 1.1$ eV and with $D_s = -0.2$ eV in tetragonal symmetry. Re and Im are the real and imaginary parts of the tensor</figcaption></figure>
<figure><img src="image_47.png" /><figcaption>**Fig. 4.4** Fundamental spectra $R(0, 0)$ and $R(2, 0)$ for a dipole transition for a $d^9$ ion in a tetragonal crystal field ($10D_q = 1.1$ eV and $D_s = -0.2$ eV)</figcaption></figure>

In this case $\epsilon = [\cos(\theta), \sin(\theta), 0]$ where $\theta$ is the rotation angle defined from the $x$-axis. The expression of the polarization dependence for the term $R(2, 0)$ reveals that no angular dependence is to be expected in this case. The system is effectively $O_h$ in the $x-y$-plane and one would expect no angular dependence as discussed in the previous example. This XAS cross section for this rotation is shown in Fig. 4.5a. On the contrary, rotating the polarization vector in the $x-z$ ($\epsilon = [\cos(\theta), 0, \sin(\theta)]$) or $y-z$ ($\epsilon = [0, \cos(\theta), \sin(\theta)]$) planes should yield an angular dependence as the polarization vector probes the distortion, which is indeed observed as shown in Fig. 4.5b and c. The dependence of the XAS cross-section on the direction of the

<figure><img src="image_48.png" /><figcaption>**Fig. 4.5** Angular dependence of the $L_{2,3}$ XAS of a $3d^9$ ion in a tetragonal crystal field. Calculations are done by rotating the polarization vector in the $x-y$-plane [panel (a)], $x-z$-plane [panel (b)], and $y-z$-plane [panel (c)] as illustrated in the top panel</figcaption></figure>
linearly polarized light is an effect referred to as linear dichroism as discussed pre-
viously.

**Octahedral Crystal Field with an Exchange Field $\| z$**

Another interesting system to investigate is a magnetic 3d⁹ ion where the crystal
field is Oₕ with an exchange field aligned along the z-axis. Hence, the z-axis is
inequivalent to the x- and y-axes due to the exchange field. The conductivity tensor
of such a system is shown in Fig. 4.6. The exchange field is aligned along a high
symmetry direction in this example which preserves the C₄ rotation symmetry of
the system and consequently preserves the symmetry of the conductivity tensor. All
off-diagonal elements are zero. Note that the off-diagonal elements are zero because
we chose to calculate the tensor using the symmetry adapted transition operators.
Three fundamental spectra come into play and are plotted in Fig. 4.7:

• $R(0, 0)$ which gives the isotropic cross section,

• $R(1, 0)$ which has a polarization dependence of the form $\frac{1}{2}(i\epsilon_x^*\epsilon_y - i\epsilon_x\epsilon_y^*)$,

• $R(2, 0)$ which has a polarization dependence of the form $\frac{1}{6}(2|\epsilon_z|^2 - |\epsilon_x|^2 - |\epsilon_y|^2)$.

Nearly no angular dependence can be observed by rotating the incident linear
polarization vector in the x - y-, x - z-, and y - z-planes (see Fig. 4.8a, b, and c).
This is consistent with the fact that the fundamental spectrum R(2, 0) responsible
for the angular dependence is nearly zero [R(2, 0) is about two orders of magnitude
smaller than the other two fundamental spectra in this system]. The difference in the
absorption cross section of linear polarized light in a magnetic system is an effect
referred to as XMLD [24]. The magnitude of the XMLD effect for this system can
be seen in Fig. 4.9. Note that the magnitude of the XMLD effect in Fe₃O₄ is ~1%
of the XAS signal, which could be reliably measured on existing beamlines [25].

A strong dichroism is observed when circularly polarized light is used as in the
case for Fig. 4.10a. Here the incident polarization vector is either left or right polarized
about the z-axis leading to a difference in the absorption. This is an effect referred to
as X-ray magnetic circular dichroism (XMCD) [26]. It can be seen from the expres-
sion of the polarization part of the cross-section that if the incident wave vector is
aligned perpendicular to the exchange field, for example, for $\epsilon = [0, -\frac{i}{\sqrt{2}}, \frac{1}{\sqrt{2}}]$, and
$\epsilon = [0, -\frac{i}{\sqrt{2}}, -\frac{1}{\sqrt{2}}]$, no XMCD effect is observed. This is shown in Fig. 4.10b and c.
This dichroism can be used to quantify the ground state spin and orbital moments of
the system as given by the sum rules [27].

**Octahedral Crystal Field with an Exchange Field $\| [210]$**

As a last example, we consider a system in C₁ symmetry. Consider aligning the
exchange field along a low symmetry direction, e.g., [210]. The exchange field now
completely breaks the symmetry of the system and the conductivity tensor has off-
diagonal elements (bottom of Fig. 4.6). Contrary to the previous case (where the
exchange field was aligned to the z-axis), it is now not possible to find a rotated
<figure><img src="image_49.png" /><figcaption>**Fig. 4.6** Conductivity tensor calculated for a *d*⁹ ion in an octahedral crystal field (10Dq = 1.1 eV) with an exchange field (*B* = 0.05 eV) along the z-axis (top) and along the [210] direction (bottom). The calculations are performed using a real spherical harmonics basis set</figcaption></figure>
<figure><img src="image_50.png" /><figcaption>**Fig. 4.7** Fundamental spectra $R(0, 0)$, $R(1, 0)$, and $R(2, 0)$ for a dipole transition for a $d^9$ ion in an octahedral crystal field ($10D_q = 1.1$ eV) with an exchange field ($B = 0.05$ eV) aligned along the z-axis of the cluster</figcaption></figure>

<figure><img src="image_51.png" /><figcaption>**Fig. 4.8** Angular behaviour of the $L_{2,3}$ XAS of a $3d^9$ ion in an octahedral crystal field ($10D_q = 1.1$ eV) with the exchange field ($B_z = 0.05$ eV) aligned along the z-axis. The calculations are done by rotating the polarization vector in the **a** x – y-plane, **b** x – z-plane and **c** y – z-plane as depicted in the top panel</figcaption></figure>

basis set that diagonalizes the conductivity tensor for all excited states (i.e., the basis set becomes energy dependent). It remains possible to diagonalize the conductivity tensor for a given excited state. It is important to realize that when the exchange field is aligned along a low symmetry direction, off-diagonal elements become important and more dichroic effects come into play according to (4.39) and (4.48).

### 4.2.2 The Case of Electric Quadrupole Transitions

For electric quadrupole transitions, we will follow the same procedure as the one used for electric dipole. The transition operator is now (up to a factor of $i/2$) $T =$
<figure><img src="image_52.png" /><figcaption>**Fig. 4.9** X-ray magnetic linear dichroism of a 3d⁹ ion in an octahedral crystal field (10Dq = 1.1 eV) with the exchange field (Bz = 0.05 eV) aligned along the z-axis. The dichroism is computed by subtracting the XAS: **a** with **ϵ** || **x** from that with **ϵ** || **y**, **b** with **ϵ** || **x** from that with **ϵ** || **z** and **c** with **ϵ** || **y** from that with **ϵ** || **z**</figcaption></figure>

<figure><img src="image_53.png" /><figcaption>**Fig. 4.10** X-ray magnetic circular dichroism of a 3d⁹ ion in an octahedral surrounding with the exchange field (Bz) aligned along the z-axis. The dichroism is computed by subtracting the XAS signal calculated with right circularly polarized light from that with left polarized light. **a** The incident wave vector is aligned parallel to the z-axis. **b** The incident wave vector is aligned parallel to the y-axis. **c** The incident wave vector is aligned parallel to the x-axis</figcaption></figure>

($\epsilon \cdot r$)($k \cdot r$). It can be seen from the expression of the transition operator that the cross section will depend on the orientation of the polarization vector ($\epsilon$) and of the wave vector ($k$) with respect to the absorbing system. Two recoupling steps are required in this case. First, the transition operator can be rewritten into a combination of scalar products of two tensors: one tensor that depends only on $\epsilon$ and $k$ coupled together, and one tensor that depends only on the absorber $r$. This recoupled transition operator is expressed as follows:
$$ 
\begin{aligned}
\hat{T} &= \sum_{b=0}^{2} (-1)^b \{\epsilon^1 \otimes k^1\}^b \cdot \{r^1 \otimes r^1\}^b, \\
\hat{T}^\dagger &= \sum_{c=0}^{2} (-1)^c \{\epsilon^{1*} \otimes k^1\}^c \cdot \{r^1 \otimes r^1\}^c.
\end{aligned}
\qquad (4.49) 
$$

The next step is to recouple the two transition amplitudes of the absorption cross section. This gives the expression

$$ \sigma_{\omega} = \pi^2 \alpha \hbar \omega \times \mathrm{Im} \left[ \sum_{a=0}^{4} \sum_{b=0}^{2} \sum_{c=0}^{2} (-1)^a (-1)^b (-1)^c \{\{\epsilon^{*1} \otimes k^1\}^c \otimes \{\epsilon^1 \otimes k^1\}^b\}^a \times \{\langle I | \{r^1 \otimes r^1\}^c G^+ \{r^1 \otimes r^1\}^b | I \rangle\}^a \right]. \quad (4.50) $$

Before attempting to write out the recoupled absorption cross section in (4.50), it is useful to simplify the expression of the transition operator first. This in turn will simplify the expression of the absorption cross section. The transition operator is a rank two tensor according to (4.49) with $b = 0, 1, 2$. We shall write out the three b terms:

### Term $b=0$

$$ \begin{aligned} (-1)^0 \{\epsilon^1 \otimes k^1\}^0 \cdot \{r^1 \otimes r^1\}^0 = & \left(-\frac{1}{\sqrt{3}} \epsilon_0^1 k_0^1 + \frac{1}{\sqrt{3}} \epsilon_1^1 k_{-1}^1 + \frac{1}{\sqrt{3}} \epsilon_{-1}^1 k_1^1\right) \\ & \times \left(-\frac{1}{\sqrt{3}} r_0^1 r_0^1 + \frac{1}{\sqrt{3}} r_1^1 r_{-1}^1 + \frac{1}{\sqrt{3}} r_{-1}^1 r_1^1\right) \end{aligned} \quad (4.51) $$

The first part of the expression can be rewritten as $\frac{1}{\sqrt{3}}(-\epsilon_z k_z - \epsilon_x k_x - \epsilon_y k_y)$. This is equal to zero because the polarization vector is orthogonal to the wave vector. This means that the term $b=0$ is zero.

### Term $b=1$

This term consists of three components according to

$$ (-1)^1 \{\epsilon^1 \otimes k^1\}^1 \cdot \{r^1 \otimes r^1\}^1 = -\frac{i}{\sqrt{2}} \frac{i}{\sqrt{2}} (\epsilon \times k) \cdot (r \times r). \quad (4.52) $$

The second part of the expression is equal to zero because it is a cross product of the same vector. This means that the term $b=1$ is also zero.

### Term $b=2$

This term consists of five components. These five components can be simplified applying the orthogonality between $\epsilon$ and $k$. In addition the $r$ tensor can be expressed in terms of spherical harmonics of $l=2$ according to the relation
$\{\mathbf{r}^1 \otimes \mathbf{r}^1\}_m^2 = r_m^2 = \sqrt{\frac{8\pi}{15}} Y_{2,m}(\mathbf{r}) = \sqrt{\frac{8\pi}{15}} r^2 Y_{2,m}(\theta, \phi)$. One obtains the following five components after simplification

$$ \{\epsilon^1 \otimes k^1\}_0^2 \{\mathbf{r}^1 \otimes \mathbf{r}^1\}_0^2 = \left(\sqrt{\frac{3}{2}} \epsilon_z k_z\right) (r_0^2), \quad (4.53) $$

$$ \{\epsilon^1 \otimes k^1\}_1^2 \{\mathbf{r}^1 \otimes \mathbf{r}^1\}_{-1}^2 = \left(-\frac{k_z(\epsilon_x + i\epsilon_y) + (k_x + ik_y)\epsilon_z}{2}\right) (r_{-1}^2), \quad (4.54) $$

$$ \{\epsilon^1 \otimes k^1\}_{-1}^2 \{\mathbf{r}^1 \otimes \mathbf{r}^1\}_1^2 = \left(\frac{k_z(\epsilon_x - i\epsilon_y) + (k_x - ik_y)\epsilon_z}{2}\right) (r_1^2), \quad (4.55) $$

$$ \{\epsilon^1 \otimes k^1\}_2^2 \{\mathbf{r}^1 \otimes \mathbf{r}^1\}_{-2}^2 = \left(\frac{(k_x + ik_y)(\epsilon_x + i\epsilon_y)}{2}\right) (r_{-2}^2), \quad (4.56) $$

$$ \{\epsilon^1 \otimes k^1\}_{-2}^2 \{\mathbf{r}^1 \otimes \mathbf{r}^1\}_2^2 = \left(\frac{(k_x - ik_y)(\epsilon_x - i\epsilon_y)}{2}\right) (r_2^2). \quad (4.57) $$

The same arguments apply for the $c = 0, 1, 2$ terms of the recoupled $\hat{T}^\dagger$ operator ending up with the values $b = 2$ and $c = 2$ for the XAS cross section. The recoupled cross section writes

$$ \sigma_{\omega} = \pi^2 \alpha \hbar \omega k^2 \times \mathrm{Im} \left[ \sum_{a=0}^{4} (-1)^a \{\{\epsilon^{*1} \otimes k^1\}^2 \otimes \{\epsilon^1 \otimes k^1\}^2\}^a \times \{\langle I | \{\mathbf{r}^1 \otimes \mathbf{r}^1\}^2 G^+ \{\mathbf{r}^1 \otimes \mathbf{r}^1\}^2 | I \rangle\}^a \right]. \quad (4.58) $$

Now one can develop (4.58) in further details for $a = 0, 1, 2, 3, 4$. We shall only report the final expression here.

