J. Phys. Soc. Jpn. 85, 062001 (2016) [33 Pages]

INVITED REVIEW PAPERS

Electronic Structures of Uranium Compounds Studied by Soft X-ray Photoelectron Spectroscopy

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Shin-ichi Fujimori¹^,, Yukiharu Takeda¹, Tetsuo Okane¹, Yuji Saitoh¹, Atsushi Fujimori¹^,², Hiroshi Yamagami¹^,³, Yoshinori Haga⁴, Etsuji Yamamoto⁴, and Yoshichika Ōnuki⁵

+ Affiliations

¹Materials Sciences Research Center, Japan Atomic Energy Agency, Sayo, Hyogo 679-5148, Japan²Department of Physics, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan³Department of Physics, Faculty of Science, Kyoto Sangyo University, Kyoto 603-8555, Japan⁴Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan⁵Faculty of Science, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan

Received August 3, 2015; Accepted March 17, 2016; Published May 26, 2016

The electronic structures of uranium-based compounds have been studied by photoelectron spectroscopy with soft X-ray synchrotron radiation. Angle-resolved photoelectron spectroscopy with soft X-rays has made it possible to directly observe their bulk band structures and Fermi surfaces. It has been shown that the band structures and Fermi surfaces of itinerant compounds such as UB₂, UN, and UFeGa₅ are quantitatively described by a band-structure calculation treating all U 5f electrons as itinerant. Furthermore, the overall electronic structures of heavy-fermion compounds such as UPd₂Al₃, UNi₂Al₃, and URu₂Si₂ are also explained by a band-structure calculation, although some disagreements exist, which might originate from the electron correlation effect. This suggests that the itinerant description of U 5f states is an appropriate starting point for the description of their electronic structures. The situation is similar for ferromagnetic superconductors such as UGe₂, URhGe, UCoGe, and UIr, although the complications from their low-symmetry crystal structures make it more difficult to describe their detailed electronic structures. The local electronic structures of the uranium site have been probed by core-level photoelectron spectroscopy with soft X-rays. The comparisons of core-level spectra of heavy-fermion compounds with typical itinerant and localized compounds suggest that the local electronic structures of most itinerant and heavy-fermion compounds are close to the U 5f³ configuration except for UPd₂Al₃ and UPt₃. The core-level spectrum of UPd₂Al₃ has similarities to those of both itinerant and localized compounds, suggesting that it is located at the boundary between the itinerant and localized states. Moreover, the spectrum of UPt₃ is very close to that of the localized compound UPd₃, suggesting that it is nearly localized, although there are narrow quasi-particle bands in the vicinity of E_F.

https://doi.org/10.7566/JPSJ.85.062001

1. Introduction

Electron correlation effects are the source of the complex and remarkable physical properties of modern materials. In particular, strongly correlated materials exhibit unusual physical properties such as quantum critical behaviors or unconventional superconductivity. Among these classes of materials, uranium-based compounds have a unique place owing to their rich variety of physical properties. One of their most significant properties is the coexistence of unconventional superconductivity and magnetic ordering observed in some uranium-based metallic compounds.¹⁾ The emergence of unconventional superconductivity is closely related to the magnetic instability, and their interplay is particularly interesting. Those characteristic physical properties of uranium-based compounds originate from strongly correlated U $5f$ electrons. They exhibit both itinerant and localized properties, and their unified understanding has been a challenging subject for over thirty years in the field of condensed matter physics.

To understand the physical properties of strongly correlated materials, it is essential to reveal their electronic structures. Quantum-oscillation-type techniques, such as de Haas–van Alphen (dHvA) experiments, have been applied to uranium compounds, and the topologies of their Fermi surfaces have been revealed.²⁾ Meanwhile, the recent remarkable progress in photoelectron spectroscopy (PES) has enabled more precise and detailed information to be obtained about the electronic structures of these compounds. In particular, angle-resolved photoelectron spectroscopy (ARPES) with various incident photon energies enables us to observe the band structures and Fermi surfaces of f-electron materials. Among the PES experiments, recent soft X-ray (SX) PES experiments have provided new opportunities to understand the electronic structures of strongly correlated materials.

In this article, recent SX-PES studies on uranium compounds are reviewed and the current understanding of U $5f$ electronic structures in metallic uranium-based compounds is discussed. The present article is structured as follows. In Sect. 2, basic concepts of the electronic structures of f-electron materials are briefly summarized. In Sect. 3, general properties of PES experiments as well as the characteristics of SX-PES are briefly outlined. In Sect. 4, the physical properties of the uranium compounds discussed in this article are summarized. In Sect. 5, the details of the experimental method and the band-structure calculations are briefly summarized. In Sect. 6, the results of SX-ARPES for uranium compounds are discussed. First, the results for typical itinerant compounds (UB₂, UN, and UFeGa₅) and a localized compound (UPd₃) are shown. In the itinerant compounds, U $5f$ electrons form bands with sub-electronvolt energy dispersion in the vicinity of $E_{\text{F}}$, and the overall band structure and Fermi surface are explained by a band-structure calculation treating all U $5f$ electrons as itinerant. In the case of the localized compound UPd₃, U $5f$ electrons are located on the high-binding-energy side, and there are no contributions to the states in the vicinity of $E_{\text{F}}$. Its band structure is explained by the band-structure calculation of the non-$5f$ compound ThPd₃. Then, the results for heavy-fermion superconductors (UPd₂Al₃, UNi₂Al₃, and URu₂Si₂) and ferromagnetic superconductors (UGe₂, URhGe, UCoGe, and UIr) are discussed. In these compounds, the overall band structure in electronvolt to sub-electronvolt energy order can be explained by the band-structure calculation, although there are some deviations, especially in the states in the vicinity of $E_{\text{F}}$. In Sect. 7, the core-level spectra of uranium compounds are discussed. The core-level spectra are sensitive to the local electronic structure of specific atoms and can be utilized to determine local physical parameters such as the valence states of uranium atoms. It is shown that the core-level spectra of most heavy-fermion superconductors are similar to those of itinerant compounds, suggesting that U $5f$ electrons have an itinerant character. Finally, in Sect. 8, all of these results are summarized, and future prospects are outlined.

2. Electronic Structures of f-Electron Materials

The physical properties of actinide and rare-earth materials are governed by the f-electrons. The interactions between localized f-electrons and itinerant ligand electrons lead to their strongly correlated nature. Here, we summarize some important concepts in the electronic structures of f-electron materials.

2.1 Itinerant vs localized problem

The itinerant vs localized problem of f-electrons has been one of the central issues in the understanding of actinide and rare-earth materials. These two very different pictures of f-electrons are realistic starting points to solve complex many-body problems in f-electron materials. The hidden order (HO) problem in URu₂Si₂ is a good example for understanding the issue. As discussed in detail in later sections, URu₂Si₂ shows an enigmatic HO phase transition at $T_{\text{HO}}=17.5$ K,³⁾ and its order parameter has yet not been identified nearly 30 years after its discovery. There have been a tremendous numbers of theoretical proposals to address the issue, and they can principally be classified into localized models based on crystalline electric field (CEF) schemes and itinerant models based on Fermi surface instability.⁴^,⁵⁾ These two pictures assume very different microscopic origins of the HO phase, and their validity can be evaluated by understanding how f-electrons are itinerant or localized. Here, we examine the relationship between this problem and the information obtained by PES experiments.

What is the main criterion to distinguish between localized and itinerant states? An important point is that itinerant and localized states are understood in terms of the dynamical properties of f-electrons. In the localized case, the f-electrons do not make major contributions to the states at $E_{\text{F}}$, and the local f-electron configuration is stable. On the other hand, when f-electrons make finite contributions to the states at $E_{\text{F}}$, the local f-electron configurations can dynamically fluctuate upon thermal excitation. They make major contributions to various transport properties such as the electrical resistivity, magnetic susceptibility, and electronic specific heat. Experimentally, when f-electrons are itinerant, the quasi-particle bands with large f-components have sizable energy dispersions in the momentum space and make finite contributions to the Fermi energy. Therefore, the fundamental difference between the itinerant and localized f-electron pictures is reflected in not static but dynamical properties of f-electrons.

2.2 Theoretical frameworks

A further important question is what is an appropriate theoretical treatment for describing the electronic structure of f-electron materials. In the modern theoretical framework, there are two major approaches to describing the electronic structures of f-electron materials. One approach is based on a localized picture, where the Coulomb energy between f-electrons $U_{ff}$ is more than one order of magnitude larger than the hybridization energy between f and ligand states $V_{fc}$. For Ce-based compounds, their angle-integrated photoemission spectroscopy (AIPES) spectra are basically described within the framework of the single-impurity Anderson model (SIAM).⁶⁾ In this approach, the f-electron is treated as an impurity state, and the model is solved under the assumptions of large $U_{ff}$ and small $V_{fc}$. This model can describe the local Kondo physics of f-electron materials. A straightforward extension of the SIAM to a lattice is the periodic Anderson model (PAM), where the momentum dependences of renormalized f-states can be treated. In the large-$U_{ff}$ limit, the model is reduced to the two-level mixing model.⁷⁾ Within this approximation, the renormalized f-level in the vicinity of the Fermi energy hybridizes with dispersive ligand bands. Such approaches can successfully explain the ARPES spectra of correlated Ce-based compounds.⁸⁾ Meanwhile, it should be noted that the Fermi surface based on the PAM is often very similar to that of non-f compounds except for the heavy renormalized bands.⁹⁾ Therefore, it is very important to reveal not only the Fermi surface but also the band structure to comprehend the nature of f-electrons.

Another approach is a band-structure calculation based on the local density approximation (LDA), with all f-electrons treating as itinerant Bloch states. In the band-structure calculation, the Kohn–Sham equation is solved by assuming a local exchange correlation potential, and one-electron excitation energies are numerically solved. Although calculations can quantitatively describe the band structures and Fermi surfaces of non-interacting metallic materials, clear disagreements exist in the case of strongly correlated materials. This is due to the nature of the LDA, which underestimates the electron correlation effects. In strongly correlated materials, the effects appear as qualitative deviations from the results of the band-structure calculation. The most well-known effect is the emergence of a correlation satellite on the high-binding-energy side. For example, there is the correlation satellite at $E_{\text{B}}\sim 6$ eV in the ARPES spectra of Ni metal, which has been interpreted as the two-hole bound state.¹⁰⁾ Experimental band structures are also modulated by the strong electron correlation effects. They appear as energy-dependent line-width broadening,¹¹⁾ narrowing of the bandwidth,¹²⁾ a kink structure in the vicinity of $E_{\text{F}}$,¹³⁾ or water-fall-like spectral features.¹⁴⁾

For uranium-based compounds, the overall electronic structures of itinerant or heavy-fermion compounds are generally explained by a band-structure calculation treating all U $5f$ electrons as itinerant as discussed in the present paper. For some $5f$ compounds with simple electronic structures, the renormalization of bands due to the electron correlation effect has been observed and formulated.¹⁵^,¹⁶⁾ On the other hand, the band-structure calculations of heavy-fermion compounds suggest many weakly dispersive bands in the vicinity of $E_{\text{F}}$, and their quantitative descriptions is still challenging even if the electron correlation effect is not strong. In the case of metallic $4f$ compounds, their band structures are generally very different from those indicated by the LDA calculation assuming $4f$ electrons as itinerant, and they cannot be accounted for even by introducing the electron correlation effects to the calculation. This suggests that Coulomb interactions between $4f$ electrons are too strong to be understood within these frameworks based on the LDA.

2.3 Band structure and Fermi surface

These itinerant and localized treatments are based on very different approximations, and therefore predict very different electronic structures. Meanwhile, it was pointed out that the topologies of Fermi surfaces derived from the two approaches are often very similar.⁹⁾ This is explained by the fact that non-f bands determine the overall band structure of f-electron materials. Even in these cases, the band structures in the vicinity of the Fermi energy are very different. In the nearly localized case, renormalized f-bands are hybridized with other states only in the vicinity of $E_{\text{F}}$. On the other hand, in the itinerant case, f-electrons form strongly dispersive bands. Therefore, it is essential to reveal not only the Fermi surface but also the band structure to understand the electronic structures of f-electron materials since the Fermi.

Accordingly, the following information is required to comprehend the itinerant or localized nature of f-electrons as well as the applicability of each theoretical approach: (i) the energy position of f-states, (ii) whether they make finite contributions to $E_{\text{F}}$, and (iii) the applicability of the band-structure calculation. SX-PES experiments are a very suitable tool for clarifying these points as discussed in the next section.

