In XDMR, not only the frequency of the microwave pump field has to be adjusted in order to satisfy the usual conditions of magnetic resonance, but also the energy of the monochromatic, circularly polarized X-ray photons has to be carefully tuned in order to maximize the XMCD probe signal. This can happen only in the close vicinity of one of the multiple absorption edges which characterizes the photoionization of the core shells for a given absorbing element. In this respect, XDMR can be seen as an unusual double resonance experiment since it combines microwave Zeeman spectroscopy with an atomic, core level spectroscopy that is inherently element-selective. Recall that the X-ray absorption K-edge refers to the photoionization of a 1s deep core level, L_2,3-edges being similarly associated with the photoionization of the spin-orbit split 2p core levels.

Edge-selectivity of XDMR measurements is a further advantage linked to the conservation of angular momentum in the photoionization process of deep atomic core levels, e.g., the 1s electrons of a K-shell or the 2p electrons of a L-shell. In the so-called two-step model [26], the angular momentum carried by a circularly polarized X-ray photon (+ħ for a right-handed circular polarization, −ħ for a left-handed circular polarization) is transferred to the excited photoelectron in a way that primarily depends on spin-orbit coupling in the excited core level. For L_2,3 absorption edges that are split by spin-orbit, the Fano effect implies that part of the photon angular momentum is converted into spin via spin-orbit coupling (ℓ + s at the L₃ edge; ℓ – s at the L₂ edge). No such conversion is possible at a K-edge (or L₁-edge) due to the absence of spin-orbit coupling in the core states and the photon angular momentum is entirely converted into ±ħ orbital moments. The magnetic properties of the sample drive the second step: XMCD spectra simply reflect the difference in the density of final states that are allowed by the electric dipole (E1) or electric quadrupole (E2) selection rules owing to the symmetry of the initial core state. It immediately appears that an XMCD signal measured at a K-edge can only be assigned to the orbital polarization of the valence band. The interpretation of the XMCD spectra recorded at spin-orbit split edges is more complicated since one has to disentangle the respective contributions of spin and orbital polarizations. By summing up all integrated dichroisms of electrons originating from conjugated sub-levels, one would again measure the orbital polarization of the final states; in contrast the difference in the integrated dichroic intensities measured at the spin-orbit split L₃ and L₂ edges can only be caused by a spin imbalance in the empty states because the orbital momentum transferred to the photoelectron has implicitly the same sign at both edges. This is all the physical content of what is known as the magneto-optical sum rules for XMCD [1,2,26].

In XDMR, it is more appropriate to use a differential formulation of the XMCD sum rules as proposed by Strange [27] and others [28,29]. Ignoring first the contribution of electric quadrupole (E2) transitions, one may write for a K-edge [7,10] :

in which [Δσ]_K is the difference in the absorption cross-sections of left- and right-circularly polarized X-rays in the vicinity of the K-edge, C_p being a constant factor. Note that such a differential formulation refers to a fixed energy of the photoelectron: ΔE = E_RX + E₀ – E_F in which E_RX, E₀ and E_F respectively denote the energy of the X-ray photons, the binding energy and the reference Fermi level. Whereas 〈L_z 〉_p is the expectation value of the orbital angular momentum operator integrated over all excited states featuring p-type symmetry, 〈ℓ_z 〉_p defines the orbital polarization of p-projected densities of states (DOS) at energy ΔE. Taking into account the weaker electric quadrupole (E2) transitions simply results in mixing final states associated with atomic orbitals of different symmetry [7,10]:

One could easily extend this result to molecular complexes where the final states are described with molecular orbitals, e.g., when the absorbing atom is in a tetrahedral ligand field. In all cases, however, K-edge XDMR experiments will refer to the forced precession of some energy-dependent orbital magnetization component proportional to 〈ℓ_z〉. XMCD cross-sections at L_2,3-edges can similarly be related to differential operators involving spin and orbital magnetization DOS [7]:

in which N_b ≃ 2 is the statistical branching ratio. Two distinct operators (s_z, t_z) are now needed to describe the spin dynamics. Recall that 〈T_z〉 and 〈t_z〉 reflect (to the lowest order) the asphericity of the spin magnetization due either to spin-orbit interactions or anisotropic charge distributions [26,30,31]. Though T_z does not contribute to the Zeeman free energy, it contributes to the magnetocrystalline anisotropy free energy as pointed out by Van der Laan [31]. As long as the spin-orbit splitting remains small with respect to the crystal field splitting, T ≃ −2/7 Q · u_S in which u_S denotes the unit vector along the direction of the spin moment, whereas Q = L² – 1/3L² is the electric quadrupole operator which is a traceless spherical tensor of rank 2. In cubic crystals, Q vanishes so-that 〈T_z〉 can reliably be neglected, especially for 3d transition elements. Anyhow, since the magnetocrystalline anisotropy free energy of ferrimagnetic iron garnets is fairly small, one can hardly expect a large contribution of 〈T_z〉 and 〈t_z〉 in these magnetic systems.

As opposed to electron magnetic resonance experiments which can be pumped with poorly energetic photons (E ≤ 10 cm⁻¹), XMCD measurements require highly energetic photons with wave numbers in excess of 10⁷ cm⁻¹. Typically, the transition-time towards X-ray excited states is considerably shorter than the time-scale of spin-orbit interactions and of collective magnetic excitations involving exchange and dipole-dipole interactions. XMCD thus delivers a snapshot picture of the precession dynamics of spin and orbital magnetization components in excited states in the immediate surrounding of the X-ray absorbing atom.

This has some interesting implications illustrated with Figure 1A in which we have reproduced ab initio simulations of the spin-polarized DOS of yttrium iron garnet (YIG) in the ferrimagnetic state. The latter simulations carried out with the PY-LMTO-LSDA code [32] show that the spin-polarized DOS have the opposite sign at the (24d) and (16a) crystal sites in which the iron ions have slightly distorted tetrahedral ( ) and octahedral ( ) coordination symmetry respectively. This is fully consistent with the antiferromagnetic coupling of the Fe ions in (24d) and (16a) sites [33]. Let us draw attention, however, onto the opposite sign of the spin polarized DOS in the filled states (ΔE < 0) and in the excited states (ΔE > 0) which belong to the same irreducible representation of groups or : this implies that the corresponding magnetization components should precess with the opposite helicity in the ground and excited states.

