Takuji Nomura1 and Akihiro Koide2
1Condensed Matter Theory Group, 2 Magnetism Research Group,
Synchrotron Radiation Research Centerand emitted x-rays in the hard x-ray regime. Thus, XMCPE canbe a powerful new tool to investigate magnetization
distributions deep inside bulk ferromagnets.

Spectroscopic methods utilizing synchrotron radiation toanalyze magnetic properties in materials have attracted muchinterest. One of the most widely used methods is x-ray magnetic
circular dichroism (XMCD). XMCD is a kind of x-ray
absorption spectroscopy. In XMCD, the incident absorbed x-ray
is circularly polarized, carrying a finite angular momentum.
When a circularly polarized x-ray is absorbed by an electron in amaterial, the electron is excited to an unoccupied state above theFermi level (EF) with changing its angular momentum, where
the sum of the angular momenta of the electrons and absorbed
x-ray is conserved. In ferromagnets, electrons are polarized inspin angular momentum. In other words, electron unoccupancy
as well as electron occupancy differs between spin up and downstates. This difference leads to the difference in the electrontransition probability or equivalently absorption intensity, forright- or left-handed circularly polarized x-rays.
Recently, Inami succeeded in observing magnetic circularly
polarized emissions (XMCPE) at the K edge in magnetized iron[1]. This provides a new magnetically-sensitive spectroscopic
method utilizing synchrotron radiation. In Fig. 1, a typical
process of XMCPE at the transition-metal K edge is illustrated.
In XMCPE, an incident linearly polarized x-ray (i.e., carrying
no angular momentum) is absorbed, promoting a 1s electron to afree conduction state and leaving a hole on the inner K-shell (asshown on the left side of Fig.1). At this stage, since the incident
photon brings no angular momentum, the free electron takes thesame spin angular momentum as in the 1s state before promoted.
This means that the electron system still maintains the same
total angular momentum. Subsequently one of the 2p electrons
goes down to the unfilled 1s state, emitting a circularly
polarized x-ray. Since the total angular momentum of theelectron system and outgoing x-ray is conserved, the electronsystem must have a different angular momentum from that in
the initial state. In other words, the electronic system loses anequal amount of angular momentum to that carried away by thecircularly polarized outgoing photon. Thus, in the final state of
XMCPE, the electron system will be in an excited state with adifferent angular momentum from that of the initial ground state,
as in the final state of XMCD. In ferromagnets, such anexcitation will occur with a different probability, depending onright- or left-handed x-ray emission. In fact, for magnetized iron,Inami observed a clear difference in emission spectra for thetwo helicity states of emitted x-rays [1]. Here we should notethat in contrast to the XMCD technique that utilizes thedifference in intensity between the two helicity states ofabsorbed x-rays, the XMCPE technique utilizes the difference
between the two helicity states of emitted x-rays. Comparedwith XMCD, XMCPE has the following advantages: (i) theflipping ratio (i.e., the relative difference in intensity between
the two helicity states of [absorbed or emitted] x-rays) is large.
(ii) High bulk sensitivity can be achieved, by setting the incident