### 4.2.3 Term $a = 0$

Let us substitute $a = 0$ in (4.58). This is the zero rank of the tensor, $\sigma(0, 0)$, describing an isotropic spectrum

$$ \sigma(0, 0) = \pi^2 \alpha \hbar \omega k^2 \times \mathrm{Im} \left[ \frac{1}{10} \left( \langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,0} | I \rangle + \langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,-1} | I \rangle \right. \right. \\ \left. \left. + \langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,1} | I \rangle + \langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,-2} | I \rangle \right. \right. \\ \left. \left. + \langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,2} | I \rangle \right) \right] . \quad (4.59) $$
### 4.2.4 Term a = 1

The term $a = 1$ consists of three components, namely, $\sigma(1, 0)$, $\sigma(1, 1)$ and $\sigma(1, -1)$

$$ \sigma(1, 0) = \pi^2 \alpha \hbar \omega k^2 \times \mathrm{Im} \left[ \frac{i}{20} \left( (2k_x^2 + 2k_y^2 + k_z^2) \epsilon_x \epsilon_y^* - (2k_x^2 + 2k_y^2 + k_z^2) \epsilon_x^* \epsilon_y \right. \right. \\ \left. \left. + k_y k_z \epsilon_x \epsilon_z^* - k_y k_z \epsilon_z \epsilon_x^* + k_x k_z \epsilon_z \epsilon_y^* - k_x k_z \epsilon_y \epsilon_z^* \right) \right. \\ \left. \times \left( \langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,1} | I \rangle - \langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,-1} | I \rangle \right. \right. \\ \left. \left. + 2 \langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,2} | I \rangle - 2 \langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,-2} | I \rangle \right) \right], \quad (4.60) $$

$$ \sigma(1, 1) = \pi^2 \alpha \hbar \omega k^2 \times \mathrm{Im} \left[ \frac{1}{40} \left( (2k_x^2 + k_y^2 + 2k_z^2 - ik_x k_y) \epsilon_x^* \epsilon_z \right. \right. \\ \left. \left. -(2k_x^2 + k_y^2 + 2k_z^2 - ik_x k_y) \epsilon_x \epsilon_z^* - i(k_x^2 + 2k_y^2 + 2k_z^2 + ik_x k_y) \epsilon_y^* \epsilon_z \right. \right. \\ \left. \left. + i(k_x^2 + 2k_y^2 + 2k_z^2 + ik_x k_y) \epsilon_y \epsilon_z^* + (k_y + ik_x) k_z \epsilon_x^* \epsilon_y - i(k_x - ik_y) k_z \epsilon_x \epsilon_z^* \right) \right. \\ \left. \times \left( \sqrt{6} \langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,0} | I \rangle + \sqrt{6} \langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,1} | I \rangle \right. \right. \\ \left. \left. + 2 \langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,-1} | I \rangle + 2 \langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,2} | I \rangle \right) \right], \quad (4.61) $$

$$ \sigma(1, -1) = \pi^2 \alpha \hbar \omega k^2 \times \mathrm{Im} \left[ \frac{-1}{40} \left( (2k_x^2 + k_y^2 + 2k_z^2 + ik_x k_y) \epsilon_x^* \epsilon_z \right. \right. \\ \left. \left. -(2k_x^2 + k_y^2 + 2k_z^2 + ik_x k_y) \epsilon_x \epsilon_z^* + i(k_x^2 + 2k_y^2 + 2k_z^2 - ik_x k_y) \epsilon_y^* \epsilon_z \right. \right. \\ \left. \left. -i(k_x^2 + 2k_y^2 + 2k_z^2 - ik_x k_y) \epsilon_y \epsilon_z^* + k_z(k_y - ik_x) \epsilon_x^* \epsilon_y + ik_z(k_x + ik_y) \epsilon_x \epsilon_y^* \right) \right. \\ \left. \times \left( \sqrt{6} \langle I | r^2 C_{2,1}^* G^+ r_{2,0}^C | I \rangle + \sqrt{6} \langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,-1} | I \rangle \right. \right. \\ \left. \left. + 2 \langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,1} | I \rangle + 2 \langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,-2} | I \rangle \right) \right]. \quad (4.62) $$

A quick check of (4.60), (4.61) and (4.62) reveals that the term $a = 1$ is zero for linear light. This implies that these fundamental spectra can only be probed with circular or elliptically polarized light. It is also clear that if the conductivity tensor has no off-diagonal terms, and satisfies

$$ \langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,1} | I \rangle = \langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,-1} | I \rangle, $$

$$ \langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,2} | I \rangle = \langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,-2} | I \rangle, $$

then the term $a = 1$ is again zero.
### 4.2.5 Term a = 2

The term $a = 2$ consists of five components, namely, $\sigma(2, 0)$, $\sigma(2, 1)$, $\sigma(2, -1)$, $\sigma(2, 2)$, and $\sigma(2, -2)$

$$ \sigma(2, 0) = \pi^2 \alpha \hbar \omega k^2 \times \mathrm{Im} \left[ \frac{1}{84} \left( \begin{aligned} & (4k_x^2 \epsilon_x - 2k_x k_y \epsilon_y + k_x k_z \epsilon_z + 6k_y^2 \epsilon_x - 3k_z^2 \epsilon_x) \epsilon_x^* \\ & + (6k_x^2 \epsilon_y - 2k_x k_y \epsilon_x + 4k_y^2 \epsilon_y + k_y k_z \epsilon_z - 3k_z^2 \epsilon_y) \epsilon_y^* \\ & + (-3k_x^2 \epsilon_z + k_x k_z \epsilon_x - 3k_y^2 \epsilon_z + k_y k_z \epsilon_y - 8k_z^2 \epsilon_z) \epsilon_z^* \end{aligned} \right) \times \left( \begin{aligned} & 2\langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,2} | I \rangle + 2\langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,-2} | I \rangle \\ & -2\langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,0} | I \rangle - \langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,1} | I \rangle \\ & -\langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,-1} | I \rangle \end{aligned} \right) \right], \quad (4.63) $$

$$ \sigma(2, 1) = \pi^2 \alpha \hbar \omega k^2 \times \mathrm{Im} \left[ \frac{1}{168} \left( \begin{aligned} & \left( [4k_x \epsilon_x - 6ik_y \epsilon_x + ik_x \epsilon_y - ky \epsilon_y] k_z + (2k_x^2 + ik_x k_y + 3k_y^2 + 2k_z^2) \epsilon_z \right) \epsilon_x^* - i \left[ (-k_x \epsilon_x - ik_y \epsilon_x + 6ik_x \epsilon_y + 4ky \epsilon_y) k_z + (3k_x^2 - ik_x k_y + 2(k_y^2 + k_z^2)) \epsilon_z \right] \epsilon_y^* + \left[ 2k_z^2 (\epsilon_x - i\epsilon_y) + k_y^2 (3\epsilon_x - 2i\epsilon_y) + k_x^2 (2\epsilon_x - 3i\epsilon_y) - 4ik_y k_z \epsilon_z + k_x (ik_y \epsilon_x - k_y \epsilon_y + 4k_z \epsilon_z) \right] \epsilon_z^* \right) \\ & \times \left( \sqrt{6}\langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,1} | I \rangle - \sqrt{6}\langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,0} | I \rangle + 6\langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,2} | I \rangle - 6\langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,-1} | I \rangle \right) \end{aligned} \right], \quad (4.64) $$

$$ \sigma(2, -1) = \pi^2 \alpha \hbar \omega k^2 \times \mathrm{Im} \left[ \frac{1}{168} \left( \begin{aligned} & \left( 6ik_y k_z \epsilon_x - k_y k_z \epsilon_y + 2k_x^2 \epsilon_z + 3k_y^2 \epsilon_z + 2k_z^2 \epsilon_z - ik_x (4ik_z \epsilon_x + k_z \epsilon_y + k_y \epsilon_z) \right) \epsilon_x^* + i \left[ -k_x k_z (\epsilon_x + 6i\epsilon_y) + k_y k_z (i\epsilon_x + 4\epsilon_y) + 3k_x^2 \epsilon_z + ik_x k_y \epsilon_z + 2(k_y^2 + kz^2) \epsilon_z \right] \epsilon_y^* + \left[ 2k_z^2 (\epsilon_x + i\epsilon_y) + k_y^2 (3\epsilon_x + 2i\epsilon_y) + k_x^2 (2\epsilon_x + 3i\epsilon_y) + 4ik_y k_z \epsilon_z + k_x (-ik_y \epsilon_x - k_y \epsilon_y + 4k_z \epsilon_z) \right] \epsilon_z^* \end{aligned} \right) \\ \times \left( \sqrt{6}\langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,-1} | I \rangle - \sqrt{6}\langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,0} | I \rangle + 6\langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,-2} | I \rangle - 6\langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,1} | I \rangle \right), \quad (4.65) $$
$$
\begin{equation}
\begin{aligned}
\sigma(2, 2) = \pi^2 \alpha \hbar \omega k^2 & \times \operatorname{Im} \left[ \frac{1}{4\sqrt{21}} \left( -2(k_x - ik_y)(\epsilon_x^* - i\epsilon_y^*)(k_x \epsilon_x + k_y \epsilon_y - 2k_z \epsilon_z) \right. \right. \\
& \left. \left. -3(\epsilon_z(k_x - ik_y) + k_z(\epsilon_x - i\epsilon_y))(\epsilon_z^*(k_x - ik_y) + k_z \epsilon_x^* - ik_z \epsilon_y^*) \right. \right. \\
& \left. \left. -2(k_x - ik_y)(\epsilon_x - i\epsilon_y)(k_x \epsilon_x^* + k_y \epsilon_y^* - 2k_z \epsilon_z^*) \right) \right. \\
& \left. \times \left( \sqrt{\frac{2}{7}} \langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,2} | I \rangle + \sqrt{\frac{3}{7}} \langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,1} | I \rangle \right. \right. \\
& \left. \left. + \sqrt{\frac{2}{7}} \langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,0} | I \rangle \right) \right],
\end{aligned}
\tag{4.66}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\sigma(2, -2) = \pi^2 \alpha \hbar \omega k^2 & \times \operatorname{Im} \left[ \frac{1}{4\sqrt{21}} \left( -2(k_x + ik_y)(\epsilon_x^* + i\epsilon_y^*)(k_x \epsilon_x + k_y \epsilon_y - 2k_z \epsilon_z) \right. \right. \\
& \left. \left. -3(\epsilon_z(k_x + ik_y) + k_z(\epsilon_x + i\epsilon_y))(\epsilon_z^*(k_x + ik_y) + k_z \epsilon_x^* + ik_z \epsilon_y^*) \right. \right. \\
& \left. \left. -2(k_x + ik_y)(\epsilon_x + i\epsilon_y)(k_x \epsilon_x^* + k_y \epsilon_y^* - 2k_z \epsilon_z^*) \right) \right. \\
& \left. \times \left( \sqrt{\frac{2}{7}} \langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,-2} | I \rangle + \sqrt{\frac{3}{7}} \langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,-1} | I \rangle \right. \right. \\
& \left. \left. + \sqrt{\frac{2}{7}} \langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,0} | I \rangle \right) \right].
\end{aligned}
\tag{4.67}
\end{equation}
$$

4.2.6 Term a = 3

The term $a = 3$ consists of seven components, namely, $\sigma(3, 0)$, $\sigma(3, 1)$, $\sigma(3, -1)$, $\sigma(3, 2)$, $\sigma(3, -2)$, $\sigma(3, 3)$, and $\sigma(3, -3)$

$$
\begin{equation}
\begin{aligned}
\sigma(3, 0) = \pi^2 \alpha \hbar \omega k^2 & \times \operatorname{Im} \left[ \frac{i}{20} \left( (k_x^2 + k_y^2 - 2k_z^2)\epsilon_x^* \epsilon_y^* - (k_x^2 + k_y^2 - 2k_z^2)\epsilon_y^* \epsilon_x^* \right. \right. \\
& \left. \left. + 2k_x k_z \epsilon_y^* \epsilon_z^* - 2k_x k_z \epsilon_z^* \epsilon_y^* + 2k_y k_z \epsilon_z^* \epsilon_x^* - 2k_y k_z \epsilon_x^* \epsilon_z^* \right) \right. \\
& \left. \times \left( 2\langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,-1} | I \rangle - 2\langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,1} | I \rangle \right. \right. \\
& \left. \left. + \langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,2} | I \rangle - \langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,-2} | I \rangle \right) \right],
\end{aligned}
\tag{4.68}
\end{equation}
$$
$$
\begin{equation}
\begin{aligned}
\sigma(3, 1) = \pi^2 \alpha \hbar \omega k^2 & \times \operatorname{Im} \left[ \frac{1}{40\sqrt{6}} \left( i(3k_x^2 - 2ik_x k_y + k_y^2 - 4k_z^2) \epsilon_z^* \epsilon_y^* \right. \right. \\
& \left. \left. \qquad + 8k_z(k_y + ik_x)\epsilon_x^* \epsilon_y^* - (k_x^2 + 2ik_x k_y + 3k_y^2 - 4k_z^2)\epsilon_x^* \epsilon_z - 8k_z(k_y + ik_x)\epsilon_y^* \epsilon_x^* \right. \right. \\
& \left. \left. \qquad + \epsilon_z^*[k_x^2(\epsilon_x - 3i\epsilon_y) + 2ik_x k_y(\epsilon_x + i\epsilon_y) + k_y^2(3\epsilon_x - i\epsilon_y) - 4k_z^2(\epsilon_x - i\epsilon_y)] \right) \right. \\
& \left. \qquad \times \left( 2\langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,1} | I \rangle + 2\langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,0} | I \rangle \right. \right. \\
& \left. \left. \qquad - \sqrt{6}\langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,2} | I \rangle - \sqrt{6}\langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,-1} | I \rangle \right) \right] ,
\end{aligned}
\tag{4.69}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\sigma(3, -1) = \pi^2 \alpha \hbar \omega k^2 & \times \Im \left[ \frac{1}{40\sqrt{6}} \left( i(3k_x^2 + 2ik_x k_y + k_y^2 - 4k_z^2) \epsilon_y^* \epsilon_z \right. \right. \\
& \left. \left. \qquad + 8ik_z(k_x + ik_y)\epsilon_x^* \epsilon_y^* + (k_x^2 - 2ik_x k_y + 3k_y^2 - 4k_z^2)\epsilon_x^* \epsilon_z + 8k_z(k_y - ik_x)\epsilon_x^* \epsilon_y^* \right. \right. \\
& \left. \left. \qquad + \epsilon_z^*[k_x^2(-\epsilon_x + 3i\epsilon_y) + 2k_x k_y(\epsilon_y + i\epsilon_x) - k_y^2(3\epsilon_x + i\epsilon_y) + 4k_z^2(\epsilon_x + i\epsilon_y)] \right) \right. \\
& \left. \qquad \times \left( 2\langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,-1} | I \rangle + 2\langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,0} | I \rangle \right. \right. \\
& \left. \left. \qquad - \sqrt{6}\langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,-2} | I \rangle - \sqrt{6}\langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,1} | I \rangle \right) \right] ,
\end{aligned}
\tag{4.70}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\sigma(3, 2) = \pi^2 \alpha \hbar \omega k^2 & \times \operatorname{Im} \left[ \frac{1}{4\sqrt{6}} (k_y + ik_x) (\epsilon_x^*(k_x \epsilon_y - ik_y \epsilon_y + 2ik_z \epsilon_z) \right. \\
& \left. \qquad + \epsilon_y^*(-k_x \epsilon_x + ik_y \epsilon_x + 2k_z \epsilon_z) - 2ik_z \epsilon_z^*(\epsilon_x - i\epsilon_y)) \right. \\
& \left. \qquad \times (\langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,2} | I \rangle - \langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,0} | I \rangle) \right] ,
\end{aligned}
\tag{4.71}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\sigma(3, -2) = \pi^2 \alpha \hbar \omega k^2 & \times \operatorname{Im} \left[ \frac{1}{4\sqrt{6}} (ik_y + k_x) (\epsilon_x^*(ik_x \epsilon_y - k_y \epsilon_y + 2k_z \epsilon_z) \right. \\
& \left. \qquad + \epsilon_y^*(-ik_x \epsilon_x + k_y \epsilon_x + 2ik_z \epsilon_z) - 2k_z \epsilon_z^*(\epsilon_x + i\epsilon_y)) \right. \\
& \left. \qquad \times (\langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,0} | I \rangle - \langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,-2} | I \rangle) \right] ,
\end{aligned}
\tag{4.72}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\sigma(3, 3) = \pi^2 \alpha \hbar \omega k^2 & \times \operatorname{Im} \left[ \frac{1}{8} (k_x - ik_y)^2 \left( -\epsilon_z^*(\epsilon_x - i\epsilon_y) + \epsilon_z(\epsilon_x^* - i\epsilon_y^*) \right) \right. \\
& \left. \times \left( -\langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,2} | I \rangle - \langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,-1} | I \rangle \right) \right] ,
\end{aligned}
\tag{4.73}
\end{equation}
$$
$$
\sigma(3, -3) = \pi^2 \alpha \hbar \omega k^2 \times \mathrm{Im} \left[ \frac{1}{8} \left( (k_x + ik_y)^2 \right) \left( -\epsilon_z^*(\epsilon_x + i\epsilon_y) + \epsilon_z(\epsilon_x^* + i\epsilon_y^*) \right) \times \left( \langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,-2} | I \rangle + \langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,-1} | I \rangle \right) \right]. \quad (4.74)
$$