We have also reproduced in Figure 1B the XMCD spectra that can be deduced from the spin-polarized DOS of Figure 1A for both electric dipole (E1) and quadrupole (E2) transitions. The site-resolved XMCD spectra confirm the existence of a quite significant orbital magnetization component 〈ℓ_z〉 at the tetrahedral iron sites: this is because E1 transitions are allowed from the 1s core level to final states which belong to the same representations (b, e) of group as the 4p atomic orbitals. In contrast, transitions from the 1s core level to final states belonging to the a_g or e_g representations are allowed only by E2 transitions which have a much lower contribution to the XMCD cross-sections. In other terms, XDMR spectra recorded at the maximum intensity of the Fe K-edge XMCD spectrum will benefit from a strong site-selectivity favoring the Fe sites: this is a considerable advantage over FMR. Recall, however, that site-selectivity is lost if the XDMR spectra are collected at the L_2,3 edges of iron since 2p → 3d transitions are dipole allowed at both sites.

Let us associate a fictitious, local magnetization component M^(ℓ,s) to 〈ℓ_z〉 and 〈s_z〉 respectively. The precession of M^(ℓ,s) can indeed be described either by Equation 3 or a classical torque equation. Regarding ferromagnetically ordered materials, the Landau–Lifshitz–Gilbert or the Bloch–Bloembergen torque equations are most appropriate even though they are purely phenomenological. The Landau–Lifshitz–Gilbert equation has a rather simple formulation for infinite thin films, i.e., slabs with vanishing aspect ratio. Defining the local magnetization vectorMby its polar and azimuthal angles (θ, φ) and its norm M, one obtains the coupled equations [7]:

in which the magnetic free energy F encompasses the Zeeman term (F_Z), the magnetocrystalline anisotropy (F_A) as well as the demagnetizing (F_D) free energy responsible for the shape anisotropy. In equations 8 and 9, α_G classically denotes the Gilbert damping parameter, γ = gμ_B/ħ, whereas M_s is the length of the fictitious magnetization component at equilibrium. Since the norm of the magnetization vector is not invariant in the Bloch–Bloembergen model, a third equation is needed [7]:

in which ω_r refers to a longitudinal relaxation frequency traditionally related to the spin-lattice relaxation time T₁. It will become clear later on that the transverse relaxation time T₂ is affected by additional relaxation processes involving a redistribution of the absorbed energy among all degrees of freedom of the whole spin system.

As illustrated with Figure 2, XDMR spectra can be recorded in two distinct geometries:

For a ferromagnetic thin film with uniaxial anisotropy and perpendicular magnetization, the opening angle of precession θ₀ is a constant of motion that characterizes the precession dynamics: it can be determined by combining together XDMR and static XMCD cross-sections [7,10]. For XDMR measurements in LOD geometry:

whereas in the TRD geometry:

The Landau–Lifshitz–Gilbert equations lead to the following resonance equations for θ₀:

in which b_cp denotes the circular component of the microwave pump field and :

where B_u and B_A1 refer to the uniaxial demagnetizing field and the cylindrical component of the magnetocrystalline anisotropy respectively. On the other hand, Q₀ = −α_Gω_p in which the Gilbert’s damping factor α_G is of the order of 6×10⁻⁶ for high quality YIG films. Similarly, one would show that:

In the low microwave power limit, i.e., when cos θ₀ → 1, the resonance condition P₀ = 0 will converge towards the usual Lorentzian lineshape whereas φ₀ → π/2 if P₀ → 0. However, for pumping fields in excess of 100 mG or higher, Equation 13 yields a biquadratic equation in cos θ₀ with multivalued (unstable) solutions that characterizes the so-called foldover regime in which field-swept spectra exhibit a strongly hysteretic shape with a large foldover jump. This is very often the case in XDMR experiments carried out in LOD geometry because the pumping power has to be increased in order to get a chance to detect the weak XDMR signal which is only a second order effect.

Clearly, in TRD geometry, the oscillating signal Δσ_XDMR(k_RX ^⊥) varies linearly with the microwave pump field b_p, whereas, in LOD geometry, the steady state signal Δσ_XDMR(k_RX ^||) is proportional to the incident microwave power ∝ b_p ². At this stage, one should keep in mind that Δσ_XDMR as well as Δσ_XMCD are time-reversal odd observable properties which change their sign when the direction of B₀ is inverted, independently of the oscillating pump field. Indeed, the normalized ratios (11)–(12) are time-reversal even: regarding the TRD geometry, this implies that, under time-reversal, the precession helicity (as well as B₀) should be inverted in order to keep the product ωt invariant.

In magnetically ordered systems, the precession is driven by non-uniform, time-dependent fields h(r, t) which do not simply reduce to the external pump field. This causes a number of orthogonal eigen modes to be excited in which all spins precess with the same frequency (ω) but not with the same phase. In this respect, dipole-dipole interactions -which are responsible for the demagnetizing field- open a whole band of magnons and make the degeneration of the uniform mode possible [38]. Let us emphasize that orbital magnetization componentsM^(ℓ) directly experience dipole-dipole interactions.

