X-ray magnetic circular dichroism

The effect of magnetic X-ray dichroism was first observed for the Tb M_4,5 edge of Tb₃Fe₅O₁₂ at the Laboratoire pour l'Utilization du Rayonnement Electromagnetique (LURE) at Université Paris-Sud in France (van der Laan et al., 1986) only one year after its prediction (Thole, van der Laan & Sawatzky, 1985). Sum rules relate the integrated intensities of the X-ray magnetic circular-dichroism (XMCD) spectrum to the ground-state expectation values of the orbital and spin magnetic moments (Thole et al., 1992; Carra, Thole et al., 1993). These rules offer a theory-independent analysis which has transformed XMCD into a powerful standard technique, providing invaluable insight into the element-specific microscopic origin of anisotropic magnetic properties, such as the magnetocrystalline effect, the easy direction of magnetization, magnetostriction, coercivity and magnetic hysteresis.

Dichroism is defined as the polarization dependence of the light absorption by a material. Circular dichroism arises when either space-inversion or time-reversal symmetry is broken in the material. Noncentrosymmetric symmetry allows natural circular dichroism (NCD). In this chapter, we only treat X-ray magnetic circular dichroism (XMCD, also known as CMXD in the older literature) originating from time-reversal symmetry breaking by a net magnetic field.

The XMCD is given as the difference between the two X-ray absorption (XA) signals with the light helicity vector parallel and antiparallel to the net magnetic field. XMCD resembles magneto-optical phenomena in the visible region, such as the Faraday effect. However, since XMCD arises from the excitation of a core electron into an unoccupied valence state, the technique is element-specific. The XMCD originates from electric dipole (or quadrupole) transitions, in which the spin is conserved. It is not caused by magnetic dipole transitions, which are notoriously weak. The sensitivity to the magnetic moment comes from the combined action of the Pauli principle for the valence band and the spin–orbit interaction, which is usually large for the core level.

Because of the optical selection rules, the dipole transitions from the ground state can only reach a limited subset of final states, which means that XA provides a fingerprint of the specific ground state. This makes it a local probe with a high sensitivity to the crystal field, spin–orbit interaction, site symmetry and spin configuration of the valence states (van der Laan & Thole, 1991). The element specificity of XMCD can be used to distinguish between the magnetic contributions from different layers in multilayers or capped particles composed of different elements. It also permits the generation of spatially and magnetically resolved elemental maps (van der Laan & Figueroa, 2014). XMCD offers a means to separate different sites of the same element, especially in ferrimagnets such as ferrites and garnets, where the distinction between different iron sites is facilitated by their antiferromagnetic alignment (Pattrick et al., 2002).

The various absorption edges of interest can show rather different characteristics. It is useful to distinguish between the soft and hard X-ray regions, which are separated at around 2–3 keV.

Although the absorption coefficients for right- and left-circularly polarized X-rays are measured in the same way as for standard XA, the linearity of the detected signals is important to acquire an undistorted difference spectrum. For hard X-rays the common detection modes are either transmission or fluorescence yield (FY), depending on the relative concentrations of the absorbing atoms. Only transmission measurements can give absolute absorption cross sections. Unfortunately, transmission is often not practical for the soft X-ray region due to the high absorbance of the sample. However, the decay of the created core holes after X-ray absorption results in an avalanche of electrons, photons and ions emitted from the sample surface, all of which can be used to measure samples of arbitrary thickness in vacuum.

We will group the absorption edges according to their similar behaviour. Actually, XMCD has also been reported for non­magnetic elements, such as copper and gold, due to hybridization with a magnetic atom, but here we restrict ourselves to the edges of practical interest. Although the 4d and 5d metals are a growing field of interest, so far fewer measurements have been reported.