3. Soft X-ray Photoelectron Spectroscopy

PES is a very powerful experimental tool for observing the detailed electronic structures of materials, and it is suitable for clarifying the nature of f-electrons. Furthermore, ARPES experiments can directly observe the energy dispersion of f-electrons, and the obtained band structure and Fermi surface can be directly compared with the results of various theoretical models.

Various incident photon energies have been utilized for PES experiments on uranium compounds.¹⁷^,¹⁸⁾ By using different incident photon energies, the photoemission spectra reflect different information of the electronic structures. In the present study, soft X-rays ($h\nu=400{\text{--}}1200$ eV) are employed as incident photons. In recent years, there has been remarkable progress in the experimental techniques of SX-PES. One of the most important breakthroughs is that ARPES experiments with soft X-rays are now possible.¹⁹⁾ This is due to the development of soft X-ray beamlines in the third-generation synchrotron light sources²⁰^–²³⁾ as well as the innovative photoelectron analyzers with two-dimensional position sensitive detectors. Here, we discuss various important factors in PES experiments with emphasis on the characteristics of SX-PES.

3.1 Probing depths of PES experiments

One of the important points of PES experiments is that they are essentially surface-sensitive techniques. The escape depths of photoelectrons in conventional PES experiments are typically 5–20 Å, and one needs to consider the contributions from surface layers appropriately. Figure 1(a) shows the inelastic mean free path of electrons in solids.²⁴⁾ In vacuum-ultra violet (VUV)-PES experiments ($h\nu< 100$ eV), the kinetic energies of photoelectrons are typically 10 eV order, and their escape depths are less than 10 Å. The most commonly used photon energies in VUV-ARPES experiments are $h\nu=20{\text{--}}50$ eV, and the expected probing depths are about 5 Å. Figure 1(b) shows the crystal structure of URu₂Si₂. The interval between uranium layers in URu₂Si₂ is about 5 Å, and ARPES spectra measured by VUV photons are dominated by the signals from the topmost uranium layer at the surface. Since the surface uranium layer has fewer ligand atoms than the other layers, the U $5f$ electrons are more localized than those in the bulk. Furthermore, surface atomic layers often have structural reconstructions. For example, topographic images of the topmost uranium layers obtained by a scanning tunneling microscopy study of URu₂Si₂ indicated that the first uranium layer has a large structural reconstruction.²⁵⁾ Such reconstruction should drastically alter the surface electronic structures, and the spectra are remarkably different from those of the bulk. Therefore, in VUV-ARPES experiments, it is critical to distinguish the contributions from surface layers to discuss the bulk electronic structures.

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Figure 1. (Color online) Comparison of VUV-PES and SX-PES experiments, based on Refs. 17 and 18. (a) Inelastic mean free path of electrons in solids. (b) Crystal structure of URu₂Si₂ and the probing depth of VUV-PES and SX-PES experiments. (c) Photoionization cross sections of Si $3s$, Si $3p$, Ru $4d$, and U $5f$ orbitals.³⁵⁾ (d) Comparison of angle-integrated PES spectra of URu₂Si₂ measured at $h\nu=21.2$ eV³⁶⁾ and $h\nu=800$ eV.³⁷⁾ Reprinted figure 1 with permission from Ref. 18. © 2016 IOP Publishing.

On the other hand, in SX-PES experiments, the mean free paths of photoelectrons are about 15 Å, and the signals from the second to the fourth uranium layers can be observed in the spectra. Such deeper uranium layers are less affected by the surface terminations, and the signals from bulk electronic structures are strongly enhanced in SX-PES spectra. In fact, a comparison between the VUV- and SX-ARPES spectra of UN demonstrated that they have very different shapes of their U $5f$ band structure.¹⁷⁾ A further increase in the incident photon energy results in more enhanced bulk sensitivity. For example, significant progress has been made in hard X-ray PES (HAXPES) experiments with incident photon energies of kiloelectronvolt order, and bulk sensitivities on the order of 100 Å have been achieved.²⁶⁾ HAXPES experiments have also been performed on f-electron materials, and it was pointed out that finite contributions from surface layers still exist in SX-PES experiments.²⁷⁾ However, ARPES experiments with hard X-rays are generally quite difficult owing to many limitations on the experimental conditions as well as intrinsic effects in photoemission processes.²⁸^,²⁹⁾ The very good agreements between the results of SX-ARPES and band-structure calculations for many itinerant compounds³⁰^–³²⁾ suggests that SX-ARPES experiments are generally dominated by the bulk electronic structure. Therefore, an important boundary exists between VUV- and SX-PES, and SX-ARPES provides a unique opportunity to reveal the bulk electronic structures of uranium-based compounds.

Another important aspect of the finite escape depth of photoelectrons is that it determines the broadening of ARPES spectra along the $k_{\bot}$ direction ($\Delta k_{\bot}$).³³⁾ The escape depth of photoelectrons along the surface normal direction λ and $k_{\bot}$ has the relationship $\lambda\cdot \Delta k_{\bot}\sim 1$. In VUV-ARPES experiments, a typical escape depth is $\lambda\sim 5$ Å, and the broadening is about $\Delta k_{\bot}\sim 0.2$ Å⁻¹, which is nearly half of the typical size of the Brillouin zone $k\sim 0.5$ Å⁻¹. In this case, the momentum resolution along the $k_{\bot}$ direction is not sufficient to resolve the three-dimensional shape of the band structure and Fermi surface.³³⁾ On the other hand, in SX-ARPES experiments, the broadening is $\Delta k_{\bot}\lesssim 0.1$ Å⁻¹, which is sufficient to observe the three-dimensional electronic structures of materials.

3.2 Matrix element effect

In the photoemission process, the photocurrent $I(\boldsymbol{{k}},\omega)$ is expressed by the following formula.³⁴⁾ \begin{equation} I (\boldsymbol{{k}}, \omega) \propto f (\omega) \sum_{\nu} | M_{i,f}^{\boldsymbol{{k}}} |^{2} A (\boldsymbol{{k}}, \omega), \end{equation} (1) where $f(\omega)$ is the Fermi–Dirac function, $A(\boldsymbol{{k}},\omega)$ is the photoemission spectral function, and $M_{i,f}^{\boldsymbol{{k}}}$ is the one-electron dipole matrix element expressed by \begin{equation} M_{i,f}^{\boldsymbol{{k}}} = \langle \phi_{f}^{\boldsymbol{{k}}}| H_{\textit{int}} | \phi_{i}^{\boldsymbol{{k}}} \rangle. \end{equation} (2) Here, $|\phi_{i}^{\boldsymbol{{k}}}\rangle$ and $|\phi_{f}^{\boldsymbol{{k}}}\rangle$ are initial and final one-particle states, respectively, and $H_{\textit{int}}$ represents the interaction Hamiltonian. The photocurrent is proportional to the square of $M_{i,f}^{\boldsymbol{{k}}}$, and this is called the matrix element effect. This effect often influences the appearance of experimental photoemission spectra. One of the most typical effects is the photon energy dependence of the photoionization cross section of atomic orbitals. By changing the photon energy, the photoionization cross section varies in a manner depending on the each atomic orbital. Figure 1(c) shows the atomic calculation of photoionization cross sections of Si $3s$, Si $3p$, Rh $4d$, and U $5f$ orbitals.³⁵⁾ In general, the photoionization cross sections of s, p, and d orbitals are dominant at low photon energies ($h\nu\lesssim 100$ eV). Entering the soft X-ray region, that of the f-orbital increases relative to other orbitals. For soft X-rays with $h\nu\gtrsim 400$ eV, the photoionization cross section is almost comparable to that of a d orbital. Therefore, the relative sensitivities of U $5f$ orbitals to those of d orbitals are enhanced in soft X-ray PES. This situation is demonstrated in Fig. 1(d), where the angle-integrated PES spectra of URu₂Si₂ measured at $h\nu=21.2$ eV³⁶⁾ and $h\nu=800$ eV³⁷⁾ are compared. The contribution from Ru $4d$ states located at around $E_{\text{B}}=1{\text{--}}3$ eV is dominant in the spectrum measured at $h\nu=21.2$ eV, and the energy position of the U $5f$ states is not clear in the spectrum. On the other hand, the contribution from U $5f$ states is greatly enhanced in the spectrum measured at $h\nu=800$ eV. It is clear that U $5f$ states make finite contributions to $E_{\text{F}}$, showing their itinerant nature. Therefore, the photoionization cross section of the U $5f$ orbital is a very important factor in the PES studies of uranium compounds.

The matrix element effect often alters the structures of ARPES spectra. It has been realized that the shapes of bands are often different, depending on the photon energies or polarizations of the incident photons, even if they have the same symmetry in the Brillouin zone. For example, such effects are well recognized in the case of cuprate superconductors, and they have been well studied.³⁸⁾ Furthermore, it was pointed out that the crystal structure can also alter the appearance of bands in materials that contain more than one groups of atoms in the unit cell.³⁹⁾ This is called the photoemission structure factor, which has also been observed in f-electron materials. In the SX-ARPES study of UN, the ARPES spectra measured along the X–W–X high-symmetry line are not symmetric relative to the W point,³²⁾ which might be due to these effects. Therefore, it is extremely important to measure wide areas in the momentum space to eliminate such effects.

3.3 ARPES and Brillouin zone

In the free-electron final state approximation, the momentum of electrons is expressed by \begin{align} k_{\|} &= \frac{\sqrt{2 m E_{\text{kin}}}}{\hbar} \sin\theta - k_{\| \text{photon}}, \end{align} (3) \begin{align} k_{\bot} &= \sqrt{\frac{2 m}{\hbar^{2}}(E_{\text{kin}} \cos^{2}\theta + V_{0})} - k_{\bot \text{photon}}, \end{align} (4) where $V_{0}$ is the inner potential, which is generally determined phenomenologically. The kinetic energy of photoelectrons $E_{\text{kin}}$ satisfies the relation $E_{\text{kin}} = h\nu - E_{\text{bin}} -\Phi$. Therefore, the momentum of photoelectrons is a function of the angle and photon energy.

To understand the relationship between the momentum, detection angle, and incident photon energy, we show the case of URu₂Si₂ in Fig. 2. Figures 2(a) and 2(b) display the experimental setup in the real space and the Brillouin zone in the momentum space, respectively. In this experimental setup, angle scans along the [100] and [110] planes correspond to momentum scans along the $\langle 100\rangle$ and $\langle110\rangle$ directions, respectively. Figure 2(c) shows traces of $k_{\|}$ and $k_{\bot}$ with constant $h\nu$ and θ. The right half and left half of the figure represent the scans along the $\langle 110\rangle$ and $\langle 100\rangle$ directions, respectively. It is shown that a wider area of $k_{\|}$ is covered at higher photon energies with a fixed θ. For example, the X point corresponds to $\theta\sim 30$° at $h\nu=21.2$ eV and $\theta\sim 4$° at $h\nu=760$ eV. Therefore, the electronic structure of a wider momentum space can be obtained by ARPES experiments with higher photon energies. A modern display-type photoelectron analyzer can simultaneously detect electrons in an angular range of $\theta = 15{\text{--}}40$°. This implies that the ARPES spectra along multiple Brillouin zones can be detected without rotating the sample in SX-ARPES experiments. On the other hand, in ARPES experiments with low photon energies, one needs to observe photoelectrons from very large detection angles to reach the boundary of the Brillouin zone. Furthermore, final states are considerably affected by the crystal potential, and the free-electron final state might not be an appropriate approximation. In this case, Eq. (4) is not applicable, and the position at which $k_{\bot}$ should be probed is not obvious. Moreover, the transition itself may not occur owing to the absence of an appropriate final state under the conservation of energy and momentum. Therefore, ARPES experiments with high photon energies are suitable for observing the overall band structure and Fermi surface of a wide momentum space.

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Figure 2. (Color online) Experimental setup of ARPES experiments and ARPES cuts in the momentum space. (a) Experimental configuration of ARPES experiment. (b) Brillouin zone of URu₂Si₂ in the paramagnetic phase. (c) ARPES cuts with $h\nu =\text{const.}$ (solid lines) and $\theta =\text{const.}$ (dashed lines). The Brillouin zones of URu₂Si₂ along $\langle 100\rangle$ and $\langle 110\rangle$ are also indicated. Note that the momentum of incident photons parallel to the scan direction is assumed to be zero.