It is preferable to reformulate the Landau–Lifshitz–Gilbert equations as a system of canonical Hamiltonian equations for spin wave amplitudes (c_k,c^*_k). Such a classical formulation strictly parallels the canonical transformations used by Holtstein and Primakoff in their quantum theory of spin-waves [37–39]. The equation of motion for each spin-wave amplitude c_k has the form [41]:

in which the relaxation frequency (ω_rk ) for mode k can be easily converted into the spin-wave full linewidth (ΔH_k) using the relationship: ω_rk = γΔH_k/2. Notice that the right-hand member of Equation 15 is expressed as a functional derivative of the Hamiltonian function . Usually, spin-wave excitations in ferromagnets are sufficiently small that can be expanded in a Taylor series in powers of c_k and c_k^*:

The first term in the right-hand side of Equation 16 is often denoted but it corresponds to the linear theory of spin waves: it yields the energy of non-interacting modes with the dispersion law : ω_k = [A_k² – |B_k|²]^1/2 in which:

where η_ex is the exchange stiffness constant. Moreover, θ_k and φ_k refer to the polar and azimuthal angles of the wavevector k in spherical coordinates, taking the z axis parallel to the direction of the equilibrium magnetization M_eq. Recall that a precession mode becomes elliptical as soon as |B_k| ≠ 0 [20].

The next terms refer to the perturbation induced by the microwave pump field. The first one ( ) arises from the Zeeman interaction with the transverse field component b_⊥:

which causes the resonant excitation of the uniform mode (k = 0) in FMR. The nonlinear term is responsible for the spin wave instability in parallel pumping [43]:

It causes the parametric excitation of a pair of spin waves satisfying the conservation laws: ω_p = ω₁+ω₂ and k₁ = −k₂ = k.

All information about the interaction of spin-waves between themselves is contained in the perturbation term = + with [39]:

In the quasi-particle language of second quantification, and describe three- and four-magnon interactions respectively. Three-magnon processes, in either the confluent or splitting modes, arise not only from long range dipole-dipole interactions as first recognized by Akhiezer [40], but also from short range pseudo-dipolar interactions [14]. Energy conservation implies that: ω₁ = ω₂ + ω₃ for a 3-magnon splitting process, whereas (ω₁ + ω₂) = (ω₃ + ω₄) in a 4-magnon scattering process. Even though refers to a higher order perturbation, the 4-magnon scattering process is the corner stone of all nonlinear theories of spin waves [41] because exchange may largely dominate over dipole-dipole interactions in this term [39]. Given that the exchange part is α (k₁ · k₂)², it immediately appears, however, that the exchange cross section is zero if either k₁ or k₂ is zero: in other terms, exchange cannot relax the uniform mode (k = 0). As pointed out by Keffer [39], this is consistent with the well known result (see for instance [42]) that the Heisenberg exchange Hamiltonian commutes with all vector components of the total spin operator: = ∑_i S_i so-that: [ , ] = [ , ] = [ , ] = 0. In particular, this implies that the Heisenberg exchange Hamiltonian should also commute with the longitudinal (z) component of the magnetization: [ , ] = 0.

There is no doubt that the local spin magnetization components M^(s) probed by XDMR are directly affected by collective excitations of exchange spin waves. Even though exchange interactions should have no direct effect on orbital magnetization components M^(ℓ), one should not forget that M^(ℓ) and M^(s) are dynamically coupled by spin-orbit interactions at a time-scale much shorter than the time-scale of precession. Let us insist, however, that it cannot be taken yet as firmly established that M^(ℓ) couple to exchange spin waves.

Let us make a distinction between two different processes: (i) spin-orbit and dipole-dipole interactions can be strong enough to let the local orbital magnetization components M^(ℓ) couple to spin waves; (ii) energy can be redistributed within additional degrees of freedoms involving orbital ordering (O_ord) in orbitally degenerate systems. In the first case we shall speak of pseudo-spin waves or pseudo-magnons. In the second, we have to deal with true orbital ordering waves or orbitons which can be best described using a formalism that was recently reviewed by Khaliullin [25].

The starting consideration is that both spin exchange and charge motion strongly depend on the orbital state in chemical bonding. One may easily admit that the interaction energy can hardly be optimized simultaneously for all bonds: this leads to peculiar frustrations, oscillations and quantum resonances among orbital bonds. Let us consider first the superexchange term of the Hamiltonian of a magnetic system which would be orbitally degenerate:

For simplicity, we deliberately neglected Hund’s coupling in Equation 25. Orbital ordering effects are described by the operators J_ij^γ andK_ij^γ. Their formulation depends on the crystal structure and is different for doubly degenerate (E or E_g) or triply degenerate states (T₂ or T_2g). The effects are most spectacular in the case of E_g-states in a cubic crystal in which:

whereas: K_ij^γ = 0. In Equation 26,J = 4t_h²/U, t_h and U denoting respectively the hopping integral and the on-site Coulomb repulsion in the Hubbard model for two-fold degenerate states E_g. The index γ of the operators τ^γ specifies the orientation of the bond r_ij relative to the cubic axes a, b and c. Following Kugel and Khomskii [23]:

where σ^x and σ^z are Pauli matrices. According to usual conventions, σ^z = 1 corresponds to orbital 3d_x2–y2, whereas σ^z = −1 refers to orbital 3d_3z2–r2. Equations 25–28 therefore let us expect a strong interplay between spin and orbital degrees of freedom.

Regarding triply degenerate states (e.g., T_2g) we shall again follow Kugel and Khomskii and replace the operators τ_i with linear combinations of quadratic terms: [(ℓ_z)_i² – 2/3], [(ℓ_x)_i² – (ℓ_y)_i²] and [(ℓ_xℓ_y)_i + (ℓ_yℓ_x)_i], in which ℓ_x, ℓ_y and ℓ_z are the components of a fictitious orbital angular momentum built up from the 3d_xz, 3d_yz and 3d_xy transition metal orbitals. The superexchange Hamiltonian is modulated by an operator which looks like the orbital quadrupole-quadrupole interaction ( ) between sites i and j, i.e., the higher-order terms in the Van Vleck anisotropic exchange interaction. Note that also appears in the theory of the Jahn–Teller effect.