4.2.7 Term a = 4

The term $a = 4$ consists of nine components, namely, $\sigma(4, 0)$, $\sigma(4, 1)$, $\sigma(4, -1)$,
$\sigma(4, 2)$, $\sigma(4, -2)$, $\sigma(4, 3)$, $\sigma(4, -3)$, $\sigma(4, 4)$, and $\sigma(4, -4)$

$$
\begin{equation}
\begin{split}
\sigma(4, 0) = \pi^2 \alpha \hbar \omega k^2 \times \operatorname{Im} \left[ \frac{1}{140} \left( \right. & \left[ 3k_x^2 \epsilon_x + k_y^2 \epsilon_x - 4k_z^2 \epsilon_x + 2k_x k_y \epsilon_y - 8k_x k_z \epsilon_z \right] \epsilon_x^* \right. \\
& \left. \left. + [k_x^2 \epsilon_y + 3k_y^2 \epsilon_y - 4k_z^2 \epsilon_y + 2k_x k_y \epsilon_x - 8k_y k_z \epsilon_z] \epsilon_y^* \right. \right. \\
& \left. \left. - 4\epsilon_z^* [\epsilon_z(k_x^2 + k_y^2 - 2k_z^2) + 2k_z(k_x \epsilon_x + k_y \epsilon_y)] \right) \right. \\
& \left. \times \left( 6\langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,0} | I \rangle - 4\langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,1} | I \rangle \right. \right. \\
& \left. \left. - 4\langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,-1} | I \rangle + \langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,2} | I \rangle \right. \right. \\
& \left. \left. + \langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,-2} | I \rangle \right) \right],
\end{split}
\tag{4.75}
\end{equation}
$$

$$
\begin{equation}
\sigma(4, 1) = \pi^2 \alpha \hbar \omega k^2 \times \operatorname{Im} \left[ \frac{1}{56} \left( \epsilon_x^*[3k_x^2 \epsilon_z - 2ik_x(k_y\epsilon_z + 3ik_z\epsilon_x + k_z\epsilon_y) + \epsilon_z(k_y^2 - 4k_z^2) \right. \right. \\
\left. + 2ky_kz(\epsilon_y - i\epsilon_x)] - i\epsilon_y^*[\epsilon_z(k_x^2 + 2ik_xky + 3k_y^2 - 4k_z^2) \right. \right. \\
\left. + 2k_z(k_x\epsilon_x + ik_x\epsilon_y + ik_y\epsilon_x + 3ky\epsilon_y)] + \epsilon_z^*[k_x^2(3\epsilon_x - i\epsilon_y) \right. \\
\left. + 2k_xky(\epsilon_y - i\epsilon_x) - 8k_xk_z\epsilon_z + k_y^2(\epsilon_x - 3i\epsilon_y) + 8ik_yk_z\epsilon_z - 4k_z^2(\epsilon_x - i\epsilon_y)] \right) \\
\left. \times \left( \sqrt{6}\langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,1} | I \rangle - \sqrt{6}\langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,0} | I \rangle \right. \right. \\
\left. \left. \langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,-1} | I \rangle - \langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,2} | I \rangle \right) \right], \tag{4.76}
\end{equation}
$$
$$
\begin{equation}
\begin{aligned}
\sigma(4, -1) = \pi^2 \alpha \hbar \omega k^2 & \times \operatorname{Im} \left[ \frac{-1}{56} \left( \epsilon_x^* [\epsilon_z (3k_x^2 + 2ik_x k_y + k_y^2 - 4k_z^2) \right. \right. \\
& \left. \left. \qquad + 2k_z (3k_x \epsilon_x + ik_x \epsilon_y + ik_y \epsilon_x + k_y \epsilon_y) \right] + \epsilon_y^* [i\epsilon_z (k_x^2 - 2ik_x k_y + 3k_y^2 - 4k_z^2) \right. \right. \\
& \left. \left. \qquad + 2k_z (ik_x \epsilon_x + k_x \epsilon_y + k_y \epsilon_x + 3ik_y \epsilon_y) \right] + \epsilon_z^* [k_x^2 (3\epsilon_x + i\epsilon_y) \right. \\
& \left. \left. \qquad + 2k_x k_y (\epsilon_y + i\epsilon_x) - 8k_x k_z \epsilon_z + k_y^2 (\epsilon_x + 3i\epsilon_y) - 8ik_y k_z \epsilon_z - 4k_z^2 (\epsilon_x + i\epsilon_y)] \right) \right. \\
& \left. \times \left( \sqrt{6} \langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,-1} | I \rangle - \sqrt{6} \langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,0} | I \rangle \right. \right. \\
& \left. \left. \qquad \langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,1} | I \rangle - \langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,-2} | I \rangle \right) \right],
\end{aligned}
\tag{4.77}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\sigma(4, 2) = \pi^2 \alpha \hbar \omega k^2 & \times \operatorname{Im} \left[ \frac{1}{56} \left( -(k_x - ik_y)(\epsilon_x^* - i\epsilon_y^*)(k_x \epsilon_x + k_y \epsilon_y - 2k_z \epsilon_z) \right. \right. \\
& \left. \left. \qquad + 2[\epsilon_z(k_x - ik_y) + k_z(\epsilon_x - i\epsilon_y)][\epsilon_z^*(k_x - ik_y) + k_z \epsilon_x^* - ik_z \epsilon_y^*] \right. \right. \\
& \left. \left. \qquad -(k_x - ik_y)(\epsilon_x - i\epsilon_y)(k_x \epsilon_x^* + k_y \epsilon_y^* - 2k_z \epsilon_z^*) \right) \right. \\
& \left. \times \left( \sqrt{6}\langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,2} | I \rangle + \sqrt{6}\langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,0} | I \rangle \right. \right. \\
& \left. \left. \qquad -4\langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,1} | I \rangle \right) \right],
\end{aligned}
\tag{4.78}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\sigma(4, -2) = \pi^2 \alpha \hbar \omega k^2 & \times \operatorname{Im} \left[ \frac{1}{56} \left( -(k_x + ik_y)(\epsilon_x^* + i\epsilon_y^*)(k_x \epsilon_x + k_y \epsilon_y - 2k_z \epsilon_z) \right. \right. \\
& \left. \left. \qquad + 2[\epsilon_z(k_x + ik_y) + k_z(\epsilon_x + i\epsilon_y)][\epsilon_z^*(k_x + ik_y) + k_z \epsilon_x^* + ik_z \epsilon_y^*] \right. \right. \\
& \left. \left. \qquad -(k_x + ik_y)(\epsilon_x + i\epsilon_y)(k_x \epsilon_x^* + k_y \epsilon_y^* - 2k_z \epsilon_z^*) \right) \right. \\
& \left. \times \left( \sqrt{6}\langle I | r^2 C_{2,0}^* G^+ r^2 C_{2,-2} | I \rangle + \sqrt{6}\langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,0} | I \rangle \right. \right. \\
& \left. \left. \qquad -4\langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,-1} | I \rangle \right) \right],
\end{aligned}
\tag{4.79}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\sigma(4, 3) = \pi^2 \alpha \hbar \omega k^2 & \times \operatorname{Im} \left[ \frac{-1}{8} (k_x - ik_y) \left( (\epsilon_x^* - i\epsilon_y^*)(\epsilon_z(k_x - ik_y) \right. \right. \\
& \left. \left. \qquad + 2k_z(\epsilon_x - i\epsilon_y) \right) + \epsilon_z^*(k_x - ik_y)(\epsilon_x - i\epsilon_y) \right) \\
& \times \left( \langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,1} | I \rangle - \langle I | r^2 C_{2,-1}^* G^+ r^2 C_{2,2} | I \rangle \right),
\end{aligned}
\tag{4.80}
\end{equation}
$$
$$ \sigma(4, -3) = \pi^2 \alpha \hbar \omega k^2 \times \mathrm{Im} \left[ \frac{1}{8} (k_x + ik_y) \left( (\epsilon_x^* + i\epsilon_y^*)(\epsilon_z(k_x + ik_y) + 2k_z(\epsilon_x + i\epsilon_y)) + \epsilon_z^*(k_x + ik_y)(\epsilon_x + i\epsilon_y) \right) \times \left( \langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,-1} | I \rangle - \langle I | r^2 C_{2,1}^* G^+ r^2 C_{2,-2} | I \rangle \right) \right], \quad (4.81) $$

$$ \sigma(4, 4) = \pi^2 \alpha \hbar \omega k^2 \times \mathrm{Im} \left[ \frac{1}{4} \left( (k_x - ik_y)^2 (\epsilon_x - i\epsilon_y)(\epsilon_x^* - i\epsilon_y^*) \right) \times \left( \langle I | r^2 C_{2,-2}^* G^+ r^2 C_{2,2} | I \rangle \right) \right], \quad (4.82) $$

$$ \sigma(4, -4) = \pi^2 \alpha \hbar \omega k^2 \times \mathrm{Im} \left[ \frac{1}{4} \left( (k_x + ik_y)^2 (\epsilon_x + i\epsilon_y)(\epsilon_x^* + i\epsilon_y^*) \right) \times \left( \langle I | r^2 C_{2,2}^* G^+ r^2 C_{2,-2} | I \rangle \right) \right]. \quad (4.83) $$

In the most general case, the quadrupole XAS signal can be described using 25 fundamental spectra as given in (4.59)–(4.83). Although the expression seems at first sight complicated, major simplifications and intuitive conclusions can be made when one considers the symmetry of the absorbing system. We shall illustrate this in the following section.

### 4.2.7.1 Case Study of a $d^9$ Ion

As an example, we will study again a $d^9$ ion in different local symmetries.

#### Spherical Symmetry

We shall start with an isolated $d^9$ ion (i.e., spherical symmetry). The conductivity tensor of such an ion is shown in Fig. 4.11. The tensor consists of 25 elements that form the 25 fundamental spectra through appropriate linear combinations. Only the five diagonal elements are non-zero in this case and are all equal. This means that the only possibly active fundamental spectra are of the type $\sigma(a, 0)$ with $a = 0, 1, 2, 3, 4$. However, because all the diagonal elements are equal, only the fundamental spectrum $\sigma(0, 0)$ is non-zero. This fundamental spectrum has no angular dependence, hence this system is isotropic. It is not a surprising result that for a spherical system, no angular dependence would be observed.
<figure><img src="image_54.png" /><figcaption>**Fig. 4.11** Conductivity tensor calculated for a *d*⁹ ion in spherical symmetry. Re and Im are the real and imaginary parts of the tensor</figcaption></figure>

Octahedral Crystal Field

We shall examine next a *d*⁹ ion in *O*<sub>*h*</sub> symmetry. The conductivity tensor of this ion is shown in Fig. 4.12. Several differences can be directly seen in comparison with the previous case:

* The elements with the transition operator $T = C_{-2}^2$ are mixed with those of $T = C_2^2$.

* The diagonal elements are not equal.