3.4 Energy resolution

In recent years, the energy resolution in PES experiments has been dramatically improved. There are two major contributions to the energy resolution in PES experiments. One of them is the energy line width of incident photons. This has been improved by the advances in various laboratory-based light sources as well as synchrotron radiation instruments. The other factor is the energy resolution of photoelectron analyzers. This also has been improved by the recent development of analyzers with a display-type electron detector. The highest energy resolution has been achieved by low energy (LE) PES with laser light sources, and the highest energy resolution at the present is sub-millielectronvolt order.⁴⁰⁾ Typical total energy resolutions of VUV-PES experiments are about 10–20 meV in modern experimental setups. The energy resolutions in SX-PES experiments is 80–200 meV. Although the resolutions are worse than those of VUV-PES, the resolving power $E/\Delta E$ is about ${\sim}10^{4}$ in SX-PES, while it is typically less than $5\times 10^{3}$ in VUV-PES. Even with these energy resolutions, it is still possible to follow the behaviors of quasi-particle bands of f-electron materials. The topologies of the Fermi surface can also be obtained in the case of a relatively simple band structure.

3.5 Core-level spectroscopy

The core-level spectral line shapes are very sensitive to the local electronic structures, and they have been utilized to identify the chemical state of a specific atomic site. For example, the valence states of specific atoms have been analyzed by the chemical shifts of the core level. A general review of core-level spectroscopy is given in Ref. 41. Furthermore, the core-level spectra of strongly correlated materials are accompanied by satellite structures originating from various screening channels of the core hole in the final states. By analyzing their structures, basic physical parameters can be obtained. The binding energies of core electrons used for such analysis are typically on the order of 100–1000 eV, and photon energies of $h\nu\gtrsim 100$ eV are required to excite them. Core-level spectroscopy with soft X-ray incident photons results in photoelectrons with kinetic energies of $E_{\text{kin}}\gtrsim 400$ eV, which have sufficient bulk sensitivity for probing the bulk electronic structures of materials. In Sect. 7, the core-level spectra of uranium-based compounds are discussed.

3.6 PES experiments and f electron physics

From the above, we have found that PES experiments with each incident photon-energy range have their own characteristics. Here, we consider the relationships between these PES experiments and the relevant physics of f-electron materials. The energy scales for the various physical properties of f-electron materials and the energy resolutions of LE-, VUV-, and SX-PES are schematically represented in Fig. 3. In general, the overall band structures have an energy scale of electronvolt to sub-electronvolt order. The overall shape of Fermi surfaces is also determined by the band structures with these energy scales, although there might be some effects from low-energy excitations. Magnetic ordering temperatures of f-electron materials are typically on the order of 10 K, corresponding an energy of 1 meV. The superconducting transition temperature is less than 2 K in heavy-fermion materials, corresponding to an energy of less than 0.2 meV. SX-PES experiments are suitable for the observation of overall band structures as well as Fermi surfaces since they have comprehensive coverages in the momentum space, high $k_{\bot}$ selectivity, and enhanced bulk sensitivity. In some cases, it is possible to observe the changes in electronic structures due to the magnetic transitions. LE-PES experiments are suitable for the observation of very small energy phenomena in the vicinity of $E_{\text{F}}$. However, the very narrow coverage along the $k_{\|}$ direction, low tunability along the $k_{\bot}$ direction, and low sensitivity to f-orbitals make it difficult to observe overall electronic structures. Although VUV-PES experiments can be performed to observe overall band structures as well as low-energy excitations, large contributions from surface electronic structures need to be considered in their interpretation. Therefore, it is important to combine PES experiments with various photon energy ranges to understand the multiple energy scale of U $5f$ electronic structures.

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Figure 3. (Color online) Energy scales of f-electron materials and energy resolutions of PES experiments with various photon energies. Each PES experiment has different characteristics and relevant energy scales. SX-PES is suitable for the determination of overall band structures and Fermi surfaces of f-electron materials.

4. Summary of Measured Compounds

Table I summarizes the uranium compounds measured in the present study.³^,³⁰^–³²^,⁴²^–⁶²^,⁶⁴^–⁷³⁾ The physical properties of uranium compounds range from completely localized to fully itinerant states, and we first discuss the typical itinerant and localized compounds to understand both extreme cases. On the basis of these results, the electronic structures of heavy-fermion compounds are discussed. UB₂, UFeGa₅, UPtGa₅, UN, USb₂, UGa₃, and UAl₃ are chosen as itinerant compounds. Their specific heat coefficients range from 10 mJ/(mol·K²) for UB₂ to 64 mJ/(mol·K²) for UPtGa₅. In these compounds, the U $5f$ electrons are expected to form dispersive bands and Fermi surfaces. On the other hand, UPd₃, and UGa₂ are chosen as typical examples of localized compounds. Their specific heat coefficients are on the order of 10 mJ/(mol·K²), suggesting that U $5f$ electrons make no essential contributions to the Fermi energy. The heavy-fermion superconductors studied in the present article are also included in the table. One of the most significant aspects of these uranium-based compounds is that the superconductivity coexists with various types of ordering. The most studied compound among them is URu₂Si₂, which undergoes a transition to the HO state at $T_{\text{HO}}=17.5$ K.³⁾ It further undergoes a transition to the superconducting state at $T_{\text{SC}} = 1.5$ K. UNi₂Al₃, UPd₂Al₃, and UPt₃ have coexistence of antiferromagnetic (AFM) ordering and unconventional superconducting states. Their specific heat coefficients range from 120 mJ/(mol·K²) for UNi₂Al₃ to 450 mJ/(mol·K²) for UPt₃, suggesting the existence of strong electron correlation effects in these compounds. UGe₂, URhGe, UCoGe, and UIr have coexistence of ferromagnetic ordering and unconventional superconducting states. Their specific heat coefficients are about 50 mJ/(mol·K²) except for URhGe, suggesting that the correlation effect is not as significant as in the cases of UPd₂Al₃ and UPt₃. URhGe has the largest γ of 160 mJ/(mol·K²), and the electron correlation effect plays an essential role in this compound. UCoAl is a paramagnetic compound, and it undergoes a metamagnetic transition under a magnetic field. The ferromagnetic quantum criticality of this compound has been pointed out.⁷⁴⁾

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Table I. Physical properties of uranium compounds studied in the present article.

5. Experimental Method and Band-Structure Calculation

The SX-PES experiments presented in this article were performed at the soft X-ray beamline BL23SU of SPring-8.²⁰^,²²⁾ For the AIPES measurements at $h\nu=800$ eV, the typical energy resolution was about 120 meV. The exception was for the U $4f$ spectrum of UIn₃, which was measured at $h\nu=850$ eV to avoid the contributions from Auger signals. For the ARPES measurements at $h\nu=450{\text{--}}820$ eV, the energy resolution was about 75–160 meV. The position of $E_{\text{F}}$ was determined by measurements of vapor-deposited gold films. Clean sample surfaces were obtained by cleaving the samples in situ under an ultra high vacuum (UHV) condition. The vacuum during the measurements was typically ${<}3\times 10^{-8}$ Pa, and the sample surfaces were stable for the duration of measurements (1–2 days) since no significant changes were observed in the ARPES spectra during the periods. The positions of ARPES cuts were determined by assuming a free-electron final state with an inner potential of $V_{0}=12$ eV for all compounds. The ARPES spectral shapes are often influenced by the matrix element effect, as discussed in Sect. 3.2, and some bands are not well observed depending on the photon energy or detection angle. In this case, the spectra are averaged with data points with the same symmetry in the Brillouin zone. The details of the procedure are outlined in Ref. 32.

The background contributions to the PES spectra presented in this article were subtracted in the following manner. For AIPES spectra, subtraction of the Shirley-type background was assumed to remove the contributions from inelastically scattered photoelectrons.⁷⁵⁾ For ARPES spectra, the background contributions from elastically scattered photoelectrons due to surface disorder or phonons were subtracted by assuming momentum-independent spectra. The details of the procedure are outlined in Ref. 66.

In the band-structure calculations, relativistic linear augmented plane wave (RLAPW) calculations⁷⁶⁾ within the LDA⁷⁷⁾ were performed, treating all U $5f$ electrons as itinerant. In this approach, the Dirac-type Kohn–Sham equation has been formulated, and the spin–orbit interaction is exactly taken into account. To compare the results of the calculations with the ARPES spectra, we have simulated the ARPES spectral functions on the basis of the band-structure calculations. In the simulation, (i) the broadening along the $k_{\bot}$ direction due to the finite escape depth of photoelectrons, (ii) the lifetime broadening of the photohole, (iii) the photoemission cross sections of orbitals, and (iv) the energy resolution and angular resolution of the photoelectron analyzer were taken into account. The details are outlined in Ref. 32.

6. Soft X-ray ARPES for U-Based Compounds

6.1 UB₂

We first show the results for UB₂, which is a prototypical itinerant compound. UB₂ is a paramagnetic compound with a hexagonal AlB₂-type crystal structure.⁴²⁾ Figure 4(a) shows the crystal structure of UB₂. The lattice constants are $a=3.132$ Å and $c=3.986$ Å,⁷⁸⁾ and the nearest U–U distance is 3.132 Å, which is much smaller than the Hill limit of 3.4 Å.⁷⁹⁾ Therefore, large direct f–f overlaps are expected in this compound. Its specific heat coefficient γ is as low as 10.3 mJ/(mol·K²), which is very close to that of Pd metal [9.42 mJ/(mol·K²)], implying that the electron correlation effect is as weak as that for d-electron materials. In fact, the specific heat coefficient from the band-structure calculation is 7.29 mJ/(mol·K²), which is comparable to the experimental value. The experimentally observed dHvA branches were well explained by a band-structure calculation treating all U $5f$ electrons as itinerant.⁷⁸⁾ Therefore, UB₂ can be considered as a uranium compound with bandlike U $5f$ electrons. Figure 4(b) displays the hexagonal Brillouin zone. In the present study, two-dimensional ARPES scans within the A–H–L and Γ–K–M high-symmetry planes were performed.

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Figure 4. (Color online) Crystal structure and Brillouin zone of UB₂. (a) Hexagonal crystal structure of UB₂ and (b) hexagonal Brillouin zone.

6.1.1 Valence band spectrum of UB₂

Figure 5 shows the valence band spectrum measured at $h\nu=800$ eV and the calculated U $5f$ partial density of states (pDOS) of UB₂.³⁰⁾ At this photon energy, the photoionization cross section of U $5f$ orbitals is larger than those of other orbitals, such as B $2s$ and $2p$ orbitals, by 1–2 orders of magnitude,³⁵⁾ and the spectrum is dominated by the contributions from U $5f$ states. The experimental spectrum shows a sharp peak structure slightly below $E_{\text{F}}$, and it has a long tail toward high binding energies. The lower curve in the figure represents the calculated pDOS multiplied by the Fermi–Dirac function and is broadened with the experimental energy resolution. The calculated pDOS has a similar peak structure to the experimental spectrum. The pDOS is much broader than the experimental spectrum, suggesting that the U $5f$ states are strongly hybridized with B s and p states in the calculation.

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Figure 5. (Color online) Valence band spectrum measured at $h\nu=800$ eV and the calculated pDOS of UB₂. The calculated pDOS is multiplied by the Fermi–Dirac function and is broadened with the experimental energy resolution.

6.1.2 Band structure and Fermi surface of UB₂

Figure 6 summarizes the experimental and calculated Fermi surface and band structure of UB₂.³⁰⁾ The spectra within the A–H–L high-symmetry plane shown in Fig. 6(a) and those within the Γ–K–M high-symmetry plane shown in Fig. 6(b) were measured at $h\nu=450$ and 500 eV, respectively. The ARPES spectra are divided by the Fermi–Dirac function convoluted with the instrumental energy resolution to show the states near $E_{\text{F}}$ clearly. The spectra show clear energy dispersions along all high-symmetry lines. The relatively weakly dispersive bands in the vicinity of $E_{\text{F}}$ correspond to the quasi-particle bands with large contributions from U $5f$ states. On the other hand, the strongly dispersive bands at high binding energies ($E_{\text{B}}\gtrsim 1$ eV) correspond to bands with large contributions from the B s and p states. The dispersive nature of U $5f$ states is clearly recognized, particularly in the vicinity of $E_{\text{F}}$, indicating that U $5f$ electrons have very itinerant character. Some characteristic features exist in the experimental Fermi surface. Within the A–H–L high-symmetry plane, a triangular feature centered at the H point is observed, which has a hole-like dispersion, as seen in the band structure along the A–H high-symmetry line. Within the Γ–K–M high-symmetry plane, a star like feature is realized around the Γ point. A circular feature exists inside of this feature, although its intensity is rather low. The ARPES spectra along the Γ–M and Γ–K directions indicate that the star like feature and the circular feature are the cross-section of an electron-pocket Fermi surface.³⁰⁾

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Figure 6. (Color online) Experimental and calculated Fermi surface and band structure of UB₂. (a) Experimental Fermi surface mapping and band structure within the A–H–L high-symmetry plane measured at $h\nu=500$ eV. (b) Same representation within the Γ–K–M high-symmetry plane measured at $h\nu=450$ eV. (c) Result of the band-structure calculation and the simulation of ARPES spectral functions within the A–H–L high-symmetry plane. (d) Same representation within the Γ–K–M high-symmetry plane. (e) Three-dimensional shape of calculated Fermi surface. Data replotted from Ref. 30.