Collective excitations of the orbital ordering degrees of freedom were first envisaged by Cyrot and Lyon-Caen [24] who pointed out the marked similarity between the structure of and that of the spin wave Hamiltonian for antiferromagnetic interactions. In order to solve the superexchange problem of Equation 25, Khaliullin [25] proposed to split in three terms:

in which δJ = J –〈J〉. Spin waves representations were systematically used for S_i in Equation 29 and, similarly, orbital waves were introduced for τ_i in Equation 30. From a mathematical point of view, the easiest strategy to take into account the interaction Hamiltonian of Equation 31 is by exploiting diagram methods. Restricting ourselves here to three-bosons interactions, one may anticipate that an (excited) magnon could couple to an intermediate state containing one magnon plus one orbiton, whereas an orbital excitation (orbiton) could decay by a splitting into two magnons.

Regarding magnetic systems with orbitally degenerate ground states, Khaliullin went to the following conclusions: (i) fluctuations are enhanced in both spin and orbit subspaces; (ii) given the strong magnetic anisotropy of bonds, quasi 1-D orbital order is favored; (iii) joint spin-orbital fluctuations should significantly lower the ground state as first predicted by Cyrot and Lyon-Caen [24].

With excellent insulator properties and no detectable magnetic disorder, yttrium iron garnet (YIG) has played a historical role in the promotion and understanding of FMR. It has a cubic structure (space group:Ia3̄d; group N°230), each unit cell consisting of eight formula units Y₃[Fe₂](Fe₃)O₁₂ [61]. This formulation emphasizes the role of the tetrahedral ( ) and octahedral ( ) iron sites. Below the magnetic ordering temperature (T_c ≃ 550 K), the two Fe sublattices get magnetized antiparallel one to each other, with an unbalanced magnetization (ca. 5 μ_B) in favor of the tetrahedral sites. The ferrimagnetic order is caused by a strong superexchange interaction between the two iron sublattices, the rather large Fe(16a)–O–Fe(24d) angles (126.6°) giving a clear indication that the wavefunctions of oxygen and iron have a substantial overlap. YIG is the generic term for a rich family of iron garnets in which yttrium can be substituted with rare earths in variable proportions.

In Section 3, we shall reproduce XDMR spectra collected essentially on two iron garnet films that were grown by liquid phase epitaxy on oriented gadolinium gallium garnet (GGG) substrates: 1 = Y₃Fe₅O₁₂ (YIG # 520); 2 = [Y_1.3La_0.47Lu_1.3]Fe_4.84O₁₂ (Y-La-LuIG). Note that in film 2, “diamagnetic” (¹S₀) rare earth cations (La³⁺, Lu³⁺) substitute for Y³⁺. In Section 3.4, we shall also briefly discuss XDMR spectra recorded with a thin, polished platelet of gadolinium iron garnet (GdIG) single crystal.

We like first to show that XDMR spectra of unprecedented quality can now be recorded in the transverse detection geometry (TRD) on YIG and related thin films, thanks to our superheterodyne detection scheme that relies on the powerful concepts of either bpsk or qpsk (bi- or quadrature phase shift keying) used in satellite telecommunications. A vector detection scheme also allows us to recover the phase information that is preserved in a TRD geometry. This is illustrated with Figure 4A in which the absorptive (χ″) and dispersive (χ′) XDMR components of film 1 can be identified with the real and imaginary parts of the vector detection scheme. A small (instrumentation dependent) phase-shift (ΔΦ ≃ 2°) was added in order to let χ′pass through zero when χ″is maximum as well as |XDMR|. In this experiment, the energy of the circularly polarized X-ray photons was tuned to the maximum of the Fe K-edge XMCD spectrum (E₁ = 7113.91 eV) and the film was rotated by β_Y ≃ 42⁰ in order to minimize the demagnetizing field anisotropy. We checked that the XDMR peak intensity varied linearly with the square root of the pumping power up to ca. 10 mW.

Arrows in Figure 4A point to weak satellite resonances assigned to either backward (BMSW) or forward (FMSW) magnetostatic spin waves [10]. This is a clear indication that, locally, the orbital magnetization components M^(ℓ) couple to non uniform magnetostatic spin waves through dipole-dipole interactions. Note that forward/backward MSW seem to have different phases. At this stage, one should realize that there is no chance to excite and detect standing waves resonances associated with magnetostatic modes unless there is a net transverse magnetization component interacting with both the microwave pump field and the circularly polarized X-rays: in the YIG thin film 1, this can be envisaged only for standing waves of rather low order and featuring an odd number of semiperiods [62]. In this respect, we already discussed elsewhere [10] the reasons which let us expect the relative amplitude of the forward/backward MSW satellites to be much weaker in XDMR than in conventional FMR spectra.

In Figure 4B, we compare the FMR and XDMR power spectral density (PSD) spectra that were recorded simultaneously under a pumping power of only 1 mW. It immediately appears that the two PSD spectra do not peak at the same resonance field whereas the intensity of the sharp BMSW modes are considerably more intense in the FMR spectrum. Also the XDMR linewidth is definitely broader. Unfortunately, by the time of these early experiments, the limited number of channels in our vector spectrum analyzer prevented us from simultaneously recording both the FMR and XDMR spectra in a fully coherent vector mode. This technical problem was recently solved but no beamtime was allocated as yet to further tests.

Long before we started the XDMR experiments reported below, a static XMCD study of film 2 had revealed the presence of large induced spin components 〈s_z〉 at the location of the “diamagnetic” trivalent cations La³⁺, Lu³⁺, Y³⁺ [33]. Technical arguments (e.g., circular polarization rates, X-ray fluorescence yields . . . ) convinced us that XDMR experiments carried out at the La L_2,3-edges would be less beamtime-consuming than similar experiments at the low-energy Y L_2,3-edges. Preliminary test experiments carried out on film 2 in longitudinal detection geometry (LOD) led to the rather unexpected result that, under perpendicular magnetization, the apparent precession cone angle of the orbital magnetization component M^(ℓ) measured at the Fe K-edge, i.e., θ₀^(ℓ) [Fe] = 13 – 19°, was much larger than the apparent precession cone angle of the spin magnetization component M^(s) measured at the La L_2,3-edges, i.e., θ₀^(s) [La] = 4.7°. Even more puzzling was the large precession cone angle θ₀^(ℓ) [Fe] measured for film 2, especially if we compare it to the opening angle (θ₀^(ℓ) [Fe] = 7.2°) measured for the YIG film 1 under identical experimental conditions. In contrast, θ₀^(s) [La] was found to be slightly smaller than the cone angle measured at the Y L-edges in film 1 (i.e., θ₀^(s) [Y] = 5.9°).