Let us consider the first point. This mixing leads to the same form of eigenvectors than for obtained for the 3*d* orbitals (Y<sub>2,m</sub>) for an *O*<sub>*h*</sub> crystal field [see (4.23)]. Indeed, this is exactly the same problem. In order to obtain only diagonal elements, one can apply the following rotation (4.84):
<figure><img src="image_55.png" /><figcaption>Fig. 4.12 Conductivity tensor of a d⁹ ion in an octahedral crystal field (10Dq = 1.1 eV). Re and Im are the real and imaginary parts of the tensor</figcaption></figure>

$$ Rot = \begin{bmatrix} \frac{1}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} \\ 0 & 0 & 1 & 0 & 0 \\ 0 & \frac{i}{\sqrt{2}} & 0 & \frac{i}{\sqrt{2}} & 0 \\ 0 & \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} & 1 \\ \frac{i}{\sqrt{2}} & 0 & 0 & 0 & -\frac{i}{\sqrt{2}} \end{bmatrix}. \quad (4.84) $$

The rotated conductivity tensor is shown in Fig. 4.13. Only diagonal elements exist now. These are separated into two types: three ($t_{2g}$) that are equal with transition operators, $C_{xy}^2$, $C_{yz}^2$ and $C_{xz}^2$, and two ($e_g$) that are equal with transition operators, $C_{z^2}^2$ and $C_{x^2-y^2}^2$.
<figure><img src="image_56.png" /><figcaption>**Fig. 4.13** Conductivity tensor for a *d*⁹ ion in an octahedral crystal field (*10D*<sub>q</sub> = 1.1 eV) calculated using the symmetry adapted basis set</figcaption></figure>

Five fundamental spectra come into play, namely, σ(0, 0), σ(2, 0), σ(4, 0), σ(4, 4), and σ(4, -4) as shown in Fig. 4.14. The *O*<sub>*h*</sub> symmetry implies that *R*(2, 0) is always equal to zero as confirmed by the calculation. In addition, *R*(4, 4) = *R*(4, -4) and are proportional to *R*(4, 0) as can be seen from (4.75), (4.82), and (4.83). Therefore, as can be expected from group theory, only two fundamental spectra are required to fully describe the system.

Let us investigate the angular dependence of a quadrupole transition in an *O*<sub>*h*</sub> crystal field considering two scattering geometries. In the first geometry, the wave vector (*k*) is aligned parallel to the [100] direction and the polarization (*ϵ*) is rotated in the *z* - *y*-plane as illustrated in the right panel of Fig. 4.16. Despite the presence of non-isotropic fundamental spectra [σ(4, 0), σ(4, 4), and σ(4, -4)], the XAS cross-section is constant in these settings as shown in Fig. 4.16a. In the second geometry we
<figure><img src="image_57.png" /><figcaption>**Fig. 4.14** Fundamental spectra for a quadrupole transition in a 3d⁹ ion in an octahedral crystal field. Re and Im are the real and imaginary parts of the spectra</figcaption></figure>

have $k \parallel [\frac{1}{\sqrt{2}} \frac{1}{\sqrt{2}} 0]$ and $\epsilon$ is rotated about the $[\frac{1}{\sqrt{2}} \frac{1}{\sqrt{2}} 0]$ axis as depicted in Fig. 4.16b. The XAS cross section shows a clear twofold angular dependence in these settings.

It is interesting to discuss the difference between both scattering geometries and the reason behind the absence of angular dependence in the first case. In Fig. 4.15, we plot the angular dependence of the light tensor [$E(4, 0)$, $E(4, 4)$, and $E(4, -4)$] for both cases. The contribution of the term $E(4, 0)$ is 90° out-of-phase with respect to the terms $E(4, 4)$ and $E(4, -4)$ for the first scattering geometry [see panel (a) of Fig. 4.15]. The $O_h$ symmetry implies that the ratio between the $R(4, 0)$ and the
<figure><img src="image_58.png" /><figcaption>**Fig. 4.15** Angular dependence of *E*(4, 0), *E*(4, 4), and *E*(4, -4) terms. **a** The wave vector (k) is aligned with [100]. The angular dependence is computed by rotating the polarization (ϵ) about [100] with θ = 0° for ϵ || [001]. **b** k is aligned with [-1/√2, 1/√2, 0]. The angular dependence is computed by rotating ϵ about [1/√2, 1/√2, 0] with θ = 0° for ϵ || [001]</figcaption></figure>

*R*(4, ±4) terms leads to a constant XAS cross section. On the other hand, as depicted in Fig. 4.15b, all terms are in phase which leads to an angular dependent XAS.

An important distinction between the dipole and quadrupole transitions can be concluded from these examples. While a dipole transition in an *O*<sub>*h*</sub> system exhibits no angular dependence, the quadrupole transition can show angular dependences when the scattering geometry is appropriately chosen. This difference holds because a quadrupole transition has higher multipole contributions that give rise to angular dependences not observable for dipole transitions.

Tetragonal Crystal Field

The effect of reducing the crystal field symmetry to tetragonal by applying a com-
pressive distortion along the z-axis can be directly seen in the angular dependence of
the quadrupole transition. In contrast to the case of O<sub>h</sub> crystal field (see Fig. 4.16a),
now rotating **ϵ** about the [100] axis shows angular dependence because the z- and
y-axes are not equivalent (see Fig. 4.17a). However, as could be expected, rotating **ϵ**
about the [001] axis shows no angular dependence (see Fig. 4.17b). In this projection,
the system is effective of O<sub>h</sub> symmetry.
<figure><img src="image_59.png" /><figcaption>**Fig. 4.16** Angular dependence of XAS for a quadrupole transition in a 3d⁹ *O*ₕ ion with 10Dq = 1.1 eV. **a** The wave vector (**k**) is aligned with [100]. The angular dependence is computed by rotating the polarization vector (ϵ) about [100] with θ = 0° for ϵ ≈ [001] as depicted in the sketch on the right. **b k** is aligned with [1/√2, 1/√2, 0]. The angular dependence is computed by rotating ϵ about [1/√2, 1/√2, 0] with θ = 0° for ϵ ≈ [001] as depicted in the sketch on the right</figcaption></figure>

**Octahedral Crystal Field with Exchange Field** || **z**

Consider a magnetic 3d⁹ ion where the crystal field is *O*ₕ with an exchange field aligned along the z-axis. Seven fundamental spectra come into play, namely, *R*(0, 0), *R*(1, 0), *R*(2, 0), *R*(3, 0), *R*(4, 0), *R*(4, −4), and *R*(4, 4). We have shown previously that for *O*ₕ symmetry, when **k** is aligned parallel to the [100] direction, and **ϵ** is rotated in the z − y-plane, angular dependence is observed (see Fig. 4.16a). Repeating the same calculation with an exchange field aligned along the z-axis leads to an angular dependent XAS as shown in Fig. 4.18a. The exchange field reduces the symmetry along the z-axis. The effects of rotating the incident linear polarization in the z − y-plane on the fundamental cross sections are shown in Fig. 4.19. Only the terms σ(2, 0), σ(4, 0), σ(4, 4), and σ(4, −4) are non-zero and exhibit a twofold angular
<figure><img src="image_60.png" /><figcaption>**Fig. 4.17** Angular dependence of XAS for a quadrupole transition in a 3d⁹ ion in a D₄h crystal field (10Dq = 1.1 eV and Ds = -0.2 eV). **a** The wave vector (k) is aligned with [100]. The angular dependence is computed by rotating the polarization (ϵ) about [100] with θ = 0° for ϵ || [001]. **b** k is aligned with [001] and ϵ is rotated about the [001] with θ = 0° for ϵ || [100]</figcaption></figure>

dependence. However, one notes that σ(4, 0) is 90° shifted with respect to σ(4, 4) and σ(4, -4) which implies that the angular dependence of the XAS will be small. In comparison, no angular dependence is observed when ϵ is rotated in the x - y-plane as shown in Fig. 4.18b.

Finally, the exchange field can give rise to interesting combinations of structural
and magnetic dichroism effects. Consider aligning **k** ‖ [001] and measuring XAS
using circular polarized light. Rotating the system about the [100] axis gives rise to
unconventional angular dependent XAS as shown in Fig. 4.20. This angular depen-
dence arises from a combination of structural and magnetic dichroism effects.

The magnetic contribution arises from the circular dichroism active terms which
are σ(1, 0) and σ(3, 0) (see Fig. 4.21a). On the other hand, the structural contribution
arises from the linear dichroism active terms which are σ(4, 0), σ(4, 4), and σ(4, -4)
(see Fig. 4.21b). In addition, these terms contribute weakly to the magnetic dichroism.
<figure><img src="image_61.png" /><figcaption>**Fig. 4.18** Angular dependence of a quadrupole transition for a $3d^9$ ion in $O_h$ crystal field ($10D_q = 1.1$ eV) with an exchange field aligned along the z-axis ($B = 0.05$ eV). **a** The wave vector ($\mathbf{k}$) is aligned with [100]. The angular dependence is computed by rotating the polarization vector ($\boldsymbol{\epsilon}$) about [100] with $\theta = 0^\circ$ for $\boldsymbol{\epsilon} \parallel [001]$. **b** $\mathbf{k}$ is aligned with [001] and $\boldsymbol{\epsilon}$ is rotated about the [001] with $\theta = 0^\circ$ for $\boldsymbol{\epsilon} \parallel [100]$.</figcaption></figure>

This is illustrated in Fig. 4.22 where in panel (a) we show the structural dichroism signal for the same system without an exchange field and in panel (b) the difference between the case with exchange versus without exchange. The magnetic contribution is about three orders of magnitude less than the structural contribution for this system.
<figure><img src="image_62.png" /><figcaption>**Fig. 4.19** Angular dependence of XAS fundamental cross sections $\sigma(1, 0)$, $\sigma(2, 0)$, $\sigma(3, 0)$, $\sigma(4, 0)$, $\sigma(4, 4)$, and $\sigma(4, -4)$. Here $k$ is aligned with [100] and $\epsilon$ is rotated about [100] with $\theta = 0^\circ$ for $\epsilon \parallel [001]$</figcaption></figure>

<figure><img src="image_63.png" /><figcaption>**Fig. 4.20** Angular dependence of XAS for a quadrupole transition in a $3d^9$ ion in an $O_h$ crystal field ($10D_q = 1.1$ eV) with an exchange field aligned along the z-axis ($B = 0.05$ eV). The wave vector ($\mathbf{k}$) is initially aligned to [001] and polarization ($\epsilon$) is circular. The angular dependence is computed by rotating the system about the [100] axis</figcaption></figure>
<figure><img src="image_64.png" /><figcaption>**Fig. 4.21** Angular dependence of XAS fundamental spectra for a quadrupole transition in a $3d^9$ ion in an $O_h$ crystal field with an exchange field aligned along the z-axis. **a** Circular dichroism active terms $\sigma(1, 0)$ and $\sigma(3, 0)$. **b** Linear dichroism active terms $\sigma(4, 0)$, $\sigma(4, 4)$ and $\sigma(4, -4)$. The angular dependence is computed by rotating the system about the [100] axis</figcaption></figure>

<figure><img src="image_65.png" /><figcaption>**Fig. 4.22** Angular dependence of XAS for a quadrupole transition in a $3d^9$ ion: **a** in an $O_h$ crystal field ($10D_q = 1.1$ eV). **b** The difference between calculation (**a**) and in an $O_h$ crystal field with an exchange field aligned along the z-axis ($10D_q = 1.1$ eV and $B = 0.05$ eV). The wave vector (**k**) is initially aligned to [001] and polarization (ε) is circularly polarized. The angular dependence is computed by rotating the system about the [100] axis</figcaption></figure>
## 4.3 Conclusion

We have expressed the XAS cross section using Green's function formalism and spherical tensors, which allows a tractable investigation of the different types of dichroism: on the one hand experimentally, by allowing the experimentalist to predict the smallest set of measurements required to recover the full spectroscopic information and therefore to optimize beamtime; on the other hand theoretically, using modern core level spectroscopy codes, by allowing a more efficient calculation rather than a point-by-point time-consuming treatment. In a forthcoming work, we intend to use a similar approach for resonant inelastic X-ray scattering (RIXS) spectroscopy, whose richness lies in the large number of possible spectra that can be obtained by varying the energy, direction, and polarization state of the incident and scattered beams [28]. But as a matter of fact, there is so much information in the spectra that it is difficult to know whether a specific set of experiments measures all potential information. In our opinion, the possibilities offered by angular and polarization dependent RIXS measurements have not yet been exploited to the best of their potential. Nevertheless, the technique has now become mature and popular and will benefit from the huge instrumentation progress achieved in the last years, which open new doors for the exploration of dichroisms in materials.

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<figure><img src="image_66.png" /></figure>
Chapter 5
Spintronics and Synchrotron Radiation

<figure><img src="image_67.png" /></figure>

Check for
updates

Richard Mattana, Nicolas Locatelli, and Vincent Cros

**Abstract** Having access to the electronic and magnetic properties of spintronic systems is of crucial importance in view of their future technological developments. Our purpose in this chapter is to elaborate how a variety of synchrotron radiation-based measurements provides powerful and often unique techniques to probe them. We first introduce general concepts in spintronics and present some of the important scientific advances achieved in the last 30 years. Then we will describe some of the key investigations using synchrotron radiation concerning voltage control of magnetism, spin-charge conversion and current-driven magnetization dynamics.

**5.1 General Introduction to Spintronics: From Magnetoresistive Effects to the Physics of Spin-Transfer Phenomena**

Whereas electronics relies on the charge of carriers (electrons and holes) to cre-
ate and convey information, in spintronics the spin of these carriers is also used.
The generation of spin currents, their detection as well as their manipulation are the
basic principles of any spintronic devices. Since the discovery of giant magnetoresis-
tance (GMR) effect in 1988, several breakthroughs have been achieved and strongly
impacted several domains of applications such as, e.g. magnetic sensors, hard-disk
drive read heads and more recently nonvolatile memories to cite the most emblematic
ones. In this first part, we aim at introducing the basic principles of GMR and some
of the pioneer experiments. Then we present some other important breakthroughs
such as tunnelling magnetoresistive effects and the manipulation of magnetization
without the application of a magnetic field but using spin-transfer effects.