Fig. 1: Illustration of a typical XMCPE process. Initially, anincident photon of an incoming x-ray promotes a 1s electron
to a free-electron state, leaving a core hole in the 1s level (onthe left side). Subsequently, one of the 2p electrons fills the1s hole and a circularly polarized x-ray photon is emitted (onthe right side). Electron excitations in the 3d bands can also
be induced by the 2p-3d Coulomb interaction. The six 2p
levels are split into 2p1/2 doublets (L2) and 2p3/2 quartets (L3)
due to the spin-orbit coupling.
To our knowledge, any theoretical descriptions of XMCPE
have been restricted so far to those based on atomic multiplet
calculations. One of such analyses is given in [1]. However, it is
widely known that such atomic calculations are insufficient for
describing electron excitations in itinerant electron systems suchas bulk iron, while it is appropriate for describing only electrontransitions between strongly localized states. In this report, ournew theoretical framework applicable to itinerant ferromagnets
is explained, following [2].
In transition-metal ferromagnets, the d electrons near EF
dominantly contribute to the total magnetization. Therefore, first
we performed a first-principles band structure calculation. To
extract the electronic structure near EF from the result, weconstructed the so-called maximally localized Wannier functions,assuming the Fe-s, Fe-p and Fe-d characters (Here we should
not confuse these Fe-s and Fe-p states with the inner-shell 1s
and 2p states). This produces a 9-orbital tight-binding model.
Here we should note that the Fe-3d states form dispersive broad
bands (with about 10 eV bandwidth), for which isolated-atom
descriptions are invalid. Including the onsite Coulomb44J) among the five Fe-3d orbitals, weinteraction (U, U
have a Hubbard-type model with a realistic electronic bandstructure. To determine the ferromagnetic ground state, theHartree-Fock (mean-field) approximation is applied to thismodel. The calculated magnetic moment is 2.0 B per Fe site for
U = 2.2 eV, U = 1.32 eV and J = 0.44 eV. For the inner-shell 1s
and 2p states, we can take completely flat bands, since they stilltake the isolated atomic nature even in the solid state. In the K
XMCPE, the Coulomb interaction V2p-3d between the 2p and 3dstates plays an essential role, because the 2p states are affectedby the 3d spin polarization through V2p-3d. If we take no accountof the 2p-state polarization through V2p-3d, we have no differencein the 2p-1s emission intensity between right- and left-handed
circularly polarized x-rays. We control V2p-3d, setting theso-called Slater-Condon parameters Fn (n = 0, 2) and Gn (n = 1,3). To calculate x-ray emission intensities, we used aquantum-field-theoretical method based on the Keldysh Green sfunctions for non-equilibrium processes. As a consequence, wederived an analytic formula for XMCPE intensity (See [2] formathematical details). According to the result, the XMCPE
intensity is expressed by a sum of two contributions:.
originates from the 2ps relaxation
processes leaving no excitations on the 3d bands in the finalstate, where R/L denotes the emission of right-handed (R) oris aleft-handed (L) circularly polarized x-rays.
many-body correction due to excitations where an electron holepair remains on the broad 3d bands in the final state.
canbe effectively calculated even within the atomic descriptions,because excitations on the 3d states are not included in any case.
On the other hand, as we shall see below,
substantiallyyields a broad weight as a function of emitted x-ray energy,
reflecting the broadness of the Fe-3d bands. Such a broad
weight should be treated by itinerant electron bands, notby discrete atomic energy levels. Finally, we should note thatthe subtraction spectrum
vanishes completely forparamagnets but remains finite for magnetized ferromagnets.
Calculated XMCPE spectra for metallic iron are displayed inFigs. 2 and 3. In Fig. 2, the emission spectra
and theirsubtraction
neglecting the excitations on the Fe-3dbands are compared with experimental data. We can see asignificant deviation on the lower-energy side of each of the L2and L3 emission peaks. In contrast, in Fig. 3, the emission
spectraand their subtraction
including theexcitations on the Fe-3d bands are compared with experimental
data. The tail-like spectral weight on the low-energy side can bewell reproduced by the theoretical calculation. This comparison
clearly indicates that the electron excitations on the Fe-3d bands
snatch a significant amount of energy from the emitted x-ray.
Thus, we can interpret that the tail-like weight on thelow-energy side captures the broad excitations of itinerant Fe-3delectrons, whose broadness is of the order of the Fe-3dbandwidth.
In conclusion, we give possible perspectives to advance: (i) ourframework can be applied further to a wide variety offerromagnets. (ii) In our framework, we adopted a perturbative
approach taking only the lowest-order contributions in V2p-3d(Born approximation). One possible future research direction is
to investigate the effects of higher-order terms including
multiple electron-hole excitations and spin-wave (magnon)collective modes, although this will require more complicatedanalytical and much heavier numerical computations.

Fig. 2: Calculated curves of XMCPE spectra
as a function of emitted
and the subtraction
x-ray energy for right- (R) and left-handed (L)circularlypolarizations
arecomparedwithexperimental plots. Here, the excitations on the Fe-3dbands are neglected.

Fig. 3: Calculated curves of XMCPE spectra
andthe subtraction
as a function of emitted
x-ray energy for right- (R) and left-handed (L)circularlypolarizations
arecomparedwithexperimental plots. Here, the excitations on the Fe-3dbands are included.

[1] T. Inami, Phys. Rev. Lett.119, 1337203 (2017).
[2] A. Koide, T. Nomura, and T. Inami, Phys. Rev. B 102,224425 (2020).

The authors are grateful to Dr. Toshiya Inami for close
collaborations and discussions.

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