We compare in Figures 5A the XDMR and FMR PSD spectra recorded simultaneously with the Y-La-LuIG film 2 under high pumping power (630 mW). The XDMR spectrum was again recorded in the longitudinal detection geometry (k_RX^|| || B₀). The FMR and XDMR spectra both exhibit strong foldover distortions resulting into broad lineshapes (ΔH₀ ≥ 400 Oe). Most intriguing to us was the very sharp increase of the XDMR signal just before the foldover jump in the downfield scan, in a range where the FMR absorption spectrum seems to saturate. Even far from the foldover jump, the precession cone angle (θ₀^(ℓ) [Fe] ≃ 13°) is much too large to be realistic: this definitively ruled out the crude assumption that the length of the precessing moment (|M^(ℓ)|) was invariant and convinced us to envisage the parametric excitation of nonuniform modes.

From Equation 24, one would expect the annihilation of two uniform magnons and the creation of two degenerate magnons to shorten the length of the magnetization vector |M^(ℓ)| and to increase the transverse relaxation rate 1/T₂. Regarding Equation 23, one would guess that the four-magnon scattering process of Figure 3B should leave M_z^(ℓ) unaffected since two uniform magnons are replaced by two degenerate magnons, so that the total number of magnons is left unchanged. This, however, does not hold true if the life-times of uniform and degenerate magnons are different. Schlömann [64] was the first to point out that, in the four-wave interaction process of Figure 3B, the damping rate of the spin waves (+k, −k) was artificially decreased. Typically, he introduced an effective damping rate:

in which the amplitude of the uniform mode |b₀|² directly depends on the incident microwave power. Clearly, the higher the microwave power, the longer should be the lifetime of the pair of degenerate magnons. Equation 23 then predicts a rapid decrease of M_z^(ℓ), i.e., a sharp raise of m_z^(ℓ) exactly as observed in Figure 5A. We still need to understand why this effect is so spectacular with film 2 but does not show up with film 1, and why it does not affect in the same way the magnetization components probed at the iron or lanthanum sites. It is our interpretation that this has to do with a relaxation mechanism that selectively affect the orbital magnetization component 〈ℓ_z〉 at the iron site in the Y-La-LuIG film. Owing to the fact that the orbital magnetization components 〈ℓ_z〉 are heavily quenched at the rare earth sites [33], no significant relaxation anomaly can be detected at the La or Lu L-edges. This nicely illustrates the different nature of the relaxation mechanisms at different magnetic sites.

In the Kasuya–LeCraw process which is regarded as the dominant spin-lattice relaxation process in YIG [20,63], it is postulated that the confluent scattering of a small k degenerate magnon with a phonon q can produce an excited magnon of much higher energy: the latter may decay through a cascade of three-magnon splitting processes now allowed by energy conservation. Since the net result is an increase of the total number of magnons, Equation 23 let us expect M_z to decrease or m_z to increase: of course, it would be totally erroneous to assign this effect to an (improbable) increase of the precession cone angle of the uniform mode in Figure 2. We have sketched in Figure 5B. two modified Kasuya–LeCraw mechanisms in which a cascade of two consecutive 3-boson processes may involve one orbiton either in a de-excitation process (a) or in the excitation process (b).

Mechanism (5B.b) deserves more attention in a context where valence or orbital fluctuations may cause a dynamical replacement of Fe³⁺ cations in either tetrahedral or octahedral sites with short living Jahn–Teller cations (Fe⁴⁺, Fe²⁺) that had unquenched orbital moments. Such valence fluctuations could be initiated or enhanced by the presence of small amounts of impurities (e.g., Pb²⁺) introduced during the growth of the film by liquid phase epitaxy. In other terms, the ⁶S_5/2 configuration for iron may not be time-invariant. Recall that the effects induced by Fe⁴⁺, Fe²⁺ or other Jahn–Teller ions in doped YIG films have fed long debates that are not closed. In the case of film 2, strong perturbations stem from the large size of the La³⁺ cations and of the small size of the Lu³⁺ cations which allow lutetium to partly replace iron in the octahedral sites. The non-uniform distribution of the rare earth cations induces fluctuations in the lattice parameter, i.e., dynamical strains and stresses resulting in the excitation of magneto-elastic waves. This is supported by the much larger growth anisotropy of film 2 as compared to film 1. Nevertheless, it would be desirable to lay deeper foundations to the mechanisms of Figure 5B. In particular, we are interested in establishing the link between this mechanism and the slow longitudinal relaxation theory pioneered in the early sixties by Van Vleck and others [66,67].

The replacement of all Y³⁺ ions with Gd³⁺ (⁸S_7/2) in the dodecahedral (24c) sites results in a more severe perturbation than that caused by the pseudo-diamagnetic cations La³⁺ or Lu³⁺. Even though YIG and GdIG have identical crystal structures and nearly the same Curie temperatures (T_C = 551–556 K), their magnetic properties are fairly different due to the contribution of the weakly coupled Gd sublattice, the Gd spins getting fully ordered only below a low ordering temperature (T_B ≃ 67 K) [68]. Above T_B, the Gd magnetization can be described as a temperature-dependent Brillouin function for spin 7/2 in a field proportional to the net magnetization of the strongly coupled ferric ions. The most spectacular consequence is the existence of a compensation point, i.e., a temperature (T_cp = 286 K) at which the spontaneous magnetization of GdIG passes through zero [61]. Considerable changes in the FMR spectra take place at the compensation point. One of them is the inversion of the Larmor precession helicity on passing through the compensation point [69]. This prompted us to check whether this could be seen in the Fe K-edge XDMR spectra of GdIG.