<footnote>R. Mattana (✉) · N. Locatelli · V. Cros
Unité Mixte de Physique, CNRS, Thales, Université Paris-Saclay, 1 avenue Augustin Fresnel,
91767 Palaiseau, France
e-mail: richard.mattana@cnrs-thales.fr</footnote>
### 5.1.1 Giant Magnetoresistance: An Historical Point of View

#### 5.1.1.1 Electric Conduction in Ferromagnets

In 3d ferromagnetic (FM) metals, electric conduction comes mainly from 4s band electrons, whereas magnetism originates from electrons in 3d bands. Resistivity in such materials arises from scattering events of carriers on impurities, phonons and local potentials. In particular, localized d orbitals act as diffusive centres for the s electrons which carry most of the charge current. Moreover, the non-zero net local magnetization comes up with a shift of the d-bands for the two electron spins (up and down), leading to different densities of states (DOS) at the Fermi level (Fig. 5.1 left). At low temperature, the electron spin is observed to be conserved during different sources of scattering (electron-phonon, electron-electron, electron-magnon, ...), allowing to describe the conduction properties in a ferromagnet using a "two-current" model [1]. Spin-up and spin-down electrons carry the current in two separated channels (Fig. 5.1 right). The difference in DOS of 3d electrons for spin-up and spin-down channels then leads to different number of scattering events for the two sub-bands, and thus different resistivities. Conduction in ferromagnet can hence be characterized by the spin-asymmetry coefficient $α$ defined by the ratio between the resistivity of spin-down channel over the resistivity of spin-up channel. The spin-asymmetry coefficient is then strongly correlated to the asymmetry of the DOS at the Fermi level. For example, Fe has a spin-asymmetry coefficient lower than 1 whereas for Co and Ni this coefficient is larger than 1.

The experimental demonstration of the two-current model has been provided by a series of experiments in which Fert and Campbell showed that the conduction of a spin channel can be tuned by introducing different types of magnetic impurity [2–5]. Depending on the introduction of two different impurities with similar or opposite spin asymmetries, the impact on the two spin channels will be different (Fig. 5.2). Let us take a Ni matrix ($α > 1$) doped with Co impurities ($α > 1$) and Rh impurities ($α < 1$). With the presence of only Co impurities, scattering in

<figure><img src="image_68.png" /><figcaption>**Fig. 5.1** (left) Scheme of the density of states in a ferromagnetic metal which has *d* bands shifted. (right) Scheme illustrating the two-current model: spin-up and spin-down electrons carry the current in separated channels. As resistivity arises from *s-d* transition and is proportional to the 3d DOS, the resistivity of a ferromagnet can be schematized by two resistances ($ρ_↑$ and $ρ_↓$) in parallel</figcaption></figure>
<figure><img src="image_69.png" /><figcaption>**Fig. 5.2** Tailoring the channel conduction with impurities. **a** Resistivity of Ni as a function of Co and Rh doping. Co and Rh having opposite spin asymmetries, scattering is significant for both spin-up and spin-down channels **b** and **c** leading to an increase of resistivity. The resistivity dependence as a function of Co and Rh concentration is not linear **a**. **d** Resistivity of Ni as a function of Co and Au doping. Co and Au having similar spin asymmetries, electrons of only spin-down channel are strongly scattered **e**. The spin-up channel is providing a short-circuit for the electrons **f**. A small linear dependence of the resistivity with Co and Au impurity concentrations is then obtained **d**. Adapted from [5] with permission (Copyright 2008, American Physical Society)</figcaption></figure>

the spin-down channel is enhanced, but the spin-up channel still experiences low
resistivity: the resulting global resistivity is low (short-circuit effect). As Rh impuri-
ties are introduced, scattering becomes significant for both channels [Fig. 5.2(b,c)],
progressively suppressing the short-circuit effect, and the global resistivity quickly
increases. The resistivity dependence as a function of Co and Rh concentrations is
not linear [Fig. 5.2a]. Instead, when Rh impurities are replaced by Au ones (α > 1,
i.e. more electrons of the down channel are scattered, like for Ni and Co), such an
enhancement of resistivity is not observed. Only electrons of the spin-down channel
are strongly scattered by Co and Au impurities and a small linear dependence of
the resistivity with Co and Au impurity concentrations is observed [Fig. 5.2d]. The
spin-up channel is providing a short-circuit for the electrons [Fig. 5.2e, f].

These experiments indeed brought experimental demonstration of the validity of
the two-current model and the proof that the mean resistivity can be varied by tailoring
of the channel conduction through impurities. These pioneer experiments can be
indeed considered as the pre-concept of the GMR effect in which different resistance
states will be achievable by applying a magnetic field in a single heterostructure.

Thanks to the development of molecular beam epitaxy of metallic materials in
the late 80's, it became hence possible to grow nanometre-thick magnetic metal-
lic heterostructures (multilayers) involving several separated FM layers. Controlling
the magnetic configurations of the magnetic films (parallel or antiparallel configura-
tions of their magnetizations) then grants the possibility to tailor the spin-dependent
transport of the system. This has led to the discovery of the GMR effect in 1988.
### 5.1.1.2 First Observation of GMR Effects

The first studies of GMR effects were done in Fe/Cr superlattices (Fe: FM, Cr: non-FM metal). A key point of these heterostructures was the control of the magnetic configuration. Exploiting the indirect exchange interaction between the magnetic layers transmitted by the conduction electron (called RKKY interaction) through the non-magnetic spacer layer, P. Grünberg and co-workers proved in 1986 that it is possible to align the magnetization of consecutive Fe layers in opposite direction at zero magnetic field by choosing adequate Cr interlayer thickness [6]. The parallel magnetic configuration can then be reached by applying a large magnetic field in order to overcome the RKKY coupling. First magnetoresistance curves have been obtained by the groups of A. Fert in Orsay, France (Fig. 5.3 left) and P. Grünberg in Jüllich, Germany (Fig. 5.3 right). A higher resistance state is observed at zero magnetic field at which the antiparallel magnetic configuration is stabilized. By applying a magnetic field, the parallel magnetic configuration is progressively reached inducing a large decrease of the resistance. In the experiments done in Grünberg’s group, a GMR effect of 1.5% at low temperature was obtained in the case of a Fe/Cr/Fe trilayer (Fig. 5.3 right). The striking point is that this effect is about one order of magnitude larger in amplitude than the anisotropic magnetoresistance of a single Fe thin film (also shown on Fig. 5.3 right). In Fert’s group, measurements were performed in Fe/Cr superlattices instead of trilayers, which allowed to considerably increase the amplitude of the effect that reached 80% for sixty Fe/Cr bilayers (Fig. 5.3 left).

An important development towards the implementation of the GMR systems for applications in read heads for hard-disk drives or sensors was made in 1991 by B. Dieny et al. [7] at IBM who reported the first observation of GMR in simple

<figure><img src="image_70.png" /><figcaption>**Fig. 5.3** Pioneer observation of GMR in Fe/Cr superlattices (left) and in Fe/Cr/Fe trilayer (right). At zero magnetic field, magnetization of consecutive layers is pointing in opposite direction leading to a high resistance state. By applying a magnetic field, a parallel magnetic configuration is reached and so a low resistance state is obtained. For a Fe/Cr/Fe trilayer, the GMR ratio is about 1.5% and one order of magnitude larger than the anisotropic magnetoresistance of a single Fe layer (right). By using Fe/Cr superlattices (60 Fe/Cr repetitions) the GMR ratio reaches 80% at low temperature. Reproduced from [9] (left) and [10] (right) with permission (Copyright 1988 and 1989, American Physical Society)</figcaption></figure>
trilayered structures with two distinct coercive fields, later called spin valves. In these structures, one of the layers switches using low fields while the second is stable up to large fields. It is to be emphasized that the time between the first discovery in a lab and the use of the GMR effect in cutting-edge technology devices has been extremely fast as less than 10 years after the discovery of GMR, and IBM introduced in 1997 spin valves in read heads of hard-disk drives. GMR-based magnetic sensors is now used in a multitude of applications such as monitoring wheel speed, detecting charge current, and fluid flow.

The research on magnetic multilayers and GMR became rapidly a very hot topic. It is not our purpose here to make here a review of all experimental and theoretical results that followed up the pioneer results. A complete review can be consulted in [8].

### 5.1.1.3 A Simple Model to Describe the GMR

The two-current model, described previously, was developed to explain the spin-dependent resistivity in magnetic materials. It can, in a rather simple way, be adapted to describe the giant magnetoresistive effect in these magnetic multilayers. This model is based on two assumptions: 1) $α > 1$: the minority-spin electrons (opposed to local magnetization) are more scattered than those of majority spin (aligned with local magnetization); 2) the spin is conserved when the electrons are scattered. These two conditions are fulfilled at low temperature.

Two geometries can be considered to evaluate the resistance of such a structure: either the current flows in the direction of the layer planes (known as "current-in-plane GMR", CIP-GMR), or the current flows in a direction perpendicular to the layer plane (known as "current-perpendicular-to-the-plane GMR", CPP-GMR). A similar description can be used to account for the magnetoresistive properties for both geometries, without entering into the fine discussion about the actual physical spin-transport mechanisms, as long as the layer thickness remains small compared to a characteristic length associated with each geometry: the mean free path for the CIP case and the spin-flip length for the CCP case [11].

Let us note *r* the resistance of a FM layer for the majority-spin channel and *R* the resistance for the minority-spin channel, with *r* < *R*. Here the resistance of the NM separating layer is neglected. As illustrated in Fig. 5.4, an electron will have to pass through adjacent FM layers. Depending on whether these layers have parallel (P) or anti-parallel (AP) magnetizations, the resulting resistance shall differ.

In the (P) configuration, the electrons with spin (↑) and (↓) are either the electron spin majority or minority in all magnetic layers. Spin (↑) electrons then experience identical resistances $r_↑ = r$ when crossing each magnetic layer, while spin (↓) electrons experience identical resistances $r_↓ = R$. The mean resistance then writes $r_P = \frac{Rr}{r+R}$. In case of materials with large spin asymmetry ($α ≫ 1$ and $r ≪ R$), the multilayer is short-biased by the spin (↑) channel and the total resistance is $r_P ≈ r$.

For the (AP) configuration, the electrons of the two (↑) and (↓) channels correspond alternately to electron spin majority and minority in consecutive magnetic
<figure><img src="image_71.png" /><figcaption>**Fig. 5.4** Illustration of the two-current model: conduction of an electron in a ferromagnetic metal/normal metal/ferromagnetic metal (FM/NM/FM) multilayer. The electrons with a spin aligned parallel (resp. anti-parallel) to the local magnetization see a resistance *r* (resp. *R*) through this magnetic layer. The equivalent resistance circuit is presented for two configurations: consecutive ferromagnetic layers with parallel magnetization or anti-parallel magnetization</figcaption></figure>

layers, and thus the effect of short-circuit by one of the channels is suppressed. Consequently the two channels have the same resistance $\frac{R+r}{2}$ and the total resistance is $\frac{R+r}{4}$, which is in general much larger than $r_p = r$.

This brings the possibility of switching between high and low resistance states by simply changing the relative orientation of the magnetization of consecutive layers. If we can engineer these structures to stabilize the two (P) and (AP) magnetic states in absence of any external field, then such system defines a magnetic bit to store the information, using the powerful stability of the magnets, already well known through the development of hard-disk drives. The state can then be read through a very simple process, i.e. by measuring the resistance of the stack.

Following this model, the amplitude of the GMR ratio can be simply deduced

$$GMR = \frac{R_{AP} - R_{P}}{R_{P}} = \frac{(R - r)^{2}}{4Rr} \qquad (5.1)$$

In short, the understanding and expertise acquired during the 60s and 70s for tailoring the spin-transport properties combined with the development of metallic magnetic multilayers in the 80s has led to the discovery of the GMR effect in 1988. Less than 10 years after the discovery of the GMR effect considered as the birth of spintronics, this effect has largely participated to the explosion of the amount of data storage in the mid-90s through the commercialization of the first generation of spintronic devices such as magnetic sensors used as read heads in hard-disk drives.
## 5.1.2 Tunnelling Magnetoresistance

The pioneer observation of tunnelling magnetoresistance (TMR) in magnetic tunnel junctions (MTJs) at room temperature in 1995 is considered to be the second break-through in spintronics, leading to a second generation of spintronic applications such as magnetic random access memory (MRAM). In this section, we describe the first TMR measurements, then the two standard models describing TMR effects.

### 5.1.2.1 First Experimental and Theoretical Studies

A magnetic tunnel junction is a device in which two FM electrodes are separated by an ultra-thin insulating barrier. The first MTJ exhibiting TMR has indeed been reported in 1975 by M. Jullière [12] who measured a TMR ratio of 14% at 4.2K in a Fe/Ge/Co junction. Twenty years later TMR effects have been observed at room temperature by J. S. Moodera [13] and T. Miyazaki [14] using amorphous alumina as tunnel barrier. In Fig. 5.5, the TMR curve obtained for a CoFe/Al₂O₃/Co MTJ [13] is displayed. The TMR ratio is around 12%, at room temperature. This increase in the resistance difference between the two states allows an easier detection of them and let appear the potential of such spintronic devices for memory applications.

Such TMR effect has been first explained by M. Jullière in 1975. Keeping the free-electron approximation, he proposed an additional assumption: the electron spin conservation during the tunnelling process, meaning that electrons can tunnel from one FM electrode toward the second FM electrode only into empty states having identical spins. This simple model easily allows to understand that the tunnelling current in the (P) and (AP) alignments of magnetizations will differ.