To demonstrate the validity of the band-structure calculation, the calculated band structure and Fermi surface, as well as the results of simulations of ARPES spectra, are shown in Figs. 6(c) and 6(d). The three-dimensional shape of the calculated Fermi surface is also shown in Fig. 6(e). The characteristic features in the experimental band structures and Fermi surfaces are well explained by the calculation. In the calculation, band 6 forms the hole-type Fermi surface around the H point, and band 7 forms the star like electron Fermi surface around the Γ point. These features correspond to the experimental band structures and Fermi surfaces very well. In addition to the bands in the vicinity of $E_{\text{F}}$, the strongly dispersive features on the high-binding-energy side also exhibit good agreement with the calculated band dispersions. These results demonstrate that the band-structure calculation can explain the characteristic features of ARPES spectra of UB₂. Peak fitting analysis of these spectra³⁰⁾ showed that the topologies of the Fermi surfaces are essentially very similar, although their sizes are slightly different. For example, the size of the star like feature within the Γ–K–M high-symmetry plane, which corresponds to the outer orbit of the calculated Fermi surface formed by band 7, is 10–13% larger than the calculated value.

Accordingly, it was demonstrated that the essential topology of the Fermi surface of UB₂ can be explained by the band-structure calculation, while the sizes of features are slightly different between the experiment and the calculation. This result suggests that the band-structure calculation treating all U $5f$ electrons as itinerant can describe the overall band structures as well the Fermi surfaces of uranium-based compounds with itinerant U $5f$ electrons.

6.2 UN

We further extend our investigation to the itinerant and magnetic compound UN, which is an itinerant antiferromagnet with a NaCl-type face-center-cubic (fcc) crystal structure. It undergoes a transition into a type I AFM phase at $T_{\text{N}}=53$ K with $\mu_{\text{ord}} = 0.75$ $\mu_{\text{B}}$. Figures 7(a) and 7(b) show the crystal structure and Brillouin zone of UN, respectively. Since the U–U distance of 3.46 Å in UN is close to the Hill limit of 3.4 Å,⁷⁹⁾ it is expected that U $5f$ electrons will be located at the boundary between itinerant and localized states. Previous VUV-ARPES⁸⁰⁾ and XPS⁴⁵⁾ studies suggested that U $5f$ electrons have both itinerant and localized natures. Another interesting aspect of UN is that it is a promising fuel material for the generation IV reactors since it has a high melting point (2850 °C), and very good thermal conductivity at high temperatures as well as a high fuel density (14.32 g cm⁻³). There have been a number of experimental and theoretical attempts to determine the physical properties of UN to develop better fuel materials.⁸¹⁾ Therefore, knowing its electronic structure has technical importance for future nuclear technology.

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Figure 7. (Color online) Crystal structure and Brillouin zone of UN. (a) Face-centered cubic crystal structure of UN. (b) Brillouin zone of UN in the paramagnetic phase.

6.2.1 Valence band spectrum of UN

Figure 8 shows a comparison of the valence band spectrum and the calculated U $5f$ DOS of UN. The sample temperature was 75 K and the sample was in the paramagnetic phase. The experimental spectrum has an asymmetric line shape, having a long tail toward higher binding energies. It spectral shape is very similar to that of UB₂. The calculated pDOS is multiplied by the Fermi–Dirac function and is broadened with the experimental energy resolution. It has an asymmetric shape, and the overall spectral features are well explained by the band-structure calculation.

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Figure 8. (Color online) Valence band spectrum of UN measured at $h\nu=800$ eV and the calculated U $5f$ pDOS. The sample temperature was 75 K and the sample was in the paramagnetic phase. The calculated pDOS is multiplied by the Fermi–Dirac function and is broadened with the experimental energy resolution.

6.2.2 Band structure and Fermi surface of UN

Figure 9 summarizes the results of ARPES experiments as well as the band-structure calculation for UN. Figure 9(a) shows ARPES spectra of UN measured at $h\nu=490$ eV along the X–W–X high-symmetry line. The sample temperature was 75 K, and the sample was in the paramagnetic phase. The spectra were symmetrized relative to the W point to eliminate the contributions from the matrix element effect.³²⁾ Five bands were observed in this energy region, and they are designated as bands A, B, C, D, and E. Bands A–D have strong energy dispersions and are located on the high-binding-energy side. Their intensities are rather low, suggesting that they are contributions manly from the N s and p states. Meanwhile, band E has an electron-type dispersion around the W point, and it forms a Fermi surface. Since its intensity is enhanced in the spectra, it is assigned to the quasi-particle band with large contributions from U $5f$ states. Figure 9(b) shows the energy band dispersions and the simulation of ARPES spectra based on the band-structure calculation treating all U $5f$ electrons as itinerant. Bands 4–8 exist in this energy region, and band 8 forms a Fermi surface around the W point. These bands correspond to the experimentally observed bands A–E very closely, although their energy positions are slightly different. Figure 9(c) shows an enlargement of the ARPES spectra measured in the paramagnetic phase. The solid lines represent the positions of bands obtained by the peak fitting of momentum distribution curves (MDCs). The behaviors of bands E and D can be clearly understood from this figure. Band E has an electron-like dispersion while band D has an inverted parabolic dispersion, and both bands are strongly hybridized around the W point. These structures closely correspond to the calculated bands 7 and 8 as shown in Fig. 9(d). Therefore, the band structure of UN is well described by the band-structure calculation.

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Figure 9. (Color online) Experimental and calculated band structures and Fermi surface of UN. (a) ARPES spectra measured along the X–W–X high-symmetry line. The sample temperature was 75 K and the sample was in the paramagnetic phase. (b) Result of the band-structure calculation in the paramagnetic phase and the simulation of the ARPES spectra. (c) Enlargement of the ARPES spectra in the vicinity of $E_{\text{F}}$ measured in the paramagnetic phase and the result of the peak fitting of MDCs. (d) Result of the band-structure calculation and the simulation of ARPES spectra in the paramagnetic phase. (e) Enlargement of the ARPES spectra measured in the AFM phase and the result of the peak fitting of MDCs. (f) Comparison of the fitted bands in both phases. (g) Fermi surface mapping obtained by measuring the photon energy dependence of ARPES spectra. (h) Fermi surface mapping obtained by the band-structure calculation. (i) Three-dimensional shape of the calculated Fermi surface. Reprinted figures 2, 3, and 4 with permission from Ref. 32. © 2012 American Physical Society.

Next, we consider the changes in the electronic structure due to the AFM transition. Figure 9(e) shows the ARPES spectra measured at 20 K, at which the sample is in the AFM phase. It is shown that the spectra do not change significantly owing to the AFM transition. Meanwhile, there are certain changes in the energy dispersions, particularly at around $k_{y}\sim 1.0 (\pi/a)$. To see the details of the temperature dependences, a comparison of the energy dispersions measured in the paramagnetic and AFM phases is shown in Fig. 9(f). The solid and dotted lines represent the energy dispersions in the paramagnetic and AFM phases, respectively. The comparison suggests that the crossing point at around $k_{y}\sim 1.0 (\pi/a)$ shifts toward a lower binding energy by 47 meV due to the transition. Since UN is a type-I antiferromagnet, the paramagnetic Brillouin zone is folded relative to the W point in the magnetic Brillouin zone.³²⁾ Therefore, the shift of the bands should be due to the AFM transition. The small changes in the electronic structure due to the transition are consistent with the picture of weak itinerant antiferromagnetism where spin-polarized itinerant electrons form magnetic moments in both the paramagnetic and AFM phases.

We further study the Fermi surface of UN. Figure 9(g) shows the Fermi surface of UN obtained by measuring the photon energy dependence of the ARPES spectra. The sample temperature was 20 K, and the sample is in the AFM phase. A circular shaped Fermi surface exists around the X point. Although the measurements were performed in the AFM phase, the spectra have symmetry of the paramagnetic Brillouin zone. This is explained by the very small changes in the electronic structure associated with the AFM transition. Therefore, the discussion here is based on the symmetry of the paramagnetic Brillouin zone. Figures 9(h) and 9(i) display the Fermi surface mapping based on the band-structure calculation and the three-dimensional shape of the Fermi surface, respectively. The experimental Fermi surface closely corresponds to the result of the calculation. However, the small square-shaped Fermi surface observed in the calculation is missing from the experimental Fermi surface. This is caused by the matrix element effect as discussed in Sect. 3.2.

Accordingly, it has been demonstrated that the band structure and Fermi surface of UN can be well understood by the band-structure calculation treating all U $5f$ electrons as itinerant. The changes in the electronic structure due to the AFM transition were observed in the ARPES spectra. However, the changes were very small, and they are consistent with the magnetic structure of the AFM phase of UN. The small changes suggest that this compound is a weak itinerant antiferromagnet with spin-polarized itinerant electrons.

6.3 UFeGa₅ and UPtGa₅

Next, the electronic structures of the uranium compounds with itinerant and more complex U $5f$ electronic structure are discussed. The discovery of superconductivity in CeMIn₅ (M = Co, Rh, Ir) compounds with a HoCoGa₅-type crystal structure has provided new opportunities to study the interplay between the magnetism and superconductivity in heavy-fermion compounds.⁸²^–⁸⁴⁾ They have a quasi-two-dimensional electronic structure and are located in the vicinity of the quantum critical point (QCP). Their physical properties have been controlled via various physical parameters such as the pressure, applied magnetic field, and chemical substitution. Isostructural compounds have been discovered among uranium, neptunium, and plutonium compounds.⁸⁵^–⁸⁷⁾ Superconductivity with $T_{\text{SC}}=18$ K was reported for PuCoGa₅.⁸⁷⁾ This transition temperature is much higher than that of CeIrIn₅ ($T_{\text{SC}}=0.4$ K).⁸³⁾ Therefore, this is a unique system where $T_{\text{SC}}$ can be controlled over nearly two orders of magnitude by chemical substitutions. The series of UMGa₅ (M: transition metal) compounds also has a tetragonal HoCoGa₅-type crystal structure⁸⁵⁾ as shown in Fig. 10(a).

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Figure 10. (Color online) Crystal structure and Brillouin zone of UMGa₅ (M: transition metal). (a) Tetragonal crystal structure of UMGa₅ and (b) tetragonal Brillouin zone. The ratio of the tetragonal lattice parameters $c/a$ is about 1.6 and its electronic structure is expected to be quasi-two-dimensional.

Among the series of compounds, UFeGa₅ is a paramagnet,⁴³⁾ whose lattice constants are $a=4.261$ Å and $c=6.734$ Å.⁸⁵⁾ The tetragonal Brillouin zone based on these lattice parameters is shown in Fig. 10(b). The ratio of the tetragonal lattice parameters $c/a$ is about 1.6 in this series of compounds, and their Brillouin zone is compressed along the $c^{\ast}$-axis. Therefore, quasi-two-dimensional electronic structures are expected in this series of compounds. UFeGa₅ has itinerant U $5f$ states, and its specific heat coefficient γ is 37 mJ/(mol·K²),⁴⁴⁾ which is about twice the calculated value of $\gamma = 19.6$ mJ/(mol·K²). The nature of its Fermi surfaces was studied by performing dHvA experiments,⁴³⁾ and the observed branches were well explained by the band-structure calculation. UPtGa₅ is an antiferromagnet with $T_{\text{N}}=23.5$ K.⁴⁴^,⁸⁸⁾ Its lattice constants are $a=4.3386$ Å and $c=6.8054$ Å,⁸⁹⁾ which are about 1–2% larger than those of UFeGa₅.

6.3.1 Valence band spectra of UFeGa₅ and UPtGa₅

Figure 11 summarizes the valence band spectra and the calculated DOS of UFeGa₅ and UPtGa₅.³¹^,³⁷⁾ Figure 11(a) shows the valence band spectra measured at $h\nu=400$ and 800 eV and the partial U $5f$ and Fe $3d$ DOS obtained by the band-structure calculation. The sample temperature was 20 K and the total energy resolution was $\Delta E=160$ meV. The calculated pDOS are multiplied by the Fermi–Dirac function and are broadened with the experimental energy resolution. The spectrum of UFeGa₅ consists of two broad features located at around $E_{\text{F}}$ and $E_{\text{B}}\sim 1.5$ eV. Furthermore, the peak in the vicinity of $E_{\text{F}}$ has two prominent features. One is located slightly below $E_{\text{F}}$ and the other is located at around $E_{\text{B}}\sim 0.4$ eV. The intensity of the former peak increases as the photon energy is increased. These peaks are ascribed to the contributions mainly from the U $5f$ and Fe $3d$ states since the relative photoionization cross section of the U $5f$ state to the Fe $3d$ state increases from 0.55 to 2.48 as is increased from $h\nu=400$ to 800 eV.³⁵⁾ This is consistent with the calculated DOS, although the peak position of Fe $3d$ is slightly deeper in the band-structure calculation than in the experimental spectrum. Furthermore, the spectra are accompanied by a broad shoulder around $E_{\text{B}}\sim 1.5$ eV. The main contribution to this peak is from Fe $3d$ states since a similar peak structure is observed in the calculated DOS. The overall agreement between the AIPES spectra and the result of the band-structure calculation is fairly good, indicating that the electronic structure of UFeGa₅ can be described by the band-structure calculation.