We have reproduced in Figure 6 complex vector XDMR spectra of a thin disk of GdIG cut parallel to the (110) planes. These Fe K-edge XDMR spectra were recorded in transverse detection geometry (TRD). Since the normal to the disk was rotated by 45° with respect to the direction of B₀, the contribution of the uniaxial anisotropy was heavily reduced. In Figure 6A, we display the XDMR spectra measured at T ≃ 150 K, i.e., at an intermediate temperature well above the ordering temperature T_B but below the compensation temperature T_cp; in Figure 6B, the XDMR spectra of the same sample were measured at T = 450 K, i.e., at a temperature now well above T_cp. We kept strictly the same microwave pumping power (475 mW) for both experiments whereas the X-ray photon energy (E₁ ≃ 7114 eV) was left unchanged. The phase-shifts (ΔΦ) were determined according to the same criteria: the dispersive part of XDMR spectrum had to pass through zero at resonance whereas the absorptive part had to be positive over the whole resonance spectral range. Whereas ΔΦ varies from +15° to −177°, it immediately appears that something unexpected did happen to the absorptive part which caused heavy distortions of the modulus (|XDMR|) and of the PSD spectra. In contrast, no anomaly did appear in the FMR PSD spectrum which was recorded simultaneously at T = 450 K.

For sure, the strong anomaly revealed by Figure 6B cannot be explained by a simple inversion in the precession helicity of M^(ℓ) at the iron sites (24d). It is our interpretation that this anomaly is typical of a destructive interference between two resonant modes precessing in opposite senses as illustrated (for example) with Figure 6C. In other terms, the polarization of the precession is getting strongly elliptical. Complementary experiments revealed that this anomaly strongly depended on the pumping power and decreased on increasing the temperature. This looks like a possible signature—in the transverse detection geometry—of a 4-magnon scattering process in which the two scattered magnons with wavevectors +k and −k had strictly opposite helicities. We suspect the destructive interference to be particularly spectacular because the linewidth of the non uniform modes (±k) should be (again) much shorter than the linewidth of the degenerate uniform mode.

As far as the proposed interpretation holds true, the clue to the present problem is in the scattering amplitude which depends mostly on dipole-dipole interactions. It is not totally clear as yet why no similar anomaly did show up below the compensation temperature, i.e., in Figure 6A. It is hard to anticipate what exact role the Gd spins play in the intermediate temperature range above the ordering temperature T_B and below T_cp. Note that ⁵⁷Fe spin-echo NMR spectra recorded well below T_B have established that the Gd spins were coupled to the Fe spins in the tetrahedral (24d) sites, the exchange integral J_dc amounting to only ≃ 12% of the exchange integral J_ad describing the strong ferrimagnetic coupling of the iron sites [70,71]. Further work is in progress in order to compare the saturation properties of low temperature XDMR spectra recorded at the Fe K-edge and at the Gd L-edges.

For iron garnets, the shape anisotropy is very often regarded as the main cause for elliptical precession in FMR: this is typically the case of YIG thin films when tangentially magnetized. Oblique magnetization at 45° or near the magic angle (54.74°) is a simple way to decrease the anisotropy fields and to minimize the foldover lineshape distortions at high pumping power. Nevertheless, the two experiments reported below unambiguously show that the precession still remains elliptical. It should be kept in mind, however, that the precession ellipicity is a time-reversal even observable property [7]: this implies that it can be measured only through very weak non-linear effects.

At the ESRF, quite a few proposals were concerned with static XMCD studies on paramagnetic organometallic complexes. In practice, such experiments require low temperatures (T ≤ 20 K) and a high magnetic field (B₀ ≥ 5 T). We have reproduced in Figure 8A the cobalt K-edge X-ray Absorption Near Edge (XANES) and XMCD spectra of a powdered pellet of the meso-(5,10,15,20)-tetraphenylporphyrinato-Co(II) complex 3 = TPP:Co; we have also reproduced in Figure 8B the rhenium L-edges XANES and XMCD spectra of a cyanometalate complex of Re(IV) (4 = [Re(CN)₇](Bu₄N)₃). Recall that 4 is a building block for the chemical synthesis of heterobinuclear molecular magnets [76] or chemically switchable molecular magnet [77].

Interestingly, the XMCD spectra of complexes 3 and 4 are dominated by a strong contribution of the orbital magnetization component 〈ℓ_z〉 in the X-ray excited final states. Since the integral of the XMCD signal reproduced in Figure 8A does not average out to zero, the magneto-optical sum rules let us anticipate the presence of a significant amount of unquenched orbital moment 〈L_z〉 located at the cobalt sites. This is consistent with the commonly admitted electronic structure of 3 in which there is a close lying (orbital) triplet ⁴T₁ slightly above the Kramer’s ground state ⁴A₂. The existence of a weak orbital moment is also supported by the rather large anisotropy of the EPR spectra: Δg² = g_||²− g_||²≃ −7.8 [78]. This is at variance with the case of vanadyl porphyrin complexes since the XMCD signals measured at the V K-edge were dramatically weak and close to the instrumental detection limit although the corresponding XANES spectra did show quite strong pre-edge structures [79]. This may look puzzling given that the EPR spectra of vanadyl complexes are known to be very intense and exhibit fairly narrow lines with well resolved hyperfine structures due to the ⁵¹V_7/2 nuclei. Actually, the very weak anisotropy of the EPR spectra (Δg² ≃ −0.098) provides us with a clear indication that the orbital moment are heavily quenched in these compounds. Moreover, the weakness of the XMCD signal at the V K-edge could also reflect a major delocalization of the unpaired electron far away from the vanadium site.