4.1. Soft X-ray region

4.1.1. 3d metal L_2,3, rare-earth M_4,5 and actinide N_4,5, M_4,5 edges

The strong absorption of the 3d metal L_2,3 (2p → 3d at 0.4–1 keV), rare-earth M_4,5 (3d → 4f at 0.8–1.5 keV) and actinide N_4,5 (4d → 5f at ∼750 eV for uranium) and M_4,5 (3d → 5f at ∼3.5 keV for uranium) edges is due to dipole-allowed transitions directly into the magnetic valence state. The strong dichroism arises from the large spin–orbit splitting of the core level. The instrumental resolution in the soft X-ray region has a similar width as the core-hole lifetime broadening (a few hundred meV), which allows us to resolve the detailed multiplet structure and charge-transfer satellites.

The XMCD is depicted schematically by a two-step model in Fig. 1(a). In the first step, the absorption of a right- or left-circularly polarized photon (μ₊ or μ₋, respectively) excites a spin-polarized electron from the spin–orbit split 2p level (Fano effect). For the 2p_3/2 level μ₊ and μ₋ give 62.5% and 37.5% spin-up electrons, respectively, while for the 2p_1/2 level μ₊ and μ₋ give 25% and 75% spin-up electrons, respectively. In the second step, the spin-polarized electron has to find a place in the unoccupied 3d band, obeying the Pauli exclusion theorem, which gives a dichroism when the 3d states are spin-polarized.

Figure 1 XMCD. (a) Schematic picture of the two-step model. Absorption of left (μ−) and right (μ+) circularly polarized X-rays exciting an electron from the spin–orbit split 2p core level (Fano effect) into the empty 3d band of a magnetic material (Pauli exclusion principle). (b) Top: XA at the Fe L2,3 edge for left- and right-circular polarization, together with the difference spectrum: the XMCD. Bottom: the integrated signals p, q and r over the L2,3 XMCD and XA which are required for the sum-rule analysis.

In localized many-electron systems, the excitations 3dⁿ → 2p⁵3dⁿ⁺¹ for transition metals (or 4fⁿ → d⁹fⁿ⁺¹ for f metals) are self-screening, i.e. the additional 3d electron effectively screens the core-hole potential, strongly reducing the charge transfer of valence electrons. The spectra can be adequately modelled using multiplet calculations, where the initial- and final-state configurations are calculated in intermediate coupling, where spin–orbit and electrostatic interactions are on an equal footing. Ligand-field and charge-transfer interactions can be included in these calculations. The input parameters for the spin–orbit inter­action and 2p–3d and 3d–3d Coulomb and exchange inter­actions are obtained from atomic Hartree–Fock (HF) theory (Cowan, 1981). The Slater parameters are typically scaled to 70–80% to account for configuration interaction and electronic screening. Calculated multiplet spectra for XMCD across the full row of elements are available from the literature for 3d metal L_2,3 edges (van der Laan & Thole, 1991), rare-earth M_4,5 edges (Thole, van der Laan, Fuggle et al., 1985; Goedkoop et al., 1988) and actinide M_4,5 and N_4,5 edges (Ogasawara et al., 1991; van der Laan & Thole, 1996).

4.1.2. Giant resonances

Although the absorption edges of the shallow core levels have been less well studied, recently they have regained interest as they became assessable by the high-harmonic generation of soft X-rays by laser light. The 3d metal M_2,3 (3p → 3d at 30–125 eV), rare-earth N_4,5 (4d → 4f at 100–175 eV) and actinide O_4,5 (5d → 5f at ∼100 eV for uranium) transitions have much in common. (i) The core spin–orbit interaction is smaller than the core–valence electrostatic interactions. (ii) The core and valence shell have the same principal quantum number, giving a large overlap between their wavefunctions, and hence a large radial matrix element in the transition probability, as well as a short core-hole lifetime (a few eV line-width broadening).