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Figure 11. (Color online) Valence band spectra of UMGa₅ (M = Fe, Pt) and the calculated pDOS. The calculated pDOS are multiplied by the Fermi–Dirac function and are broadened with the experimental energy resolution. (a) Valence band spectra measured at $h\nu=800$ and 400 eV and the calculated U $5f$ and Fe $3d$ pDOS of UFeGa₅. The relative ionization cross section of U $5f$ and Fe $3d$ orbitals ($\sigma_{\text{U $5f$}}/\sigma_{\text{Fe $3d$}}$) at each photon energy is also shown. (b) Valence band spectrum measured at $h\nu=800$ eV and the calculated U $5f$ and Pt $5d$ pDOS of UPtGa₅.

Figure 11(b) displays the same comparison for UPtGa₅. The sample temperature was 20 K and the sample was in the AFM phase. There are two distinct features located at around $E_{\text{B}} = E_{\text{F}}$ and 2–6 eV. A comparison with the calculated pDOS suggests that the former corresponds to the contributions mainly from U $5f$ states while the latter corresponds to the contributions mainly from Pt $5d$ states. A major difference from the case of UFeGa₅ is that the transition metal d states are well separated from $E_{\text{F}}$ in UPtGa₅. Therefore, the f–d hybridization should be smaller in UPtGa₅ than in UFeGa₅.

6.3.2 Band structure and Fermi surface of UFeGa₅

Figure 12(a) displays the experimental band structures and Fermi surface of UFeGa₅.³¹⁾ The spectra are measured at $h\nu=500$ eV, and they correspond to the scan within the Z–R–A high-symmetry plane. The Fermi surface mapping is obtained by integrating the photoemission intensity within $E_{\text{F}}\pm 50$ meV, and the image is symmetrized along the $k_{x}$- and $k_{y}$-directions. The ARPES spectra are divided by the Fermi–Dirac function convoluted with the instrumental energy resolution to show the states near $E_{\text{F}}$ clearly. The ARPES spectra have clear energy dispersions. In particular, features located in the vicinity of $E_{\text{F}}$ show strong momentum dependence, suggesting that quasi-particle bands with large contributions from U $5f$ states form Fermi surfaces. On the other hand, the main contribution to the bands on the high-binding-energy side is from the Fe $3d$ states. In the Fermi surface mapping, there is an arc of a circular shape around the A point. The ARPES spectra measured along the Z–A high-symmetry line suggest that the outer and the inner boundaries of the circular shape correspond to the lattice-like Fermi surface and cylindrical Fermi surface, respectively. In addition, there is an enhanced intensity at around the Z point in the Fermi surface mapping. There are some dispersive features around the Z point in the ARPES spectra, indicating that there is also a Fermi surface around the Z point.

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Figure 12. (Color online) Experimental and calculated Fermi surface and band structure of UFeGa₅. (a) Experimental Fermi surface mapping and band structure within the Z–R–A high-symmetry plane measured at $h\nu=500$ eV. (b) Result of the band-structure calculation and the simulation of ARPES spectra within the Z–R–A high-symmetry plane. (c) Three-dimensional shape of the calculated Fermi surface. Data replotted from Ref. 31.

Figures 12(b) and 12(c) show the band structure and Fermi surface mapping based on the band-structure calculation and the shape of the three-dimensional Fermi surface, respectively. In the band-structure calculation, band 15 forms a Fermi surface in this compound. There are two kinds of Fermi surface within the Z–R–A high-symmetry plane. One is an electron-type Fermi surface with a lattice shape, and the other is an electron-type Fermi surface with a circular shape centered at the A point, which are indicated as shaded areas in Fig. 12(c). The electron-type Fermi surface around the A point corresponds to the cross-section of the cylindrical Fermi surface along the A–M–A high-symmetry line. Comparison between the experimental ARPES spectra and the calculation suggests that there are good correspondences between them. For example, around the A point, the dispersive features in the band structure closely correspond between the experiment and the calculation. For the Fermi surface mapping, the cylinder and the lattice Fermi surfaces exist in both experiment and the calculation. Furthermore, the features at around the Z point also have a similar shape between the experiment and the calculation. In addition, the features located on the high-binding-energy side ($E_{\text{B}}\sim 0.2{\text{--}}0.5$ eV), whose main contributions are from Fe $3d$ states, also have one-to-one correspondences between the experiment and the calculation. Therefore, most of the prominent features in the experimental spectra are well described by the band-structure calculations, suggesting that the basic topology of the Fermi surface of UFeGa₅ can be explained by the calculation. Meanwhile, the agreement between the experiment and the calculation is less good than the cases of UB₂ and UN. In particular, the experimental band structures in the vicinity of $E_{\text{F}}$ have more enhanced intensities than those in the calculation. One possible reason for this is a weak but finite contribution from the electron correlation effect in this compound, which might renormalize U $5f$ bands. Weak but finite electron correlation effect is also suggested by the core-level spectra as shown in Sect. 7.

6.3.3 Comparison with isostructural Ce-based compounds

Here, we consider the relationship between the present results and other compounds with the same crystal structure. The present results showed that there is a lattice-like electron Fermi surface around the Z point and a cylindrical electron Fermi surface centered along the A–M–A high-symmetry line. Cylindrical Fermi surfaces have been observed even in the localized compound CeRhIn₅ and non f-compounds such as LaRhIn₅ by dHvA and ARPES experiments.⁹⁰^,⁹¹⁾ The band structure calculation also shows that the contribution from f-states is very small in this cylindrical Fermi surface. Therefore, the existence of the cylindrical Fermi surface is almost completely unrelated to the itinerant or localized nature of the f-electrons in these compounds. On the other hand, lattice-like Fermi surfaces have not been observed in CeRhIn₅, CeIrIn₅, and CeCoIn₅ by ARPES⁹⁰^,⁹²^,⁹³⁾ and dHvA experiments.⁹⁴^–⁹⁶⁾ The band-structure calculations suggest that the lattice-like Fermi surface has large contributions from f-states. Therefore, the existence of the lattice-like Fermi surface is an indication of the itinerant nature of f-electrons in the series of compounds. Accordingly, UFeGa₅ has very itinerant U $5f$ states, and the nature of the Fermi surface is different from that in the isostructural Ce-based compounds such as CeRhIn₅, CeIrIn₅, and CeCoIn₅.

6.4 Brief summary of itinerant compounds

We have discussed the results for three itinerant compounds, UB₂, UN, and UFeGa₅. In UB₂ and UN, the band structures as well as the Fermi surface were quantitatively explained by the band-structure calculation treating all U $5f$ electrons as itinerant. Meanwhile, the electronic structure of UFeGa₅ is somewhat complicated, and the agreement between the experiment and the calculation is qualitative. Note that the origins of the itinerant U $5f$ states in these compounds are different in terms of the U–U separation. In the case of UB₂, the U–U distance (3.124 Å) is much smaller than the Hill limit (∼3.4 Å) and the direct overlaps of U $5f$ wave functions are dominant. In the case of UN, the U–U distance is 3.46 Å, which is almost equal to the Hill limit. Therefore, both direct U–U overlaps and f-ligand hybridization are essential in this compound. In the case of UFeGa₅, the U–U distance is 4.261 Å, which is much larger than the Hill limit. This suggests that the f-ligand hybridization is dominant in this compound. All these compounds have different f–f overlaps, but they have basically itinerant U $5f$ states, and the band-structure calculation can describe their basic electronic structures.

6.5 UPd₃

In this subsection, the electronic structure of UPd₃ is discussed to understand the case of localized U $5f$ electrons. UPd₃ is a unique material among the metallic uranium compounds. Its electronic structure was identified as having the localized $5f^{2}$ configuration by observation of the CEF excitation in a neutron scattering experiment.⁹⁷⁾ There have been some ARPES studies on UPd₃,⁹⁸^,⁹⁹⁾ but the energy position of the U $5f$ states is rather controversial. Here, we discuss the electronic structure of UPd₃ studied by SX-ARPES.⁵²⁾ Figures 13(a) and 13(b) show the crystal structure and Brillouin zone of UPd₃, respectively. It has a double-hexagonal-type crystal structure with two chemically inequivalent uranium sites.¹⁰⁰⁾ The lattice constants are $a=5.757$ Å and $c=9.621$ Å.¹⁰¹⁾ The Brillouin zone of UPd₃ also has a hexagonal structure, but the $c^{\ast}$-axis is relatively shorter than that for other hexagonal uranium compounds such as UPd₂Al₃, UNi₂Al₃, and UPt₃ owing to its double hexagonal nature.

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Figure 13. (Color online) Crystal structure and Brillouin zone of UPd₃. (a) Double hexagonal crystal structure of UPd₃. (b) Hexagonal Brillouin zone of UPd₃.

6.5.1 Valence band spectrum of UPd₃

Figure 14 shows the valence band spectra of UPd₃ measured at $h\nu=1125$ and 560 eV. These spectra measured at two photon energies have different sensitivities to the U $5f$ and Pd $4d$ orbitals. The relative photoionization cross sections of U $5f$ states to Pd $4d$ states obtained by the atomic calculation ($\sigma_{\text{U $5f$}}/\sigma_{\text{Pd $4d$}}$) at $h\nu=1125$ and 560 eV are about 0.72 and 0.48, respectively.³⁵⁾ We subtracted the spectrum measured at $h\nu=560$ eV from that measured at $h\nu=1125$ eV to extract the contributions only from U $5f$ states. The two spectra are normalized to the peak height of each spectrum at around $E_{\text{B}}\sim 3.4$ eV. The difference spectrum, which represents the contribution mainly from U $5f$ states, is indicated in the middle of Fig. 14. The spectra have two main features located at around $E_{\text{B}}\sim 0.8$ and 2 eV. It should be noted that the difference spectrum does not make a substantial contributions to $E_{\text{F}}$, which is consistent with the localized nature of U $5f$ states in UPd₃. At the bottom of Fig. 14, the Pd $4d$ pDOS obtained from the band-structure calculation of ThPd₃ is shown. The calculated pDOS is multiplied by the Fermi–Dirac function and is broadened with the experimental energy resolution. Since ThPd₃ does not have $5f$ electrons, the calculation corresponds to that of UPd₃ with a localized $5f$ electron configuration. The calculated Pd $4d$ pDOS closely corresponds to the spectra except for the contributions from the U $5f$ states.

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Figure 14. (Color online) Valence band spectra of UPd₃ measured at $h\nu=1125$ (solid line) and 560 eV (dashed line). The difference spectrum represents the contribution from U $5f$ states. The calculated Pd $4d$ pDOS of ThPd₃ is also shown. The calculated pDOS is multiplied by the Fermi–Dirac function and is broadened with the experimental energy resolution.