It immediately appears from Figure 8B that the XMCD signatures measurement at the Re L_2,3 edges are considerably more intense than in the previous case. Moreover, the XMCD spectra displayed in Figure 8B are fairly unusual since the XMCD signal keeps the same sign at the L_2,3-edges: according to the magneto-optical sum rules, this is a typical signature of a large orbital moment 〈L_z〉 = 0.11 μ_B. This is also fully consistent with the strong anisotropy of the EPR spectra (Δg² ≃ 11) [76]. Much more surprising was the very weak effective spin contribution with 〈S_z〉 − 7/2〈T_z〉 ≃ ±5.10⁻⁴ μ_B. Quite the opposite situation was found when we measured the gadolinium L-edges XMCD spectra of cofacial porphyrinato-Gd(III) complexes in which no significant orbital moment was detected [79].

Unfortunately, the XDMR spectra of the paramagnetic complexes 3 and 4 cannot be recorded in the microwave X-band because the sensitivity of the XMCD probe is very poor at low bias field. A better option would be to record high-field XDMR spectra pumped at sub-THz frequencies. Even under such conditions, the Co K-edge XDMR signal may still be very weak. Nevertheless, one should not regard such a challenging experiment as hopeless because much larger precession cone angles could a priori be achieved in EPR: recall that pulsed EPR spectrometers require the magnetization to be rotated by 90° or even 180° in the rotating frame [80–83]. There is, unfortunately, a major difference with standard EPR which is that XDMR spectra will have to be recorded on pure or highly concentrated samples. Under such conditions, the X-ray detected electron paramagnetic resonance lines are expected to be very broad because the exchange narrowing effect should be considerably weaker than in FMR [84,85]. From a technical point of view, there is also the further handicap that the superheterodyne detection scheme discussed in Section 2.6 cannot be extended beyond F_p = 20 GHz: this implies that, as yet, X-ray detected EPR spectra could be recorded at sub-THz frequencies only in the longitudinal detection geometry (LOD) which suffers from a very poor sensitivity. This point justifies the need for a powerful pumping source such as a gyrotron [74,75].

What stimulates us to invest time and efforts in this challenging project is the hope that high-field X-ray detected EPR experiments should allow us to probe the precession dynamics of orbital magnetization components in paramagnetic species with a significant zero-field splitting (zfs), e.g., high spin complexes with integer spin that are EPR-silent at microwave pumping frequencies. Many examples can be found in a long list of Mn(III) complexes [86], e.g., porphyrinato-Mn(III) complexes for which preliminary XMCD measurements at the Mn K-edge already confirmed the existence of unquenched orbital moments. Keeping in mind that zfs is caused by spin-orbit interactions, there should be a correlation between a large zfs and the magnitude of M^(ℓ) in the X-ray excited final states. In other terms, spin-orbit is responsible for both the large zfs of Re(IV) complexes and the large contribution of 〈L_z〉 and 〈ℓ_z〉 in XMCD. Complexes involving 5d transition metal elements would thus appear as excellent candidates to demonstrate the precession of orbital components in X-ray detected EPR.

The intriguing case of the Van Vleck paramagnetism still deserves a few comments here: it can be best observed when the angular momentum J vanishes whereas L, S ≠ 0. If the ground state is singlet, the first order perturbation due to the Zeeman interaction vanishes but the second order term yields the Van Vleck susceptibility that becomes temperature independent at low temperatures (T < 20 K):

Whereas Curie’s paramagnetism reflects the alignment of permanent moments, Van Vleck’s paramagnetism refers to an electronic polarizability associated with induced moments. So far, van Vleck’s paramagnetism was observed mostly in crystals containing non-Kramers RE ions that (again) have an integer spin and a large zfs. What is required from the crystal field is to split the atomic ground multiplet so as to produce a singlet ground state. A typical example is Eu³⁺ with its ground state ⁷F₀ separated from the first excited state by ca. 300–400 cm⁻¹ and for which a slow longitudinal relaxation process was proposed by Van Vleck [67]. In NMR, there is also much interest in the Van Vleck paramagnetism of thulium crystals which had an exceptional potentiality regarding dynamic nuclear polarization (DNP-NMR) of ¹⁶⁹Tm. One of the best characterized crystals is thulium ethylsulphate, i.e., TmES = Tm(C₂H₅SO₄)₃.9H₂0 for which high-field EPR spectra have been reported at 1.2 K [87]. For the latter experiments, TmES was diluted in a diamagnetic crystal (LaES), the pumping frequency being as high as 1–1.5 THz. Unfortunately, we have not yet the technical capability to manage XDMR experiments under high pumping power at 1 THz while maintaining the sample temperature below 10 K.

Since no static XMCD signal can be measured on AFM materials with antiparallel ground states, it is tempting to conclude that there is no hope to detect any XDMR signal as well. The situation may not be as desperate if one looks at what happens under the conditions of antiferromagnetic resonance (AFMR). For simplicity, let us consider a crystal with two AFM ordered sublattices and uniaxial magnetic anisotropy directed along the c axis. As long as the antiparallel ground state is preserved, the combined action of a parallel bias field (B₀ || c) and a perpendicular pumping field (b_p ⊥ c) will excite two precession modes illustrated with Figures 9A,B and which satisfy the resonance condition:

in which B_E = |B_E1 | = |B_E2 | is the modulus of the exchange fields of the coupled sublattices (1, 2), whereas B_A = |B_A1 | = |B_A2 | similarly refers to the length of the corresponding anisotropy fields. Recall that Equation 44 is correct only at low temperatures (T ≪ T_N) and neglects all nonlinear terms. Further corrections taking into account the Van Vleck paramagnetism are possible when the temperature dependent susceptibility ratio χ_||/χ_⊥ is predetermined [88].