The electric dipole selection rules are ΔJ = J′ − J = −1, 0, 1, where J′ = J = 0 is forbidden. In an LS coupled basis there are additional selection rules: ΔS = S′ − S = 0 and ΔL = L′ − L = −1, 0, 1 with L′ = L = 0 forbidden. As an example, Gd N_4,5 is shown in Fig. 2(a) (Starke et al., 1997). In the limit of vanishing core spin–orbit coupling, only the three states (J′ = 5/2, 7/2, 9/2) are dipole-allowed from the ⁸S_7/2 ground state of gadolinium. These absorption peaks show an asymmetric Fano line shape due to the interference between the excitation-decay process 4fⁿ + ℏω → 4d⁹4fⁿ⁺¹ → 4fⁿ⁻¹ɛ and the direct photoemission channel 4fⁿ + ℏω → 4fⁿ⁻¹ɛ, where ɛ represents a continuum state.

Figure 2 (a) Gd N4,5 absorption spectra: (i) spectra measured for parallel and antiparallel orientations of photon helicity and magnetization and (ii) experimental and (iii) calculated XMCD spectra (μ+ − μ−) (Starke et al., 1997). (b) Relative contributions to the Fe 2p XMCD spectrum from the different ground-state moments for metallic iron. The blue spectra fulfil the sum rules for orbital and spin magnetic moments. The bottom spectrum gives the sum over all contributions and resembles the experimental Fe L2,3 XMCD spectrum. For further details, see van der Laan (1997).

Turning on the small 4d spin–orbit coupling, additional transitions become allowed to states with higher L, which have a lower energy (Hund's rule). When their energy falls below the continuum onset these transitions appear as sharp peaks in the pre-edge region.

Here, we will establish the relationship between the XMCD signal and the magnetic moment of the sample. The interaction Hamiltonian for an atom with angular quantum numbers L, S and J in a magnetic field B is = = , where g = 2.0023 and g_J is the Landé splitting factor. The Zeeman field lifts the (2J + 1) degeneracy of the J level, splitting it into sublevels M = −J, −J + 1, …, J. An electron in a sublevel M has a magnetic energy E_M = μ_Bg_JMB_z, where the Bohr magneton, μ_B = eℏ/(2mc), is a positive number, so that in a field B_z the lowest ground state has M_L = −L, M_S = −S and M = −J.

The magnetic moment is where the expectation value of the total angular moment (in units μ_B/ℏ, which are usually omitted) is negative for the moments aligned along the field, since the negative M levels are filled first. having the opposite sign to the angular moment arises from the classical definition of the magnetic moment caused by an orbiting electron with negative charge. Hence, has the same sign as B_z.

The electric dipole transition probability is given by the Wigner–Eckart theorem as (omitting the radial part) where the initial electronic state |JM〉 is the ket (in-state) and the final state |J′M′〉 is the bra (out-state); therefore, the left component in the 3j-symbol is the conjugate . The 3j-symbol gives the angular momentum selection rules for XMCD: It is usual to define the XMCD signal, Δμ, as the difference between antiparallel and parallel orientations of the sample magnetization B and the incident photon helicity, where the subscript in μ_± represents the sign of q = ±1. Because of time-reversal symmetry, Δμ is the same if both the field and helicity are reversed.

The definition of equation (5) means that, for example, for ferromagnetic iron, copper and nickel the L₃ peak is negative (see Fig. 1). For instance, consider the transition Ni 3d⁹ → 2p⁵3d¹⁰ for a spherical atom in a field B_z. At T = 0 K the electronic configuration 3d⁹ has ground state ²D_5/2 (M = −5/2). The 2p core hole in the final state has |M′| ≤ 3/2. For the transition |M = −5/2〉 → |M′ = −3/2〉 only q = M′ − M = +1 is allowed. Using the definition in equation (5), we indeed obtain Δμ = μ₋(B) − μ₊(B) < 0. At finite temperature, higher M levels will also be populated, reducing the XMCD signal. However, the opposite sign of the XMCD is only obtained for a population inversion of the magnetic levels in the ground state (Thole, van der Laan & Sawatzky, 1985).

We will present the sum rules from the general perspective of angular momentum transfer from the photon to a core electron which is excited to a partially filled valence state in a multi-electronic configuration. By summing over the transitions from all levels of the core hole its properties drop out and the total intensity becomes proportional to the spin and orbital moments of the ground state.