6.5.2 Band structure of UPd₃

Figure 15 summarizes the ARPES spectra of UPd₃. Figure 15(a) shows the ARPES spectra of UPd₃ measured along the Γ–K–M high-symmetry line with a photon energy of $h\nu=820$ eV. The prominent features located at $E_{\text{B}}\gtrsim 2.5$ eV are contributions mainly from Pd $4d$ states. In addition, there are nearly flat bands at around $E_{\text{B}}\sim 0.8$ and 2 eV. The U $5f$ partial spectrum is also shown in the right panel of Fig. 15(a). Since the energy positions of these bands coincide with the peaks in the U $5f$ partial spectrum, they are assigned as the localized U $5f$ states. Figure 15(b) shows the ARPES spectra measured along the A–H–L high-symmetry line with a photon energy of $h\nu=725$ eV. The basic structure of the ARPES spectra is similar to that of the spectra along the Γ–K–M high-symmetry line. There are similar flat bands at around $E_{\text{B}}\sim 0.8$ and 2 eV, and their positions correspond to the peaks in the U $5f$ partial spectrum shown in the right panel. To further understand the origin of these peak structures, the results of the band-structure calculation of ThPd₃ are shown in Figs. 15(c) and 15(d). The color coding of the bands represents the contributions from Pd $4d$ states. There are many bands, especially at $E_{\text{B}}\gtrsim 2$ eV, and they have strong contributions from Pd $4d$ states. Meanwhile, strongly dispersive bands exist in the region $E_{\text{B}}\lesssim 1.5$ eV, which have large contributions from Pd $5p$ states.⁵²⁾ Since there are many bands, especially in the region $E_{\text{B}}=2{\text{--}}4$ eV, comparisons between the experimental spectra and the calculation are rather difficult. Nevertheless, there are some similar features between the results of experiment and the calculation. For example, the weakly dispersive features located at $E_{\text{B}}\gtrsim 2$ eV have some similar shapes. Figures 15(e) and 15(f) display the simulations of ARPES spectra based on the LDA calculation of ThPd₃. Comparison of the ARPES spectra suggests that there is some agreement between them. The states located in the region $E_{\text{B}}\gtrsim 2$ eV have some correspondences to the ARPES spectra. In particular, the parabolic features centered at the Γ and A points agree with the ARPES spectra. Other features of Pd $4d$ states also have correspondences between the experimental spectra and the calculation. Meanwhile, nearly flat bands at $E_{\text{B}}\sim 0.8$ and 2 eV in the ARPES spectra do not appear in the calculation. This also suggests that these features are the contributions from localized U $5f$ states. It should be noted that the U $5f$ bands have a finite momentum dependence. Therefore, the localized U $5f$ states have some hybridizations with ligand Pd $4d$ states. The basic energy position of the U $5f$ states is consistent with that of the $5f^{1}$ final state multiplet state.⁵²⁾ Accordingly, the ARPES spectra of UPd₃ are consistent with its localized U $5f^{2}$ configuration.

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Figure 15. (Color online) Comparisons of ARPES spectra of UPd₃ and the the result of the band-structure calculation of ThPd₃. (a, b) ARPES spectra measured along Γ–K–M and A–H–L high-symmetry lines. The U $5f$ difference spectrum is also shown in the right panel. (c, d) Calculated band structure of ThPd₃ along Γ–K–M and A–H–L high-symmetry lines. Since ThPd₃ does not have $5f$ electrons, the calculation corresponds to that of UPd₃ with a localized $5f$ electron configuration. The color coding of the bands represents the contributions from Pd $4d$ states. (e, f) Simulations of ARPES spectra measured along Γ–K–M and A–H–L high-symmetry lines. Data replotted from Ref. 52.

6.6 UPd₂Al₃ and UNi₂Al₃

From this subsection, we now proceed to the results for uranium-based compounds with more interesting and complex physical properties. The first target materials are UPd₂Al₃ and UNi₂Al₃, which are heavy-fermion superconductors with relatively high superconducting transition temperature $T_{\text{SC}}$ among the uranium-based compounds. Figure 16(a) shows the crystal structure of UPd₂Al₃ and UNi₂Al₃. They have a common hexagonal crystal structure, which consists of alternately stacked U–Pd or U–Ni layers and Al layers along the c-axis. Their lattice constants are $a=5.365$ Å and $c = 4.186$ Å for UPd₂Al₃⁶⁰⁾ and $a=5.207$ Å and $c = 4.018$ Å for UNi₂Al₃.⁵⁷⁾ UNi₂Al₃ has smaller lattice constants than UPd₂Al₃. Figure 16(b) represents the hexagonal Brillouin zone of UPd₂Al₃ and UNi₂Al₃ in the paramagnetic phase. They undergo transition to AFM phases below Neel temperatures of $T_{\text{N}} = 14$ K (UPd₂Al₃) and 4.6 K (UNi₂Al₃) and to superconducting states below $T_{\text{SC}} = 2$ K (UPd₂Al₃) and 1 K (UNi₂Al₃).⁵⁷^,⁶⁰⁾ The coexistence of the local magnetic moment and the unconventional superconductivity in UPd₂Al₃ was confirmed by a neutron scattering experiment¹⁰²⁾ as well as by NQR experiments.¹⁰³⁾

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Figure 16. (Color online) Crystal structure and Brillouin zone of U$M_{2}$Al₃ (M = Ni and Pd). (a) Hexagonal crystal structure of U$M_{2}$Al₃. (b) Hexagonal Brillouin zone of U$M_{2}$Al₃ in the paramagnetic phase.

6.6.1 Itinerant and localized pictures for UPd₂Al₃ and UNi₂Al₃

Although there are many similarities between these two compounds, distinct differences exist in their transport properties. The electrical resistivity and magnetic susceptibility of UPd₂Al₃ can be described by assuming the CEF model with the U $5f^{2}$ electronic configuration, and it has often been argued that the U $5f$ electrons in UPd₂Al₃ have a localized nature.¹⁰⁴⁾ Meanwhile, the temperature dependence of the magnetic susceptibility of UNi₂Al₃ cannot be understood within the CEF scheme,¹⁰⁵⁾ implying that U $5f$ electrons have a rather itinerant nature in UNi₂Al₃. The natures of magnetism and superconductivity in these two compounds are also different. They show AFM ordering with a local magnetic moment of 0.85 $\mu_{\text{B}}$ (UPd₂Al₃) and 0.24 $\mu_{\text{B}}$ (UNi₂Al₃). The ordering vectors are commensurate $\boldsymbol{{Q}}=(0,0,0.5)$ for UPd₂Al₃¹⁰⁶⁾ and incommensurate $\boldsymbol{{Q}}=(\pm 0.5\pm\delta,0,0.5)$ with $\delta =0.11$ for UNi₂Al₃.¹⁰⁷⁾ The pairing symmetry of the superconductivity is considered to be a singlet in UPd₂Al₃ and a triplet in UNi₂Al₃.¹⁰⁵^,¹⁰⁸⁾ Therefore, how the electronic structures of these compounds are similar or different is also an interesting point in understanding the interplay of the magnetism and superconductivity in uranium-based compounds.

There have been many studies on the electronic-structure of these compounds. dHvA experiments were reported for UPd₂Al₃,¹⁰⁹⁾ and the results were compared with the band-structure calculation treating all U $5f$ electrons as itinerant.¹¹⁰⁾ On the other hand, the dual model, which assumes itinerant and localized subsystems in U $5f$ states, has been proposed to account for both the pronounced itinerant and localized properties.¹¹¹⁾ This model explains the coexistence of the unconventional superconductivity and the magnetic ordering. Furthermore, the calculation with the partially itinerant and localized model (itinerant U $5f^{1}$ + localized U $5f^{2}$) can reproduce the branches observed in dHvA experiments.¹¹²⁾ There has been only one dHvA study on UNi₂Al₃.¹¹³⁾ One branch was observed, but its correspondence to the calculated Fermi surface was not clear. Therefore, the electronic structure of UNi₂Al₃ is not yet well understood.

6.6.2 Valence band spectra of UPd₂Al₃ and UNi₂Al₃

Figure 17 summarizes the valence band spectra measured at $h\nu=800$ eV and the calculated DOS of UPd₂Al₃ and UNi₂Al₃.⁵⁸^,⁶¹⁾ The sample temperature was 20 K in both cases and the samples was in the paramagnetic phase. Figure 17(a) shows the valence band spectrum and the U $5f$ and Pd $4d$ pDOSs obtained by the band-structure calculation treating all U $5f$ electrons as itinerant. The spectrum has a sharp peak at around $E_{\text{F}}$ and complex features on the high-binding-energy side ($E_{\text{B}}\gtrsim 2.5$ eV). A comparison with the calculated pDOSs suggests that the peak at $E_{\text{F}}$ and the features on the high-binding-energy side correspond to the U $5f$ states and Pd $4d$ states, respectively. The overall features such as the sharp peak at $E_{\text{F}}$ as well as the multi peak structures in the Pd $4d$ states are well explained by the band-structure calculation.

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Figure 17. (Color online) Valence band spectra measured at $h\nu=800$ eV and calculated pDOS of U$M_{2}$Al₃. The sample temperature was 20 K and the samples were in the paramagnetic phase. The calculated pDOS are multiplied by the Fermi–Dirac function and are broadened with the experimental energy resolution. (a) Valence band and calculated U $5f$ and Pd $4d$ pDOS of UPd₂Al₃. (b) Valence band and calculated U $5f$ and Ni $3d$ pDOS of UNi₂Al₃.

Figure 17(b) displays the valence band spectrum of UNi₂Al₃ and the U $5f$ and Ni $3d$ pDOSs obtained by the band-structure calculation. The calculated pDOS are multiplied by the Fermi–Dirac function and are broadened with the experimental energy resolution. The shape of the spectrum in the vicinity of $E_{\text{F}}$ is very similar to that of UPd₂Al₃. On the other hand, there are complex peak structures on the high-binding-energy side ($E_{\text{B}}\gtrsim 1$ eV). A comparison with the calculated pDOSs suggests that the former and the latter correspond to the contributions mainly from U $5f$ and Ni $3d$ states, respectively. It is shown that the Ni $3d$ DOS has lower binding energies than the Pd $4d$ states. This suggests that the f–d hybridization is more enhanced in UNi₂Al₃ than in UPd₂Al₃, leading to the more itinerant nature of U $5f$ states in UNi₂Al₃. The overall agreement between the AIPES spectrum and the calculated pDOS is also fairly satisfactory, suggesting that U $5f$ electrons in UNi₂Al₃ have essentially itinerant character.

6.6.3 Band structures of UPd₂Al₃ and UNi₂Al₃

To understand the detailed band structures and Fermi surfaces of UPd₂Al₃ and UNi₂Al₃, SX-ARPES was performed for these two compounds.⁵⁸^,⁵⁹^,⁶¹⁾ Figure 18 shows the ARPES spectra of UPd₂Al₃ and UNi₂Al₃ in comparisons with the results of the band-structure calculation. The photon energies were $h\nu=600$ eV for UPd₂Al₃ and $h\nu=645$ eV for UNi₂Al₃, and the total energy resolution was about 160 meV. The sample temperatures was 20 K and the samples were in the paramagnetic phase. Figure 18(a) displays the ARPES spectra of UPd₂Al₃ measured along the A–H–L and Γ–K–M high-symmetry lines. Clear energy dispersions were observed in the ARPES spectra along both high-symmetry lines. On the high-binding-energy side ($E_{\text{B}}\gtrsim 0.5$ eV), some strongly dispersive bands were observed. In contrast, weakly dispersive bands with sharp peak structures were recognized in the vicinity of $E_{\text{F}}$. The corresponding energy band dispersions obtained by the band-structure calculation and the simulations of ARPES spectra based on the calculation are respectively shown in Figs. 18(b) and 18(c). The color coding of the bands in Fig. 18(b) represents the contributions from the U $5f$ states and Pd $4d$ states. The experimental ARPES spectra have some bands corresponding to those in the calculation. The strongly dispersive bands located at $E_{\text{B}}\gtrsim 0.5$ eV have good one-to-one correspondences between ARPES spectra and the results of the band-structure calculation along both the A–H–L and Γ–K–M high-symmetry lines. For example, the inverted parabolic band centered at the Γ point with its apex at $E_{\text{B}}\sim 1.5$ eV as well as the parabolic band centered at the M point with its bottom at $E_{\text{B}}\sim 2.0$ eV have corresponding features in the calculation. The states in the vicinity of $E_{\text{F}}$ also have very similar shapes. In particular, the feature at around the A and Γ points closely correspond to the calculation. These results indicate that the overall electronic structure of UPd₂Al₃ is explained by the band-structure calculation treating all U $5f$ electrons as itinerant.

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Figure 18. (Color online) Experimental and calculated band structures of UPd₂Al₃ and UNi₂Al₃. The sample temperature was 20 K and the samples were in the paramagnetic phase. (a) ARPES spectra of UPd₂Al₃ measured along the A–H–L and Γ–K–M high-symmetry lines. Data replotted from Ref. 61. (b) Calculated energy band dispersions. The color coding of the bands represents the contributions of U $5f$ and Pd $4d$ states. (c) Simulation of ARPES spectra based on the band-structure calculation. (d–f) Corresponding results for UNi₂Al₃.