Anyhow, two distinct precession modes (ω⁺; ω ⁻) are excited which have the opposite helicity. Moreover, due to the existence of the anisotropy field (B_A = B_A1,2 ), the precession cone angles θ₁ and θ₂ are no longer identical for the two magnetization components M₁ and M₂. In the transverse detection geometry (TRD) illustrated with Figure 9, one should then measure a difference XDMR signal proportional to Δm_⊥^±= m₁^±+ m₂^± and which should oscillate at the resonant frequencies ω⁺ or ω ⁻ with the following lineshape [20]:

in which b_cp^(±) is the relevant circularly polarized component of the pump field, α_G being again a dimensionless (Gilbert) damping parameter. There is, however, the considerable handicap that the zero-field resonance frequency, i.e., ω_C = γB_C2 is most often expected to be in the sub-THz, if not in the far-infrared range because the exchange fields (B_E1, B_E2) are much stronger than the external bias field. Recall that there is, as yet, no X-ray detector that can measure a signal oscillating at sub-THz frequencies, our superheterodyne detection scheme being restricted to F_p ≤ 20 GHz. This is where high-field XDMR could circumvent this difficulty, assuming that the external field B₀ can be strong enough to shift the resonance frequency ω ⁻ down to the microwave range. Notice that, at the ESRF, the static bias field applied at the sample location could now be increased up to 17 T.

When the external field B₀ approaches the critical field B_C1 = [2B_EB_A − B_A² ]^1/2 < B_C2, the magnetic system undergoes a first order transition (spin-flop) resulting in the new situation sketched in Figure 9C and which is characterized by a noncollinear ground state [88]. It is well documented that, in the spin-flop phase, a new resonance mode associated with the total magnetization M = M₁ +M₂ will be pumped:

in which: B_E||= 2B_E − B_A and B_E⊥= 2B_E + B_A, so-that B_C1 = [B_AB_E||]^1/2 and B_C2 = [B_AB_E⊥]^1/2. For completeness, it should be mentioned that there exists also a soft mode (ω → 0) but the latter will be undetectable by XDMR since it cannot be excited by the external pump field [20].

Given that the precession cone angles θ₁ and θ₂ are different, one may question whether a (very weak) XDMR signal could be measured as well in the longitudinal detection geometry (LOD), i.e., in a regime where non-linear terms would no longer be neglected. Since no fast detector is needed in LOD geometry, this option could still be envisaged when B_C2 is very large, i.e., when too large magnetic fields would be required to shift ω ⁻ down into the microwave X-band. Unfortunately, one expects the XDMR signal measured in LOD geometry to be much smaller than in FMR because Δm_z^± ∝ [tan θ₀]² × (B_C/B_E)², in which B_C/B_E does not exceed, at best, a few percents. This would bring us very close to (or below) the detection limit.

For XDMR experiments, it may be preferable to select AFM crystals in which the exchange field B_E is not too strong, i.e crystals with a rather low Néel temperature (e.g., T_N < 77 K). Regarding AFMR, MnF₂ (T_N = 68 K) is a good example given that a very narrow linewidth (ΔB = 5 G) was measured at T = 4 K for the uniform precession mode (ω ⁻) excited by a microwave pump field (F_p = 23 GHz) in a bias field B₀ ≃ 8.5 T [89] so that magnetostatic satellite resonances could be perfectly resolved. Narrow AFMR lines (ΔB = 15 G) were also obtained with epitaxial films grown on MgF₂ substrates [90]. Note that the AFMR linewidth rapidly increases in the sub-THz range since the damping term in Equation 45 is ∝ ωB_E. Unfortunately, MnF₂ may not be the ideal candidate for Mn K-edge XDMR experiments due to the vanishing orbital moment 〈L_z》 of Mn²⁺ ions in a high spin (⁶S) state. Recall, however, that 〈ℓ_z〉 may still be finite in excited states even though 〈L_z〉 vanishes in the ground state. There is the further handicap that, in an octahedral crystal field with ⁶A_1g ground term, only electric quadrupole (E2) transitions contribute to the pre-edge structures assigned to final states involving magnetic 3d orbitals.

The case of KCuF₃ (T_N = 39 K) looks more attractive owing to the fact that this crystal is the archetype of orbitally ordered systems [91]. In this crystal, the AFM spin order has an easy plane of anisotropy with its normal parallel to the c axis. On the other hand, careful studies of the angular dependence of the EPR spectra of KCuF₃ in the paramagnetic phase revealed the existence of a strong contribution of the Dzyaloshinskii–Moriya antisymmetric exchange interaction:

with d_jk ⊥ c [92]. Moriya established that such an antisymmetric exchange interaction could be identified with a second order perturbation which was bilinear in the spin-orbit coupling and exchange interaction [93]. Then, the coupling field d_jk could be expressed in terms of transition orbital moments:

in which λ denotes the spin-orbit coupling factor, ΔE_j(k) are the energy separations between the ground state orbital levels g_j(k) and the relevant excited states e_j(k), J_j(k) being the effective superexchange constants at sites j and k respectively. Clearly, the coupling field vanishes if the two sites transform into each other by inversion symmetry. It is well documented that a large antisymmetric exchange interaction can cause a small canting effect of the AFM-ordered magnetization vectors resulting in a weak ferromagnetism [39]. Typically, for B₀ ⊥ c, KCuF₃ should behave just like a weak ferromagnet satisfying the resonance condition [20]:

Under such conditions, one may reasonably expect a (weak) XDMR signal to be detectable at low temperature in TRD geometry.

Among transition metal oxides for which the low frequency mode ω ⁻ could be excited at microwave frequencies, single crystals of ilmenites ( TiO₃, with = Fe,Co) could also be good candidates for XDMR: they had rather low Néel temperatures [88,94] and significant orbital moments 〈L_z〉 at the sites of the Jahn–Teller cations [95]. With a much higher exchange field combined to a low anisotropy field [88,96], the case of Cr₂O₃ (T_N = 308 K) looks less favorable even though we already produced definitive evidence of orbital magnetism in this crystal, at least in its magnetoelectric phase [97]. Although the spin-flop critical field of Cr₂O₃ is rather low (B_C1 ≃ 5.9 T), it cannot be taken yet for granted that Δm_⊥^(±)will be large enough to yield a XDMR signal easily detectable at microwave pumping frequencies.

Still very little is known regarding AFMR in crystals involving 5d transition elements and for which much stronger XDMR signals could be expected at L-edges. Unfortunately, neither single crystals nor epitaxial films of K₂ReBr₆ (T_N = 15 K [39]) or ReO₂ are easily available.