As illustrated in Fig. 5.5, for the parallel (P) configuration of the magnetizations: when a small voltage is applied, majority-spin (denoted as ↑) electrons from the injector will tunnel toward majority-spin (↑) empty states of the collector, allowing "high" current. At the same time, minority-spin electrons from the injector will tunnel toward minority-spin (↓) empty states of the collector, allowing "low" current.<fnref n="1" /> The conductance then writes

$$G_P \propto D_1^\uparrow(E_F)D_2^\uparrow(E_F) + D_1^\downarrow(E_F)D_2^\downarrow(E_F). \quad (5.2)$$

In the case of antiparallel (AP) configuration of the magnetizations, majority-spin (↑) electrons from the injector will tunnel toward minority-spin (↓) empty states of the collector. Minority-spin (↓) electrons from the injector will tunnel toward majority-spin (↑) empty states of the collector. Both currents are then low. The conductance then writes

<footnote n="1">¹Note here that the arrow does not define any direction of the spin, but rather the fact that it is aligned with the local magnetization. In the (AP) case, a spin with one direction will correspond to (↑) spin in one electrode and (↓) spin in the other.</footnote>
<figure><img src="image_72.png" /><figcaption>**Fig. 5.5** TMR curve of a CoFe/Al₂O₃/Co MTJ recorded at room temperature [Adapted from [13] with permission (Copyright 1995, American Physical Society)]. Schematic of the spin-dependent tunnelling process through an insulating barrier when their magnetizations are aligned parallel (left) and antiparallel (right) to one another. The process is assumed to be purely elastic, so that no mixing of spin states occurs during the tunnelling process</figcaption></figure>

$$G_{AP} \propto D_1^\uparrow(E_F)D_2^\downarrow(E_F) + D_1^\downarrow(E_F)D_2^\uparrow(E_F). \qquad (5.3)$$

Using these results for the conductance, the TMR ratio, characterizing the resistance difference in (P) and (AP) configurations, is expressed as

$$TMR = \frac{G_P - G_{AP}}{G_{AP}} = \frac{R_{AP} - R_P}{R_P} = \frac{2P_1 P_2}{1 - P_1 P_2}, \qquad (5.4)$$

where $P_i$ defines the spin polarization for each electrode as

$$P_i = \frac{D_i^\uparrow(E_F) - D_i^\downarrow(E_F)}{D_i^\uparrow(E_F) + D_i^\downarrow(E_F)}. \qquad (5.5)$$

A key issue to get an accurate prediction of TMR is to properly estimate the actual amplitude and sign of the spin polarization for a given FM material. Obviously, the largest the spin polarization is, the highest will be the TMR amplitude. It thus explains the strong research activity in the last decade on material science to investigate novel families of materials (magnetic oxides, Heusler alloys, etc.) for which some of their
compounds are predicted to be half metallic, i.e. to have a 100% spin polarization at the Fermi level.

Jullière's model has enjoyed much success in correctly predicting TMR amplitudes in most MTJs with amorphous barriers, until the emergence of MTJs based on crystalline insulating barriers, notably magnesium oxide (MgO) barriers in the beginning of the 2000s [15, 16]. The model described previously is actually oversimplified in that it considers an identical tunnelling process of every electron, independently on their band belonging, hence forgetting about the interplay of the band structures throughout the MTJ heterostructure.

### 5.1.2.2 Coherent Tunnelling in Epitaxial Magnetic Tunnel Junctions

Another major breakthrough in the field of spintronics corresponds to the introduction of crystalline MgO tunnel barriers, that is today's standard in MTJ implemented in MRAMs, read heads and field sensors working with TMR effect. These experimental developments have been stimulated by some theoretical calculations made by W. Butler [17] predicting that huge TMR ratios as large as 1600% are anticipated for epitaxially grown Co or Fe electrodes on crystalline, instead of amorphous MgO. These calculations, developed with ab initio methods, derive the tunnelling probability of each kind of electrons, depending on their orbital symmetry.

Contrary to amorphous alumina [Fig. 5.6a left], in crystalline MgO tunnel junctions, the electron wave functions in the FM material are coupled with evanescent wave functions having the same symmetry in the barrier [Fig. 5.6a right]. Through ab initio calculations, it was then predicted that the tunnelling probability of an electron strongly depends on the orbital symmetry of the electron (of the band it belongs to). Beyond the two-current (up-spin and down-spin) model, the system behaves as if it exists an independent current channel for each band and each spin, leading to a possible effective symmetry filtering of the tunnelling current. The tunnel barrier can, therefore, filter the wave functions and thus select the spins in the electronic transport.

This mechanism of orbital selection for the tunnel conductance is presented in Fig. 5.6. In Fig. 5.6b, band dispersion of bcc Fe(001) for the minority and majority spins is shown. $\Delta_1$ Bloch states are present at the Fermi level only for the majority spins. In Fe(100)/MgO/Fe(100) systems, band structure calculations have demonstrated that majority-spin electrons [see Fig. 5.6c] are mainly filling $\Delta_1$ symmetry states (hybridized states with spd characters), whereas minority-spin electrons [see Fig. 5.6d] are filling $\Delta_2$ symmetry states (d type states) and $\Delta_5$ symmetry states (hybridized pd states). Moreover, this is crucial for getting a large TMR ratio, the tunnelling exponential decay is much stronger for $\Delta_2$ and $\Delta_5$ states compared to $\Delta_1$ states. For $d_{MgO} = 8$ monolayers, which is a typical barrier thickness that can be achieved experimentally, the probability of transmission of $\Delta_1$ electrons is larger than for $\Delta_5$ electrons by 10 orders of magnitude. Ultimately, only $\Delta_1$ electrons contribute significantly to the current. It is this filtering effect which can explain the large values of TMR expected on epitaxial or highly textured structures and that has made
<figure><img src="image_73.png" /><figcaption>**Fig. 5.6** **a** Schemes illustrating electron tunnelling through an amorphous Al-O barrier (left) and through a crystalline MgO barrier (right). **b** Band dispersion of bcc Fe(001) for the minority and majority spins. $\Delta_1$ Bloch states are present at the Fermi level only for the majority spins. Tunnelling DOS for Fe/MgO/Fe at $k_\parallel = 0$ for majority **c** and minority **d** spins. Decay of $\Delta_1$ Bloch states in MgO is less attenuated than the $\Delta_1$ Bloch states. **e** TMR of 600% obtained at room temperature for a CoFeB/MgO/CoFeB MTJ. Adapted from [18] (**a**, **b**) with permission (Copyright 2007, Institute of Physics), from [17] (**c**, **d**) with permission (Copyright 2001, American Physical Society) and from [19] (**e**) with permission (Copyright 2008, American Institute of Physics Publishing)</figcaption></figure>

the MgO-based tunnel junctions at the core of the development of new spintronic devices like the MRAMs.

After these theoretical predictions, a strong research effort has been made to obtain epitaxial growth of structures for Fe/MgO/Fe or CoFeB/MgO/CoFeB [20, 21]. These efforts have resulted in a TMR of about 600% obtained in 2009 [19] in CoFeB/MgO/CoFeB MTJs at room temperature [see Fig. 5.6e]. An excellent review on the physics of tunnelling transport in MgO-based systems and the experimental state of the art for TMR has been written by S. Yuasa [18].

Since the measurements of TMR effects at room temperature in 1995, a lot of work has been done in order to both increase the TMR ratio and reduce the MTJ resistance. This has been achieved by developing high-quality crystalline MgO-based MTJs. This effort has led to the development of a new class of magnetic memories called MRAMs.
### 5.1.3 Magnetization Manipulation without Magnetic Fields

For practical applications, e.g. MRAMs, the possibility to manipulate the magnetization of FM electrodes in spin valves or MTJs without using magnetic field is required. This corresponds to the next breakthrough with the prediction and the observation of spin-transfer effects, first providing a new way to reverse the magnetization in a nanostructure but also to generate in some cases steady precession of the magnetization. In this section, we briefly present the physics of spin-transfer torque (STT) and also discuss an alternative approach consisting in the manipulation of the magnetization through the application of a voltage.

#### 5.1.3.1 Spin-Transfer Torque

In the first generation of MRAM devices developed in 2004, the magnetization reversal between the two possible magnetic states was realized by using local magnetic fields generated by electrical currents flowing in lines close to each magnetic element. This writing process rapidly suffered from both the large energy consumption needed to generate large enough magnetic fields and from cross-talk problems due to the difficulty to write a single bit. A solution came out from a major breakthrough in spintronics in 1996 when it has been proposed that a spin-polarized current flowing through magnetic multilayers provides a new way to manipulate the magnetization of a ferromagnet. This new effect, which was called spin-transfer effect, has become rapidly a very hot topic. Moreover, it is at the basis of a new generation of magnetic memories, called spin-transfer torque-MRAMs (STT-MRAMs).

The concept of spin-transfer effect was proposed in 1996, concomitantly by J. Slonczewski [22] and L. Berger [23]. To describe this effect, one can take a standard spintronic structure composed of a fixed FM electrode $F_1$ and a free FM layer $F_2$ separated by a NM spacer. When the electrons flow from layer $F_1$ to $F_2$, the current becomes spin-polarized after passing through $F_1$. This non-zero spin polarization is aligned along the magnetization direction in $F_1$ and propagates into the NM metallic spacer or tunnels in the case of an insulating barrier, so that it arrives at the $NM/F_2$ interface. If the magnetization $\vec{M}_1$ and $\vec{M}_2$ are non-collinear, it results that a component of the spin current transverse to $F_2$ exists at the $NM/F_2$ interface [green arrows in Fig. 5.7a]. When the electrons penetrate into $F_2$, the spin of the conduction electron becomes aligned over a very short distance (within a few atomic distances) toward the magnetization direction in $F_2$ because of a strong exchange interaction between conducting (s) electrons and localized (d) electrons, the latter being responsible of the magnetic moments.

During all this process, the total spin angular momentum is conserved. Thus the transverse component of the spin current $\Delta\vec{m}$ lost by the electrons when passing through $F_2$ is indeed absorbed and transferred to the local magnetization of $F_2$. This transfer of spin angular momentum results in a torque exerted by the spin-polarized current on the local magnetization. For this current polarity, when the
<figure><img src="image_74.png" /><figcaption>**Fig. 5.7** a Scheme of a magnetic trilayer structure for illustrating the concept of spin-transfer torque. b The transverse component of the angular moment $\Delta m$ of the spin-polarized current is transferred to the magnetization. It results in a torque exerted on the magnetization $\vec{M}_2$, of the thin magnetic layer, which aligns along magnetization $\vec{M}_1$ for positive current. c Schematic representation of the torques acting on the magnetization including spin-transfer effect</figcaption></figure>

current increases enough, it tends to align the magnetization $\vec{M}_2$ along the direction of the spin polarization of the current, i.e. along the magnetization $\vec{M}_1$ [Fig. 5.7b]. As all the process occurs in the first atomic planes after the interface, the spin-transfer mechanism is an interfacial effect.

After having introduced the spin-dependent transport mechanisms at the origin of STT, our aim is now to address the influence of this torque on the magnetization dynamics of the layer $F_2$. A classical approach to describe the dynamical motion of a magnetization is a differential equation, named the Landau–Lifschitz–Gilbert equation, to which we add Slonczewski’s component of spin torque.<fnref n="2" /> This equation has the

<footnote n="2">²For simplicity we have only introduced the main STT called "in-plane torque". A second spin torque, called "field-like torque" or "out-of-plane torque", is similar to a torque exerted by a field along the magnetization of $F_2$ and its action might, therefore, be included into $\vec{H}_{eff}$.</footnote>
following form, taking magnetizations normalized to the saturation magnetization, $m_{i=1,2}$ for the two FM layers

$$ \frac{d\vec{m}_2}{dt} = -\gamma_0(\vec{m}_2 \times \vec{H}_{eff}) + \frac{\alpha}{\gamma_0 M_{S2}}(\vec{m}_2 \times \frac{d\vec{m}_2}{dt}) - P_{spin} \frac{Jh}{2} \frac{g\mu_B}{M_{S2}^2 t} [\vec{m}_2 \times (\vec{m}_2 \times \vec{m}_1)] \quad (5.6) $$

The first term corresponds to the tangential force describing the magnetization precession around effective field $\vec{H}_{eff}$ [green arrow in Fig. 5.7c], which takes into account the external applied magnetic field, magnetic anisotropy fields, coupling fields, etc. The second term gives a phenomenological description of magnetization dissipation of the system. The coefficient $\alpha$, named Gilbert damping, (about $10^{-2}$ for standard FM materials), describes the damping rate of the motion of $\vec{m}_2$ towards the equilibrium position oriented along $\vec{H}_{eff}$ [blue arrow in Fig. 5.7c]. The magnetization relaxation is induced by a damping force tangential to the magnetization trajectory. The third term is the so-called Slonczewski torque (or in-plane torque) where $\mu_B$ is the Bohr magneton, $t$ the layer thickness, $J$ the injected current density, and $P_{spin}$ the amplitude of spin polarization at the interface NM/F$_2$ and $M_{S2}$ the magnetization of the ferromagnet F$_2$. This simplified description allows making clear the nature of the main contribution of the spin-transfer force that can be described as a non-conservative force acting in the same direction than the natural damping, i.e. perpendicularly to the magnetization trajectory. Depending on the sign of the injected current, the spin-transfer torque decreases or increases the effective damping [purple and red arrows in Fig. 5.7c]; it favours the stability of the parallel or the antiparallel magnetic configuration.

For large current density (~$10^7$ A cm$^{-2}$ for a typical material system), the STT fully compensates the damping torque, steady magnetization precession occurring at the ferromagnetic frequency (typical in the GHz range for the ferromagnetic materials and in the THz range for antiferromagnetic ones) can be established. The spin transfer induced magnetization dynamics can convert into oscillations of resistance through the magnetoresistive effect described previously, and in turn into a radiofrequency voltage signal. Spin-torque effect thus makes it possible to convert a dc current into a rf voltage and so to build microwave oscillators at the nanoscale.

For larger current density, the torque can become sufficient to commute the magnetization between the two stable configurations. A negative current will, for instance, destabilize the parallel magnetization configuration, while stabilizing the anti-parallel configuration, allowing to commute from P to AP configuration. Identically, a positive current allows to commute from AP to P configuration.