Figures 18(d)–18(f) show the ARPES spectra and the results of the band-structure calculation for UNi₂Al₃. It is shown that the ARPES spectra of UNi₂Al₃ also display clear energy dispersions. One of the major differences from the case of UPd₂Al₃ is that Ni $3d$ states are located at around $E_{\text{B}}\sim 1{\text{--}}2$ eV. The smaller f–d energy separation in UNi₂Al₃ than that in UPd₂Al₃ leads to stronger f–d hybridization, and this results in the more itinerant nature of U $5f$ states in UNi₂Al₃ than in UPd₂Al₃. The agreement between the experiment and the calculation is fairly good, similar to the case of UPd₂Al₃. For example, the features originating from the Ni $3d$ bands located at around $E_{\text{B}}\gtrsim 1$ eV and the U $5f$ bands located in the vicinity of $E_{\text{F}}$ closely correspond to the calculated band dispersions. These results indicate that the overall electronic structures of UPd₂Al₃ and UNi₂Al₃ are explained by the band-structure calculation treating all U $5f$ electrons as itinerant. In addition, there are certain similarities between the ARPES spectra of UPd₂Al₃ and UNi₂Al₃, although the energy positions of the d bands are different. In particular, the band structures in the vicinity of $E_{\text{F}}$ are very similar between these two compounds.

6.6.4 Near-$E_{\text{F}}$ band structures of UPd₂Al₃ and UNi₂Al₃

To discuss in detail the band structures in the vicinity of $E_{\text{F}}$, enlargements of the ARPES spectra and the calculations are shown in Fig. 19. Figures 19(a) and 19(b) display a comparison of ARPES spectra and the calculated band dispersions, as well as the simulation of ARPES spectra based on the band-structure calculation. These spectra are divided by the Fermi–Dirac function broadened by the instrumental energy resolution to eliminate the effect of the Fermi–Dirac cutoff. The behaviors of bands in the vicinity of $E_{\text{F}}$ can be observed in these figures. Figure 19(c) shows the three-dimensional shape of the Fermi surface of UPd₂Al₃ obtained by the band-structure calculation. In the band-structure calculation, bands 18–20 form Fermi surfaces. Band 18 forms the cylindrical Fermi surface centered along the A–Γ–A high-symmetry line and the water-drop-like Fermi surface centered around the H point. Bands 19 and 20 form small electron pockets around the A point.

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Figure 19. (Color online) Experimental and calculated band structures of UPd₂Al₃ and UNi₂Al₃ in the vicinity of $E_{\text{F}}$. (a) Enlargements of ARPES spectra measured along the A–H–L and Γ–K–M high-symmetry lines. The spectra are divided by the Fermi–Dirac function broadened by the total energy resolution. The inset shows the spectra normalized by the area of the MDC. (b) Band structures and the simulations of ARPES spectra based on the LDA. (c) Three-dimensional shape of the Fermi surface of UPd₂Al₃. (d–f) Corresponding results for UNi₂Al₃.

An electron-type Fermi surface was observed around the A point in the experimental ARPES spectra measured along the A–H–L high-symmetry line, and there is a similar feature in the band-structure calculation. Although the features are very similar between the experimental ARPES spectra and the band-structure calculation, their sizes and the number of Fermi surfaces are different. The inset of Fig. 19(a) shows the spectra at around the A point, but their intensities are normalized by the area of the MDC. It is shown that there is only one electron-type Fermi surface around the A point. This is a part of the cylindrical Fermi surface, whose shape is very similar to that of band 18 in the band-structure calculation. On the other hand, small electron pockets, which correspond to bands 19 and 20, do not exist in the experimental ARPES spectra. In addition, the water-drop-like Fermi surface around the H point was also not observed. These results suggest that although a cylindrical Fermi surface exists, the overall topology of experimental Fermi surface is considerably different from that indicated by the band-structure calculation. Blackburn et al. pointed out that these small electron pockets around the A point are responsible for the superconductivity in UPd₂Al₃.¹¹⁴⁾ Experimentally, small electron pockets around the A point, such as those formed by bands 19 and 20, have been not observed in ARPES experiments, and therefore the scenario does not appear to be applicable for UPd₂Al₃.

Figure 19(d) shows the ARPES spectra of UNi₂Al₃ measured along the A–H–L and the Γ–K–M high-symmetry lines. The essential structures of the spectra in the vicinity of $E_{\text{F}}$ are very similar to those of UPd₂Al₃. There is an electron pocket around the A point, whose size is larger than that of UPd₂Al₃. This characteristic feature is consistent with the results of the band-structure calculation shown in Fig. 18(e). On the other hand, the small electron pockets around the A point in the band-structure calculation were not observed in the experimental ARPES spectra. The inset of Fig. 19(d) shows the spectra around the A point, where the intensities are normalized with the area of the MDC. There is one electron-like Fermi surface around the A point, as in the case of UPd₂Al₃. This is inconsistent with the band-structure calculation, and the situation is very similar to the case of UPd₂Al₃. The electronic structure around the Γ point is also very similar to that of UPd₂Al₃, and therefore these small electron pocket are part of the cylindrical Fermi surface centered along the A–Γ–A high-symmetry line.

Accordingly, it was shown that the topologies of the Fermi surface are very similar between UPd₂Al₃ and UNi₂Al₃. The overall electronic structures of both compounds are explained by the band-structure calculation treating all U $5f$ electrons as itinerant. The cylindrical Fermi surfaces centered along the A–Γ–A line were observed in both compounds, and they are consistent with the band-structure calculation. On the other hand, there is some disagreement between the experiment and the calculation, particularly in the vicinity of $E_{\text{F}}$. For example, there are no small electron pockets around the A point in both compounds as predicted by the band-structure calculation. The bottoms of these bands in the calculation are less than 50 meV, and even very small electron correlation effects should remarkably change their behaviors. Therefore, although the band-structure calculation is an appropriate starting point to describe the electronic structures of UPd₂Al₃ and UNi₂Al₃, there are some deviations in the topologies of the Fermi surfaces, which might originate from the electron correlation effect.

6.6.5 Itinerant-to-localized transition in UPd₂Al₃

We have shown that the U $5f$ electrons in UPd₂Al₃ have an itinerant character at low temperatures ($T=20$ K). On the other hand, the transport properties of UPd₂Al₃ shows dense Kondo-like behaviors with the coherent temperature $T_{\text{coh}}\sim 50$ K, suggesting that U $5f$ electrons have localized character at high temperatures. For example, the behaviors of magnetic susceptibility¹¹⁵⁾ and $1/T_{1}$ in the nuclear quadrupole resonance (NQR) of ²⁷Al¹¹⁶⁾ follow the Curie–Weiss law above $T\gtrsim 40{\text{--}}50$ K. To understand the nature of U $5f$ states at high temperatures, ARPES spectra were measured not only at 20 K but also at 100 K.

Figure 20 summarizes the temperature dependence of the ARPES spectra of UPd₂Al₃.⁵⁸⁾ Figure 20(a) shows the ARPES spectra of UPd₂Al₃ measured along the A–H–L high-symmetry line at a low temperature ($T=20$ K). The photon energy was $h\nu=595$ eV and the total energy resolution was 120 meV. The spectra are essentially identical to those measured at $h\nu=600$ eV shown in Fig. 19(a), and itinerant quasi-particle bands exist in the vicinity of $E_{\text{F}}$. Figure 20(b) shows the same ARPES spectra measured at a high temperature ($T=100$ K). The spectra are normalized with the area of the EDC spectra, and they are shown with the same intensity scale as the spectra measured at 20 K. It is shown that the essential band structures did not show significant changes with the temperatures. For instance, the features on the high-binding-energy side ($E_{\text{B}}\gtrsim 1$ eV) are essentially the same as those at the low temperature. On the other hand, some distinct differences were observed in the vicinity of $E_{\text{F}}$.

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Figure 20. (Color online) Temperature dependence of ARPES spectra of UPd₂Al₃. (a) ARPES spectra of UPd₂Al₃ measured along the A–H–L high-symmetry line with $h\nu=595$ eV at $T=20$ K. (b) ARPES spectra measured at $T=100$ K. (c) Enlargement of ARPES spectra measured at $T=20$ K. The spectra are divided by the Fermi–Dirac function broadened by the instrumental energy resolution. (d) Enlargement of ARPES spectra measured at $T=100$ K. Data replotted from Ref. 58.

To examine these changes in detail, the enlargements of the ARPES spectra measured at $T=20$ and 100 K are shown in Figs. 20(c) and 20(d), respectively. These spectra have been divided by the Fermi–Dirac function broadened by the instrumental energy resolution to show the structure in the vicinity of $E_{\text{F}}$. Some distinct differences can be recognized between the spectra measured at the two temperatures. In particular, the behaviors of the quasi-particle bands in the vicinity of $E_{\text{F}}$ are considerably different. For example, there is an electron pocket around the A point at the low temperature but its intensity is suppressed, and its spectral weight is shifted toward the higher-binding-energy side at the high temperature. Furthermore, a flat feature appears around the H point below $E_{\text{F}}$ at the high temperature, which was not observed at the low temperature. These results imply that itinerant U $5f$ bands in the vicinity of $E_{\text{F}}$ become nearly flat bands below $E_{\text{F}}$ at high temperatures. This result can be interpreted as the transition of itinerant quasi-particle bands at low temperatures into localized bands at high temperatures. The changes were observed not only in the vicinity of $E_{\text{F}}$ but also on the high-binding-energy side. For example, the features around the H point located at $E_{\text{B}}\sim 0.5{\text{--}}0.8$ eV have different shapes at the two temperatures. The peak centered at the L point located at $E_{\text{B}}\sim 1$ eV also shifted toward the higher-binding-energy side for the high temperature. These changes are caused by the changes in the nature of U $5f$ states, but their microscopic mechanism is not yet well understood. An important point to note is that these changes were not observed in the ARPES spectra of UNi₂Al₃ between 20 and 100 K.⁵⁸⁾ The transport properties of UNi₂Al₃ do not show significant temperature dependences between these two temperatures, and the absence of these changes supports the hypothesis that the temperature dependences observed in UPd₂Al₃ originate from the changes of its electronic structure.

6.7 URu₂Si₂

URu₂Si₂ is also a heavy-fermion superconductor, for which $T_{\text{SC}} = 1.5$ K.³⁾ It exhibits a second-order phase transition at $T=17.5$ K, but its order parameter is not yet well understood. This transition is called the HO transition and has been one of the long-standing mysteries in the field of condensed matter physics. Figure 21 shows the crystal structure and Brillouin zones of URu₂Si₂ in the paramagnetic and the virtual AFM phases. URu₂Si₂ has a tetragonal ThCr₂Si₂-type crystal structure with $a = 4.124$ Å and $c = 9.582$ Å³⁾ as shown in Fig. 21(a). Its body-centered tetragonal Brillouin zone is shown in Fig. 21(b). Since the HO state is adjacent to the AFM state with the magnetic ordering vector of $\boldsymbol{{Q}}=(0,0,1)$, some studies utilized the virtual AFM Brillouin zone indicated in Fig. 21(c).¹¹⁷^,¹¹⁸⁾ The relationship between these two representations can be understood from these figures. An important point to note is that the position of the X point is different in the PM and AFM phases. This often causes confusion, and one needs to pay special attention to the type of Brillouin zone utilized in each study.

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Figure 21. (Color online) Crystal structure and Brillouin zones of URu₂Si₂. (a) Body-centered tetragonal crystal structure of URu₂Si₂. (b) Brillouin zone of URu₂Si₂ in the paramagnetic phase. (c) Brillouin zone of URu₂Si₂ in the virtual AFM phase with the ordering vector of $\boldsymbol{{Q}}=(0,0,1)$. It should be noted that the position of the X point is different in the paramagnetic and AFM phases.

6.7.1 ARPES studies on URu₂Si₂

Although there has been an enormous number of experimental and theoretical studies on the order parameter of the HO phase, it has not yet been identified.⁴^,⁵⁾ A number of photoemission studies have also been carried out, especially in recent years, to reveal its electronic structure,¹¹⁹^–¹²⁵⁾ as recently reviewed by Durakiewicz.¹²⁶⁾ The first ARPES experiment on URu₂Si₂ in the paramagnetic phase was reported by Itoh et al., who used the He I ($h\nu=21.2$ eV).¹²⁷⁾ Immediately after this study, a more detailed ARPES study in the paramagnetic phase using synchrotron radiation with photon energies of $h\nu=21{\text{--}}120$ eV was reported by Denlinger et al.⁹⁾ These experiments were performed at the most surface-sensitive photon energies, and the contributions from the bulk electronic structures were not clearly distinguished. The electronic structure in the HO phase was first reported by Santandar-Syro et al., who performed ARPES experiments with the He I ($h\nu=21.2$ eV) with energy resolution of $\Delta E\sim 5$ eV.¹¹⁹⁾ They observed that a narrow band appears in the vicinity of $E_{\text{F}}$ in the HO phase. More detailed behavior of this band was clarified by Yoshida et al. using a laser light source ($h\nu=7$ eV) with energy resolution of $\Delta E\sim 2$ eV.¹²⁰⁾ Similar changes in the ARPES spectra in the HO phase have also been reported by other group, who used synchrotron radiation with $h\nu= 7{\text{--}}69$ eV.¹²²^–¹²⁵⁾ Therefore, it is now well accepted that there are certain changes in the electronic structure in the vicinity of $E_{\text{F}}$ due to the HO transition. On the other hand, these studies utilized surface-sensitive photon energies, and the intrinsic bulk electronic structure has not yet been identified. In particular, the electronic structure in the paramagnetic phase is not well understood, and this makes it difficult to establish a theoretical model of the HO transition. Here, we discuss the electronic structure of URu₂Si₂ in the paramagnetic phase measured by soft X-ray ARPES.⁵⁴⁾

6.7.2 Valence band spectrum of URu₂Si₂

Figure 22 shows the valence band spectrum measured at $h\nu=800$ eV and the calculated U $5f$ and Ru $4d$ pDOS broadened with the instrumental energy resolution. The sample temperature was 20 K and the sample was in the paramagnetic phase. The spectrum has a sharp peak at $E_{\text{F}}$ and complex features on the high-binding-energy side. Comparison with the calculated pDOS suggests that the former and latter correspond to contributions from U $5f$ and Ru $4d$ states, respectively. The overall spectral shape is well explained by the band-structure calculation, suggesting that U $5f$ electrons in URu₂Si₂ essentially have an itinerant character.