For a magnetic material, the electronic shell l is characterized by a complete set of LS-coupled multipole moments 〈w^xyz〉, in which the orbital moment x and the spin moment y are coupled to a total moment z. Moments with even x describe the shape of the charge distribution and those with odd x describe the orbital motion. The tensors for the d and f shells up to z = 2, expressed as standard tensors, are given in Table 1. We furthermore define coupled moments over the hole state, denoted by an underscore, where = −〈w^xyz〉, except that = 4l + 2.

Table 1 Relations between LS-coupled tensor operators wxyz and standard ground-state operators, , , = , , and Operator name wxyz d shell f shell Number operator w000 = n n n Spin–orbit coupling l · s Spin moment −2Sz −2Sz Orbital moment Magnetic dipole term −3Tz Quadrupole moment Pzz Rzz

We define the fundamental spectra I^z as linear combinations over the polarized I_q spectra, where I⁰ = I₋₁ + I₀ + I₁, I¹ = I₋₁ − I₁ and I² = I₋₁ − 2I₀ + I₁ are the isotropic spectrum, XMCD and X-ray magnetic linear dichroism (XMLD), respectively (q = 0 corresponds to linear polarization along the quantization axis). These spectra will depend on the ground-state tensors . Fig. 2(b) shows the contributions from each to the Fe L_2,3 XMCD spectrum of metallic iron, which were calculated by assuming an independent particle model for the 3d states and a core level j_c split by an effective exchange interaction (van der Laan, 1997). It is seen that for each tensor the integrated intensity of the XMCD vanishes, except for . The separate integrals over the L₃ and L₂ edges vanish for all tensors expect those with z = 1. Nevertheless, the other tensors still show an asymmetric line shape for each edge. Summing over all contributions gives a spectrum that resembles the iron XMCD in Fig. 1.

The above result can be generalized by formulating the sum rules. For an electron excited from a core level with orbital, spin and angular quantum numbers c, ½, j_c to a valence shell l = c + 1, the intensity of the core levels can be written as (Carra, König et al., 1993; van der Laan, 1999) where [z] is shorthand for (2z + 1). Taking the reduced matrix elements to be equal and omitting them in the following, the integrated signals of the sum, ρ^z, and weighted difference, δ^z, over the two edges are related to spin-independent (y = 0) and spin-dependent (y = 1) ground-state moments, respectively: From this general formulation we obtain the specific sum rules such as, for p → d,where ρ⁰ = ρ₁ + ρ₀ + ρ₋₁ and ρ¹ = ρ₁ − ρ₋₁ are the integrated intensities of the isotropic spectrum and XMCD over the L₃ and L₂ edges. The right-hand sides are obtained using the standard operators for the d shell given in Table 1. It is seen that the ratio ρ¹/δ¹ is independent of n_h.

By a simple change in notation (Chen et al., 1995), namely m_L = −〈L_z〉, = −2〈S_z,eff〉 = −2〈S_z〉 − 7〈T_z〉, p = , q = and r = , the sum rules for p → d can also be written as

To apply these sum rules to the experimental data, we take the integrals over the absorption edges after the removal of a background due to transitions into higher unoccupied and continuum states. This procedure is shown in Fig. 1(b), where for both L₃ and L₂ edges a step function, convoluted by a Voigt line shape, is subtracted from the spectrum. The integrals over the XA and XMCD give the values of p, q and r that are used in equation (12). In the case of less pronounced absorption peaks a reference spectrum of a similar element with a well known number of holes should be subtracted. For instance, a relative comparison between the XA of gold and platinum allows us to separate the unoccupied platinum 5d states (or the XA of silver to separate the palladium 4d states; van der Laan & Figueroa, 2014).