First experimental results allowing to confirm the theoretical predictions of the existence of STTs have been obtained by M. Tsoi et al., using point contact geometry for injection of a large current into a magnetic layer [24]. Then after, it has been demonstrated that magnetization commutation can be achieved back and forth by the STT effect in Co/Cu/Co spin valves [25]. Figure 5.8a represents magnetization rever-
<figure><img src="image_75.png" /><figcaption>**Fig. 5.8** a Current-induced magnetization switching in a Co/Cu/Co nanopillar spin valve. Jumps in the curves correspond to the magnetization reversal of the ferromagnetic free layer. b Current-induced magnetization switching in a CoFeB<sub>2.5</sub>/AlOx/CoFeB<sub>2.5</sub> MTJ. c Domain wall motion induced by a current in Co/Cu/Py spin valves. State 1 and 2 correspond to different positions of the domain walls in the strip. By applying a large enough current it is possible to move the domain walls and thus reach the parallel (P) or antiparallel (AP) magnetic configuration. Reproduced from [25] (a) with permission (Copyright 2000, American Physical Society), from [26] (b) and [27] (c) with permission (Copyright 2005 and 2002, American Institute of Physics Publishing)</figcaption></figure>

sal induced by a current for two different applied magnetic fields. Current density in the 10⁷ A cm⁻² range corresponding to few mA for their nanopillars is needed to switch the Co free electrode magnetization. Few years later similar results have been obtained in MTJs using AlOx and MgO tunnel barriers [26]. Figure 5.8b illustrates the magnetization reversal by the current for a CoFeB/AlOx/CoFeB magnetic tunnel junction. A smaller current (less than 1 mA) is needed to commute the magnetization and thus to switch between the parallel and antiparallel magnetic configurations. This result shows that STT effects can be used not only in metallic spin valves but also in MTJs and has opened the door for the development of new STT-MRAM technologies. Finally, STT has been also used to induce magnetic domain wall motion. Figure 5.8c represents the first observation of domain wall motion by spin transfer in a Co/Cu/Py trilayer, where Py stands for permalloy (Ni₈₀Fe₂₀) [27]. Starting from configuration 1 or 2 (the domain wall is located at the two-third of the strip) it is possible to move the domain wall by a current to reach the parallel (P) or antiparallel (AP) magnetic configuration. The current density needed to move these domain walls is about 10⁷ A cm⁻². Note that for all these pioneer experiments, a magnetic field is applied. Since then, a lot of work has been done and now magnetization manipulation without applying magnetic field is feasible. Later, STT has been also used to manipulate other magnetic textures such as magnetic vortices [28] or skyrmions, or generate dynamics [29, 30]. To conclude, STT is now used to write the information of a single bit in magnetic memories.<fnref n="3" />

<footnote n="3">³For more information on the latest MRAM development, see https://www.mram-info.com/</footnote>
<figure><img src="image_76.png" /><figcaption>**Fig. 5.9** (left panel) Voltage can be used to modify magnetic anisotropy, coercive fields, magnetization values, exchange bias, Curie temperature or magnetoresistance. This can be achieved by different mechanisms such as charge, strain exchange coupling, orbital ordering or electromigration (right panel). Reproduced from [31] with permission (Copyright 2017, Elsevier)</figcaption></figure>

### 5.1.3.2 Voltage Control of Magnetism

Although STT is very efficient to manipulate magnetization, alternative or complementary approaches have been explored such as the voltage control of magnetism. Magnetic anisotropy, coercive fields, magnetization magnitude, exchange bias or Curie temperature can be tuned by a voltage [Fig. 5.9 (left panel)]. Different mechanisms such as charge accumulation/depletion, electromigration, strain or orbital ordering are involved [Fig. 5.9 (right panel)].

The samples to study the effect of an electric field on the magnetization are generally composed of an ultra-thin FM film in contact with either dielectric, ferroelectric or piezoelectric materials. A wide variety of FM materials (metals, oxides or diluted magnetic semiconductors) have been also studied [31]. Depending on the mechanism used to control magnetism through an applied voltage, the ferromagnet thickness has to be adjusted. As charge, orbital ordering or electrochemistry are mainly interfacial effects, the thickness of the ferromagnet should be close to the screening length which is in the angstrom range for metal and in nanometre range for dilute magnetic semiconductors. Thicker ferromagnets, up to the micrometre range, can be used if strain is involved. In the following, several examples illustrating the variety of devices studied are displayed. Readers are invited to refer to the article of C. Song et al. [31] for a complete review.

In Fig. 5.10, we show different examples where the Curie temperature, the coercive field and the magnetic anisotropy are modified by an electric field. This magnetization control involves different mechanisms such as charge or strain effects. One of the first demonstration of magnetization manipulation by an electric field has been done using a dilute magnetic semiconductor [32]. In Fig. 5.10a, b, Hall effect curves recorded at different gate voltages for a (In,Mn)As-based field-effect transistor are presented. Magnetic properties and notably the Curie temperature of the diluted magnetic semiconductor (In,Mn)As being dependent on the hole density, it is thus possible to tune the Curie temperature by an electric field modulating the carrier density [33]. Setting the sample temperature close to its Curie temperature, the switching
<figure><img src="image_77.png" /><figcaption>**Fig. 5.10** Examples of voltage control of the magnetization. **a, b** Hall effect curves recorded for different gate voltages in a dilute magnetic semiconductor (In, Mn) As field-effect transistor. The gate voltage induces a modulation of the hole concentration in (In, Mn)As and thus a modulation of the Curie temperature. **c, d** Variation of the coercive field of a FePt thin film as a function of bias voltage. The bias voltage modifies the number of 3d electrons and thus changes the magnetocrystalline anisotropy. **e, f** Modification of the magnetic anisotropy induced by strain in a FeGaB/PZN-PT multiferroic heterostructure. The voltage leads to lattice modulation of the PZN-PT ferroelectric inducing a strain modulation of the magnetostrictive FeGaB layer. Under an electric field of 8 kV cm⁻¹ a magnetic field of 70 mT is needed to align the magnetization along the [100] direction. Adapted from [32] (**a, b**) with permission (Copyright 2000, Nature Publishing Group), from [34] (**c, d**) with permission (Copyright 2007, American Association for the Advancement of Science), from [31] (**e**) with permission (Copyright 2017, Elsevier) and from [35] (**f**) with permission (Copyright 2009, John Wiley and Sons)</figcaption></figure>

between ferromagnetic and paramagnetic states can be realized by applying, respectively, a negative and a positive gate voltage (125 V). A similar approach has been investigated to tune the coercive field of the FePt FM intermetallic compound [34]. Using an electrolyte to modify electron density at the FePt interface, a modification of 4.5% of the coercive field is observed [Fig. 5.10c, d]. The variation in the number of 3d electrons directly affects the magneto-crystalline anisotropy and thus the coercive field. Interestingly, this device geometry allows tuning the magnetic properties of a ferromagnet by applying a low bias voltage (below 1 V). Although the modification of the coercitive film is quite small, this first demonstration of voltage control of a FM metal was quite encouraging as it opens the door to room temperature modulation of magnetism by applying small voltage. Another approach to modify the magnetic anisotropy is to apply a strain. By combining ferroelectric and magnetostrictive materials, it is possible to modify the magnetic properties by a voltage. Under voltage, the lattice of the ferroelectric layer is modulated through the inverse piezoelectric effect and will thus induce a strain modulation in the ferromagnet. In Fig. 5.10(e,f), we show an example where the in-plane magnetic anisotropy is modified by an electric field. Whereas at zero electric field, the remanent magnetization is along the [100] direction, an electric field of 8 kV cm⁻¹ allows rotating the magnetization and so a magnetic field of 70 mT is now needed to saturate it along the [100] direction.
The voltage control of magnetization is obviously less mature than STT or spin-orbit torque (SOT) technologies. However, the large variety of ferromagnets (metals, semiconductors, oxides) and gating materials (dielectric, ferroelectric, electrolyte) that can be used, makes this field fascinating and promising to reduce power consumption in memory technologies, which remains still a crucial issue.

### 5.1.4 Summary

We have introduced some breakthroughs in the field of spintronics achieved during the last 30 years. Giant magnetoresistance has emerged rapidly as a promising effect to build efficient magnetic sensors working at room temperature. The ability to manipulate magnetization by STT in MTJs has allowed developing new magnetic memories such as MRAMs that are now commercially available. This is of course not exhaustive and there are a lot of other exciting and promising research fields such as spin-orbitronics, magnonics, molecular spintronics or antiferromagnetic spintronics, to cite only few of them. Spin-orbitronics is certainly the most active field nowadays. Emblematic topics are conversion between charge and spin currents, spin-polarized surface and interface states or novel chiral magnetic textures (skyrmions and domain walls). See for example A. Soumyanarayanan et al. for a review [36].

In all the spintronic effects that we have just introduced, like in many other fields, interfaces play a key role, and therefore a deep knowledge of the electronic and magnetic properties of these interfaces is desired. Synchrotron radiation-based measurements such as absorption and photoelectron spectroscopies and microscopies are powerful techniques to probe these interfaces.

## 5.2 Examples of Synchrotron Radiation Contribution to Spintronics

In the following, several examples for which synchrotron radiation-based measurements have allowed a better understanding of spintronic devices are presented. We have selected three spintronics topics: (i) voltage control of magnetism; (ii) spintronics with pure spin currents; (iii) current-driven magnetization dynamics.
## 5.2.1 Voltage Control of Magnetism

### 5.2.1.1 Effect of Charge Accumulation/Depletion and Electromigration

Perpendicular magnetic anisotropy (PMA) has been widely investigated in NM/FM/oxide heterostructures. For example, by varying the oxidation time of top Al layer in Pt/Co/AlOx devices, it has been demonstrated that the PMA is induced by Co–O bonds [37]. Figure 5.11(a,b) represent extraordinary Hall effect curves and X-ray absorption spectroscopy (XAS) measurements at the Co $L_{2,3}$ edges for Pt/Co/AlOx samples with different oxidation times. The extraordinary Hall effect allows detecting the magnetic easy axis and XAS measurements probe the electronic properties of the cobalt layer. A metallic Co/Al interface resulting from Al under-oxidation, or the formation of a CoO layer due to over-oxidation, both produce magnetization aligned within the plane of the film. Hence optimized oxidation, leading to Co–O–Al bonds, induces a perpendicular magnetic anisotropy. Instead of playing with oxidation time, D. Chiba et al. [38] have shown that it was possible to switch the easy magnetic axis from in-plane to out-of-plane by applying a bias voltage. In Fig. 5.11c, we show that a bias voltage of 10 V allows to modify the magnetic

<figure><img src="image_78.png" /><figcaption>**Fig. 5.11** a Extraordinary Hall effect as a function of oxidation time measured in a Pt/Co/AlOx trilayer. b XAS spectra recorded at the Co $L_{2,3}$ edges for different oxidation times. These measurements show that optimized Al oxidation induces perpendicular magnetic anisotropy due to Co–O bonds. c Switching between in-plane magnetization to out-of-plane magnetization by applying a bias voltage for a Pt/Co/MgO sample. Reproduced from [37] (a, b) with permission (Copyright 2008, American Institute of Physics Publishing), and from [38] (c) with permission (Copyright 2011, Nature Publishing Group)</figcaption></figure>
anisotropy in Pt/Co/MgO [38]. This magnetic anisotropy modification with voltage can originate from charge accumulation or oxygen electromigration.

C. Bi et al. have performed Hall resistivity and XAS measurements at the Co $L_3$ edge on a Pt/Co/GdOx sample [39]. A clear correlation between the evolution of the magnetic anisotropy and the Co oxidation was demonstrated [Fig. 5.12(a,b)]. In Fig. 5.12b, one can see that the shape of the Co XAS spectra are modified with voltage. Whereas a negative voltage induces some fine structures in the XAS spectra indicating a Co oxidation, a positive voltage reduces the Co layer, and therefore Co spectra become similar to that of metallic Co. Thus the Co magnetic anisotropy in this Pt/Co/GdOx structures can be reversibly controlled by voltage via Co oxidation and reduction. This result confirms the earlier experiments performed by F. Bonell et al. [40] on Au/CoFe/MgO samples. Hence, in these examples, the modification of magnetic anisotropy is rather due to oxygen electromigration than charge accumulation.

In V/Fe/MgO devices, S. Miwa et al. [41] have indeed observed a different behaviour. A slight change of coercive field with bias voltage has been measured [see Fig. 5.12c] but without any modification of the Fe $L_{2,3}$ XAS and X-ray magnetic circular dichroism (XMCD) spectra with the applied voltage, suggesting that Fe is not oxidized [Fig. 5.12d]. This is quite surprising since the electric field applied is similar in Pt/Co/GdOx and V/Fe/MgO experiments. This result shows that modifications of the coercive field is not induced by electromigration but is rather due to charge accumulation.

To conclude, this careful investigation of the electronic properties of the FM/oxide interface has allowed to unveil the origin of voltage control of magnetic anisotropy in each NM/FM/oxide structures and thus to discriminate between electromigration and charge effects.

### 5.2.1.2 Effect of Strain

Strain has been also used to tune magnetic anisotropy and can be very efficient if magnetostrictive materials are used. By combining piezoelectric and magnetostrictive materials, it is possible to control the magnetic anisotropy by voltage. Here we show an example where photoelectron emission microscopy (PEEM) measurements have been performed to highlight the correlation between strain and magnetism [42]. A magnetostrictive CoFe₂O₄ layer has been deposited on a piezoelectric BaTiO₃ substrate. BaTiO₃ can have domains with different unit cell parameters, and therefore can induce different domains with different strains in the CoFe₂O₄ layer. X-ray linear dichroism (XLD) being sensitive to local coordination, different XLD spectra will be obtained. In Fig. 5.13d, XLD-PEEM<fnref n="4" /> image recorded at the Fe $L_3$ edge are

<footnote n="4">⁴XAS spectra can be also recorded on the different domains allowing to measure the local electronic properties.</footnote>
<figure><img src="image_79.png" /><figcaption>**Fig. 5.12** a Hall resistance for a Pt/Co/GdOx sample as deposited (red) and after an applied bias voltage leading to an electric field of –625 kV cm⁻¹ (blue) and +625 kV cm⁻¹ (purple) showing an in-plane to out-of-plane magnetization transition. b XAS and XMCD spectra recorded at the Co L₃ edge. A negative voltage induces an oxidation of the Co layer, whereas a positive voltage reduces the Co layer. c Magnetic hysteresis loop of a V/Fe/MgO sample obtained for two bias voltages showing a modification of the coercive field. d Fe L₂,₃ XAS and XMCD spectra recorded at +4 V and –4 V. Fe oxidation/reduction is not observed demonstrating that the coercive field modification is rather attributed to a charge effect. a, b Adapted from [39] with permission (Copyright 2014, American Physical Society) and c, d from [41] with permission (Copyright 2015, American Institute of Physics Publishing)</figcaption></figure>

presented showing different strain domains for the CoFe₂O₄ layer induced by the BaTiO₃ substrate. By recording XMCD-PEEM images at the Fe L₃ edge, domains with different magnetic signals are observed. In Fig. 5.13, it appears clearly that a correlation exists between the strain state and the magnetic signal. Black domains in the XLD-PEEM [see Fig. 5.13d] correspond to a magnetization along the [100] direction [see Fig. 5.13a], whereas for white domains, the magnetization is along the [010] direction [see Fig. 5.13c]. Hence, by performing XLD-PEEM and XMCD-
<figure><img src="image_80.png" /><figcaption>**Fig. 5.13** XMCD-PEEM (a–c) and XLD-PEEM (d) images recorded at Fe $L_3$ edge for a BaTiO$_3$/CoFe$_2$O$_4$ sample. XMCD-PEEM images show magnetic stripes along the [100] and [010] directions. These magnetic stripes are correlated to domains observed in the XLD-PEEM image *d* which indicates different strain states of CoFe$_2$O$_4$ induced by the BaTiO$_3$ substrate.

Reproduced from [42] with permission (Copyright 2012, American Physical Society)</figcaption></figure>

PEEM measurements, it is possible to locally observe the influence of strain on magnetism.

We have shown that XAS has allowed determining the origin of the voltage control of magnetism in NM/FM/oxide structures. In particular, it is possible to disentangle charge and electromigration effects. By performing PEEM measurements with both circular and linear polarized X-rays, the correlation between strain and magnetism can be directly probed. Hence, synchrotron radiation-based measurements are useful to better understand the mechanisms of the voltage control of magnetism.

### 5.2.2 Spintronics with Pure Spin Current

The generation of pure spin current from heat, charge current, light or vibration is an active and promising research field [43]. A wide variety of materials (ferromagnets,
heavy metals, semiconductors, insulators, topological insulators, etc.) and devices are currently investigated. In this paragraph, we aim at discussing the spin-charge conversion using topological insulators as well as the heat-spin current conversion in ferromagnetic insulator/paramagnetic metal devices.

### 5.2.2.1 Spin-Charge Conversion

Conversion between charge and spin currents has been a very active branch of spintronics in the last couple of years. Such a conversion can be achieved in bulk materials (the so-called spin Hall effect) or at interfaces (Edelstein-Rashba effect) relying on the spin-orbit interaction and/or extrinsic effects. The studied materials are usually NM metals with a large spin-orbit coupling. A second approach is to rely on spin-orbit properties at the interfaces either at Rashba interfaces or through the surface states of topological insulators (TIs). A complete review of this approach can be found in J. Sinova et al. [44] or A. Soumyanarayanan et al. [36].

In the following, we provide an example where angle-resolved photoemission spectroscopy (ARPES) measurements have allowed to understand the spin-charge conversion from the α-Sn TI [see Fig. 5.14a]. In this spin-charge conversion study, the spin current is generated through the magnetization dynamics induced at the magnetization resonance of a Fe layer by an external *rf* field. The generated spin current then diffuses to the α-Sn top surface. Injection of a spin current into a TI induces a spin accumulation on one side of the Fermi contour of the Dirac cone as well as a spin depletion on the other side [see Fig. 5.14b]. As a consequence, this spin injection results in a charge current. Importantly, note that the spin-charge conversion is not observed when Fe is deposited directly on α-Sn, but is observed when a thin Ag layer is inserted at the interface [see Fig. 5.14c]. By performing ARPES measurement to probe the DOS, it has been shown that the deposition of a sub-monolayer of Fe on α-Sn indeed suppresses the Dirac cone, that is a signature of the TI, whereas it is still observable after deposition of a Ag layer [see Fig. 5.14d]. Hence the absence of spin-charge conversion at Fe/α-Sn interfaces is clearly ascribed to the loss of TI surface states after the Fe deposition. This study shows that characterization of the DOS by ARPES measurements is a very useful and unique technique to understand spin-charge conversion at such spinorbitronic interfaces.

### 5.2.2.2 Heat-Spin Conversion

Similarly to the Seebeck effect, the spin Seebeck effect describes the generation of a spin voltage from a temperature gradient in a FM conductor or insulator. Longitudinal spin Seebeck effect (LSSE) refers to experiments where the spin current generated is parallel to the temperature gradient. Materials used are FM materials (conducting
<figure><img src="image_81.png" /><figcaption>**Fig. 5.14** a Scheme of the device studied for the spin to charge conversion by spin pumping into the topological insulator α-Sn. b Illustration of the inverse Edelstein effect. A spin current injected in the topological insulator induces an accumulation of charge for one spin direction inducing a shift of the Fermi contour creating a charge current. c Ferromagnetic resonance and dc charge current signals for α-Sn/Fe/Au and α-Sn/Ag/Fe/Au samples. Only the second sample shows a dc current signal. d ARPES measurement along the [100] or [110] directions on the free surface of α-Sn (top) and after deposition of Fe (left) or Ag (right). The Dirac cone subsists after deposition of Ag but not for Fe. Adapted from [45] with permission (Copyright 2016, American Physical Society)</figcaption></figure>

or insulating) in contact with heavy metals. In order to extract LSSE efficiency, it is necessary to discard the possible artefacts such as an anomalous Nernst effect that arises from the potential induced magnetization in the NM layer [46].

Prototype material systems contain a thin film of yttrium iron garnet $Y_3Fe_5O_{12}$ (YIG) as a FM insulator and a thin layer of Pt as high spin-orbit non-FM material. In order to detect and quantify the magnetic proximity effect (MPE) at the YIG/Pt interface, XMCD measurements appear to be the most accurate method. In Fig. 5.15, XMCD measurements at the Pt $L_{2,3}$ edges performed on YIG/Pt samples are shown. Whereas no MPE have been detected for one study [see Fig. 5.15a] [47, 48], a clear XMCD signal has been observed for another experiment [Fig. 5.15b] [49]. This discrepancy can be attributed to different interface qualities. Actually, x-ray absorption near-edge spectra (XANES) have clearly different shapes. The white line intensity (XANES step height) depends on the number of holes in the 5d band and is, therefore, sensitive to the oxidation state of Pt. In the case where no MPE has been detected, the white line intensity is about 1.3 which is similar to the Pt metal value. On the contrary, when MPE is observed, the white line intensity is about 1.45 which is close to what is observed for $PtO_{1.36}$ samples [50]. These observations mean that the observed MPE is probably due to intermixing at the YIG/Pt interface.
<figure><img src="image_82.png" /><figcaption>**Fig. 5.15** XANES and XMCD recorded at the Pt *L*₂,₃ edges for YIG/Pt samples. For apparently similar samples a clear XMCD signal is observed for one sample (right), whereas no induced magnetic moment is measured for the other (left). The higher intensity of the step edge observed for the sample exhibiting an XMCD signal (right) meaning an increase of the number of holes in the 5d band indicates a possible Pt oxidation. Hence different interface qualities induce different XMCD signals. Reproduced from [48] (left) and from [49] (right) with permission (Copyright 2013, American Physical Society)</figcaption></figure>

This behaviour has been clearly evidenced by further experiments on insulating ferrite/Pt samples. In Fig. 5.16 (left panel), XMCD measurements performed at the Pt *M*₃ edge on CoFe₂O₄/Pt samples are presented [51]. A clear XMCD is observed for samples where Pt has been deposited at high temperature (HT in Fig. 5.16), whereas no XMCD signal is detected when Pt is grown at room temperature (RT in Fig. 5.16). By performing XAS and XMCD measurements at Co and Fe *L*₂,₃ edges [see Fig. 5.16 (right panel)] it appears clearly that the deposition of Pt at high temperature induces some intermixing at the CoFe₂O₄/Pt interface. Indeed, Co XAS spectra are close to those of metallic Co films for the sample with Pt grown at high temperature, whereas for the sample with Pt deposited at room temperature, spectra shapes are similar to those of CoFe₂O₄ thin films. Hence the presence of MPE in ferrite/Pt interface comes from interface alloying, whereas for clean interfaces, the induced magnetization by proximity effect is absent. In fact, XAS is not the only measurement to detect MPE; X-ray resonant magnetic reflectivity is also a powerful technique. Thanks to the latter, T. Kuschel et al. have confirmed that no MPE occurs at the clean NiFe₂O₄/Pt interface [52]. Finally, it has been shown that no MPE occurs for another ferrite (MnFe₂O₄/Pt) and for magnetite, both in the conducting and insulating states [53]. This suggests that the absence of MPE at the magnetic oxide/Pt clean interface is a general rule.
<figure><img src="image_83.png" /><figcaption>**Fig. 5.16** XAS and XMCD spectra recorded at the Pt $M_3$ edge for CoFe$_2$O$_4$/Pt where Pt is grown at room temperature (RT) and high temperature (HT). A clear XMCD signal is observed at the Pt $M_3$ edge when Pt is grown at high temperature. A clear difference can be also seen in the Fe and Co $L_{2,3}$ edges. For Pt grown at HT Co $L_{2,3}$, spectra look like Co metal signifying an intermixing at the CoFe$_2$O$_4$/Pt interface. When Pt is grown at RT, Co and Fe $L_{2,3}$ XAS spectra are similar to those of CoFe$_2$O$_4$ thin films. Adapted from [51] with permission (Copyright 2018, American Chemical Society)</figcaption></figure>

These experiments hence highlight that synchrotron radiation-based spectroscopies are powerful to probe the electronic and magnetic properties of interfaces and can be very useful to discard artefacts in LSSE experiments.

### 5.2.3 Current-Induced Magnetization Dynamics

Besides STT-MRAMs, other conceptual memory devices have been proposed and largely studied in the decade. A flagship example is the domain racetrack memory, proposed by S. S. P. Parkin [54], which is a nanoscale shift register memory in which bits are defined by magnetic domains separated by domain walls. Another version of such devices has been recently introduced where the bits are stored thanks to the presence of magnetic skyrmions [55]. In this section, we show that magnetization dynamics imaging using synchrotron radiation-based measurements that can be useful to better understand these STT-MRAM and racetrack memory devices.

#### 5.2.3.1 Spin-Orbit Torque Driven Magnetization Reversal

SOT is an efficient tool to manipulate magnetization in spintronic devices integrating only one FM electrode (see, for example, R. Ramaswamy et al. for a recent review [56]). M. Baumgartner et al. [57] have studied Co nanodot magnetization reversal by SOT. In Fig. 5.17a, the magnetization reversal [measured by XMCD at the Co $L_3$ edge (black)] induced by a current pulse (red) is shown. Then, by performing time-resolved scanning transmission X-ray microscopy (STXM) measurements, it has
<figure><img src="image_84.png" /><figcaption>**Fig. 5.17** a Magnetization reversal probed by XMCD measurements at Co $L_3$ edge (black) induced by a current pulse (red). b STXM images recorded at intervals of 100 ps during a 2 ns injected current pulse. The four rows correspond to different applied current (yellow arrows) and magnetic field (blue arrows) conditions. Red dots indicate domain wall nucleation and green arrows the direction of the domain wall propagation. Adapted from [57] with permission (Copyright 2017, Nature Publishing Group)</figcaption></figure>

been possible to perform a direct observation of the actual path leading to the magnetization reversal during the current pulse injection. Figure 5.17b represents STXM images recorded at the Co $L_3$ edge with 25 nm spatial resolution at intervals of 100 ps during a 2 ns current pulse. From these images, nucleation (red dot) and propagation (green arrow) of magnetic domain walls can be clearly observed. Depending on the applied magnetic field direction and the current polarity, the main characteristics of these nucleation and propagation might change. These measurements have allowed demonstrating that this diagonal motion of the domain wall originates from a combination of the damping-like and field-like SOTs and the Dzyaloshinskii–Moriya interaction.

### 5.2.3.2 Current-Induced Domain Walls and Skyrmions Motion

Racetrack memory-based on domain wall or skyrmion motion is an attractive field for future magnetic memories. Synchrotron radiation-based techniques such as PEEM and STXM or even X-ray magnetic resonant scattering (XMRS) have allowed to study the fine structures of magnetic textures such as chiral domain walls [58], vortex [59] and skyrmions [60, 61], see, for example, X. Cheng et al. for a review [62]. A key point is also to optimize the motion velocity and determine the factors that could
<figure><img src="image_85.png" /><figcaption>**Fig. 5.18** (left panel) XMCD images of Co and NiFe electrodes of a NiFe/Cu/Co spin valves recorded at the Co $L_3$ edge **a** and Fe $L_3$ edge **b**, **c**. Different magnetic domains are observed depending on the parallel **b** or antiparallel **c** magnetic configuration (right panel). **d** Sketch of the Pt/Co/Ta stripe deposited on a $Si_3N_4$ membrane. **e** STXM images recorded after current pulses showing that three of the four skyrmions move (orange, yellow and red circles) after the current pulse. The fourth skyrmion (white circle) is pinned. **f** Skyrmion velocity as a function of current density injected for Pt/Co/Ta and Pt/CoFeB/MgO devices. from [63] (left panel) with permission (Copyright 2010, American Physical Society) and from [29] (right panel) with permission (Copyright 2016, Nature Publishing Group)</figcaption></figure>

limit this velocity. X-ray imaging is a powerful tool to investigate magnetic object motion induced by a current. In the following, we show two examples of domain wall and skyrmion motions probed by XMCD-PEEM and STXM.

Large domain wall velocities (~600 m s⁻¹) have been measured in NiFe/Cu/Co spin valves [63]. Unexpectedly, the domain wall motion is altered when longer electrical pulses or higher current densities are applied [63]. By performing XMCD-PEEM measurements, it has been shown that the dipolar interaction between the NiFe and Co electrodes was a source of domain wall pinning. Thanks to the chemical selectivity of X-ray photoemission, magnetic configuration of the NiFe and Co electrodes can be probed by recording images at the Fe and Co $L_3$ edges [see Fig. 5.18a, c]. In the parallel magnetic configuration, the stray field of the Co domain wall locally reverses the magnetization in the NiFe layer, leading to the three domain walls [white circle in Fig. 5.18b]. On the other hand, in the antiparallel magnetic configuration, the magnetic flux closes naturally and a single domain wall is formed [Fig. 5.18c]. This magnetic imaging of the Co and NiFe electrodes demonstrates that the stray field prevents domain wall motion across the corners in the NiFe layer.

An important property of the racetrack memories is that the magnetic objects (domain walls or skyrmions) move all together. In Fig. 5.18(d-f), we show that STXM measurements can be used to probe current-driven skyrmion motion in a Pt/Co/Ta stripe as shown by K. Woo *et al.* [29]. Each image is recorded after injection of current pulses. Three of the four skyrmions (red, yellow and orange circles) move forward and backward depending on the current polarity [see Fig. 5.18e]. Note that the fourth skyrmion (white circle) is not showing any motion under current injection