The spin sum rule requires a separate integration over the L₃ and L₂ edges. Therefore, this rule is only strictly valid if j_c is a good quantum number, which is fulfilled when the core spin–orbit interaction is much larger than the core–valence interaction. This holds reasonably well for the Fe, Co and Ni L_2,3 spectra. However, a correction factor is needed for vanadium, chromium and manganese, which not only depends on the element but also on the 3d (de)localization, crystal field and hybridization (Thole & van der Laan, 1988). The rare-earth M_4,5 spectra also require a large correction factor.

Theoretical limitations for the sum rule are the assumption of constant and spin-independent radial matrix elements, the need to estimate the number of holes, the possible mixing of dipole and quadrupole moments and the contribution from both l = c ± 1 channels. Experimental errors can occur due to saturation effects, the choice of integration limits, using the sum spectra, (I₋₁ + I₁), instead of the isotropic spectrum for normalization or using the asymmetry, (I₋₁ − I₁)/(I₋₁ + I₁), which depends on the linear dichroism, instead of the XMCD.

XMCD offers the unique possibility of element-specific magnetic hysteresis measurements, which allow the separation of the contributions from different chemical species. This consists of measuring the magnetization M using the XMCD at a fixed photon energy as a function of the applied magnetic field H. The results can be converted into an Arrott plot, which shows the isothermal variation of M² as a function of H/M. By comparing Arrott plots measured at different temperatures, an accurate estimation of the Curie temperature can be obtained (Baker et al., 2015).

The TEY signal can be sensitive to the applied magnetic field, which can be verified and corrected by repeating the hysteresis measurement at an off-resonance energy. Good results are also obtained with FY detection, if saturation effects are taken for granted.

XA, and hence XMCD, acts as the initial step in coherent second-order processes such as resonant photoemission (RESPES) and resonant inelastic X-ray scattering (RIXS). Monitoring the decay processes in the second step gives information about the de-excitation of the intermediate core-hole state reached from the ground state. This can provide an unambiguous assignment of the different configurations in mixed-valence and hybridized compounds. In itinerant systems, such as the 3d transition metals, the femtosecond time scale of the core-hole clock can be used to monitor the electronic screening processes. The XMCD in RIXS shows that the 2p core hole in metallic iron and cobalt is screened within the ∼1 fs lifetime, whereas in metallic nickel, which has a narrow 3d band, the core hole is only incompletely screened.

In localized systems the X-ray transition is faster than the time it takes for the valence electrons to rearrange. For example, in the Fe L_2,3 XMCD of magnetite (Fe₃O₄) the Fe²⁺ and Fe³⁺ cations in the octahedral sites are perceived as `frozen', allowing their relative occupations to be measured (Pattrick et al., 2002). This is in contrast to the much slower time scale of Mössbauer spectroscopy, which detects an average valence of +2.5 for the octahedral sites.

On the picosecond time scale of the precessional motion of the spin, the magnetization dynamics can be recorded using X-ray-detected ferromagnetic resonance (XFMR), where the XMCD effect is used as a stroboscopic probe to measure the precessing spins (Marcham et al., 2013). By exploiting the time structure of the pulsed synchrotron radiation from the storage ring, the relative phase of precession in the individual magnetic layers of a multilayer stack or spin valve can be determined (van der Laan et al., 2017).

The advent of X-ray free-electron lasers enables the study of the element-specific magnetization dynamics on a femtosecond time scale using XMCD and XMLD.

XMCD offers large advantages, such as element specificity, spin and orbital moment analysis, magnetic anisotropy determination, surface sensitivity and time-domain resolution. The XMCD effect has found widespread application in magnetic imaging techniques, such as PEEM using TEY, Fourier transform holography (FTH) using X-ray scattering, or TXM with spatial resolutions of typically tens of nanometres.

XMCD has become one of the most versatile techniques to study the electronic and magnetic structure of diverse materials, ranging from nanoparticles, molecular magnets, organometallic complexes and dilute magnetic semiconductors to doped topological insulators.

Time-resolved XMCD with femtosecond laser excitation can be used to study magnetic switching, giving information on the driving force behind femtosecond spin–lattice relaxation.