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Figure 22. (Color online) Valence band spectrum measured at $h\nu=800$ eV and the calculated U $5f$ and Ru $4d$ pDOS of URu₂Si₂. The sample temperature was 20 K and the sample was in the paramagnetic phase. The calculated pDOS are multiplied by the Fermi–Dirac function and are broadened with the experimental energy resolution.

6.7.3 Band structure and Fermi surface of URu₂Si₂

Figure 23 summarizes the results of our SX-ARPES study as well as the band-structure calculation of URu₂Si₂. Figure 23(a) shows the ARPES spectra of URu₂Si₂ measured along the Γ–(Σ)–Z–X high-symmetry line at a photon energy of $h\nu=760$ eV. Clear energy dispersions were observed in the ARPES spectra. Strongly dispersive bands located at around $E_{\text{B}}\gtrsim 0.5$ eV are the contributions mainly from Ru $4d$ states. Furthermore, there are some bands that cross the Fermi energy. The positions of bands in the vicinity of $E_{\text{F}}$ deduced from the MDC spectra⁵⁴⁾ are shown by dashed lines in Fig. 23(a). Three bands exist, which form Fermi surfaces, and they are designated as A, B, and C. A two-dimensional scan of the ARPES spectra within the Γ–(Σ)–Z–X high-symmetry plane suggests that bands A and B form a hole-type Fermi surface around the Z point while band C forms an electron-type Fermi surface around the Γ point.

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Figure 23. (Color online) Experimental and calculated band structures and Fermi surfaces of URu₂Si₂ in the paramagnetic phase. (a) ARPES spectra of URu₂Si₂ in the paramagnetic phase measured along the Γ–(Σ)–Z–X high-symmetry line. The positions of bands deduced from MDC curve analysis are shown as bands A, B, and C. Data replotted from Ref. 54. (b) Three-dimensional shapes of Fermi surfaces deduced from the experimental ARPES spectra. (c) Energy band dispersions obtained from the band-structure calculation and the simulation of ARPES spectra. (d) Three-dimensional shapes of Fermi surfaces obtained by the band-structure calculation. (b, d) Reprinted figures 20(e) and 20(f) with permission from Ref. 18. © 2016 IOP Publishing.

The behavior of these bands was traced within the entire Brillouin zone, and the three-dimensional shapes of the Fermi surfaces were revealed. A schematic figure of the experimental Fermi surfaces obtained by the present ARPES study is shown in Fig. 23(b). Bands A and B form hole-type Fermi surfaces with a spheroidal shape around the Z point, while band C forms an electron-type Fermi surface with a similar shape. The two-dimensional Fermi-energy intensity map shows that band C was observed in very limited part of the Brillouin zone, while bands A and B were clearly observed in the entire Brillouin zone.⁵⁴⁾ This might be due to the matrix element effect. Although the shape and size of the Fermi surface formed by band C could not be fully determined experimentally, its volume should be equal to the total volume of the hole Fermi surfaces formed by bands A and B since URu₂Si₂ is a compensated metal with equal volumes of electron and hole Fermi surfaces. Accordingly, an ellipsoidal electron-type Fermi surface was assumed for band C as shown in Fig. 23(b).

Figure 23(c) displays the results of the band-structure calculation and the simulation of ARPES spectra based on the band-structure calculation. The color coding of the bands represents the contributions from U $5f$ and Ru $4d$ states. In the band-structure calculation, three bands form a Fermi surface along this high-symmetry line. Bands 17 and 18 form hole-type Fermi surfaces around the Z point, while band 19 forms an electron-type Fermi surface around the Γ point. Their basic topologies are very similar to the experimentally obtained ARPES spectra, although their sizes are different. The other features in the spectra on the high-binding-energy side ($E_{\text{B}}\gtrsim 0.5$ eV) have large contributions from Ru $4d$ states, and they have features corresponding to the experimental ARPES spectra. Figure 23(d) shows the three-dimensional shapes of the Fermi surfaces of URu₂Si₂ obtained by the band-structure calculation. There are two hole-type Fermi surfaces around the Z point and one electron-type Fermi surface around the Γ point. Although their sizes are different, the basic topologies of the Fermi surfaces are very similar to the experimental ones. Accordingly, the overall band structure as well as the Fermi surfaces of URu₂Si₂ were well explained by the band-structure calculation. On the other hand, it should be noted that the features in the vicinity of $E_{\text{F}}$ are much more complicated in the experimental spectra than in the calculation. For example, there is a nearly flat feature around the Z point slightly below $E_{\text{F}}$. This might have originated from a small electron pocket with an energy dispersion of less than 10 meV observed by very high resolution ARPES,¹²⁸⁾ or from the renormalization of bands due to finite electron correlation effects.

6.7.4 Itinerant nature of U $5f$ state in URu₂Si₂

These results lead to the following two important conclusions about the electronic structure of URu₂Si₂. First, U $5f$ electrons in URu₂Si₂ have an itinerant character in the paramagnetic phase, suggesting that the HO transition also originates from the itinerant U $5f$ electrons. The itinerant nature of U $5f$ electrons in URu₂Si₂ is consistent with other ARPES experiments as discussed in the recent review article.¹²⁶⁾ Second, the band-structure calculation can reproduce the basic topology of the Fermi surface of URu₂Si₂ in the paramagnetic phase, indicating that it is an appropriate starting point to describe its electronic structure. All these results suggest that the U $5f$ electrons essentially have an itinerant character, and they form Fermi surfaces in the paramagnetic phase as well as the HO phase. Therefore, the nature of the HO transition should take into account the itinerant nature of U $5f$ electrons.

6.8 UGe₂, URhGe, and UCoGe

The coexistence of ferromagnetic ordering and superconductivity is one of the significant characteristics of uranium-based compounds. Up to now, four such compounds, UGe₂,⁶⁴⁾ UIr,⁷¹⁾ URhGe,⁶⁷⁾ and UCoGe,⁶⁹⁾ have been discovered. UGe₂ and UIr have a superconducting state under high pressures while URhGe and UCoGe have a superconducting state at ambient pressure. Recently, unconventional critical behaviors of magnetization have been reported for UGe₂ and URhGe, suggesting that ferromagnetism and superconductivity are closely related in these compounds.¹²⁹⁾ One of the characteristics of these compounds is the low-symmetry nature of their crystal structures. UGe₂, URhGe, and UCoGe have orthorhombic structures and UIr has a monoclinic structure. Furthermore, UGe₂, URhGe, and UCoGe have zig-zag chains of uranium atoms, which have been pointed out to be responsible for their characteristic physical properties.¹³⁰^,¹³¹⁾ Although it is generally considered that U $5f$ electrons in these compounds have itinerant characters, other scenarios, such as a completely localized model¹³²⁾ and the dualism of U $5f$ states,¹³³⁾ have also been proposed. In a Shubnikov–de Haas experiment carried out on UCoGe, a small electron pocket with a large cyclotron effective mass ($25m_{0}$) was observed.¹³⁴⁾ However, the electronic structures of the above compounds are not yet well understood. In this section, we discuss the results on SX-ARPES studies of UGe₂, URhGe, and UGe₂.⁶⁶^,⁶⁸⁾

These compounds are ferromagnets with $T_{\text{C}} = 53$ K (UGe₂), 9.5 K (URhGe), and 3 K (UCoGe). UGe₂ is in the superconducting state below $T_{\text{SC}} = 0.8$ K under a pressure of 1.2 GPa,⁶⁴⁾ while URhGe and UCoGe are in the superconducting state below $T_{\text{SC}} =0.25$ and $0.8$ K at ambient pressure,⁶⁷^,⁶⁹⁾ respectively. Figure 24 shows the crystal structures and Brillouin zones of UGe₂, URhGe, and UCoGe. These compounds have orthorhombic crystal structures. UGe₂ has a body-centered orthorhombic structure with a larger lattice constant along the b-axis ($b = 15.0889$ Å) than along the a- and c-axes ($a=4.0089$ Å and $c = 4.0950$ Å)¹³⁵⁾ as shown in Fig. 24(a). Its Brillouin zone is shown in Fig. 24(b). URhGe and UCoGe have a simple orthorhombic structure with smaller lattice constants along the b-axis ($b = 4.330$ Å for URhGe and 4.206 Å for UCoGe) than those of a- and c-axes ($a=6.873$ Å and $c = 7.506$ Å for URhGe and $a=6.845$ Å and $c = 7.222$ Å for UCoGe)⁶⁹^,¹³⁶⁾ as shown in Fig. 24(c). Its Brillouin zone is indicated in Fig. 24(d).

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Figure 24. (Color online) Crystal structures and Brillouin zones of ferromagnetic superconductors UGe₂, URhGe, and UCoGe. (a) Body-centered orthorhombic crystal structure of UGe₂. The zig-zag chains of uranium atoms are along the a-axis. (b) Brillouin zone of UGe₂. (c) Simple orthorhombic crystal structure of URhGe and UCoGe. The zig-zag chains of uranium atoms are along the a-axis. (d) Brillouin zone of URhGe and UCoGe.

6.8.1 Valence band spectra of UGe₂, URhGe, and UCoGe

Figure 25 shows the valence band spectra measured at $h\nu=800$ eV and the calculated pDOS of these compounds.⁶⁶^,⁶⁸⁾ The sample temperatures were 120 K (UGe₂) and 20 K (URhGe and UCoGe), and all compounds were in the paramagnetic phase. The calculated pDOS are multiplied by the Fermi–Dirac function and are broadened with the experimental energy resolution. The valence band spectra of all three compounds show sharp peak structures at $E_{\text{F}}$, and comparisons with the calculated pDOS suggest that their main contributions are from itinerant U $5f$ electrons in these compounds. In the spectra of URhGe and UCoGe, large contributions exist from Rh $4d$ or Co $3d$ states on the high-binding-energy side, but their positions are somewhat different. The Co $3d$ states in UCoGe are located at around $E_{\text{B}}\sim 1$ eV while the Rh $4d$ states are located at around $E_{\text{B}}\sim 3$ eV in URhGe. Therefore, f–d hybridization should be enhanced in UCoGe compared with in URhGe. A Co $2p$–$3d$ resonant photoemission experiment suggested that U $5f$ states are strongly hybridized in UCoGe and that Co $3d$ states make a substantial contribution to $E_{\text{F}}$.⁶⁶⁾ The overall agreement between the valence band spectra and the calculated pDOS is fairly good, suggesting that the U $5f$ electrons in these compounds have itinerant character.

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Figure 25. (Color online) Valence band spectra measured at $h\nu=800$ eV and the calculated DOS of ferromagnetic superconductors (a) UGe₂, (b) URhGe, and (c) UCoGe. The sample temperatures were 120, 20, and 20 K for UGe₂, URhGe, and UCoGe, respectively, all of which are in the paramagnetic phase. The calculated pDOS are multiplied by the Fermi–Dirac function and are broadened with the experimental energy resolution.

6.8.2 Band structures of UGe₂, URhGe, and UCoGe

Figure 26 summarizes the results of the ARPES studies on UGe₂, URhGe, and UCoGe.⁶⁶^,⁶⁸⁾ Figure 26(a) shows a comparison of the ARPES spectra of UGe₂ measured along the T–Y–T high-symmetry line and the corresponding calculated band dispersions. The color coding of the calculated bands represents the contribution of U $5f$ states. The experimental ARPES spectra show complex energy dispersions. On the high-binding-energy side, strongly dispersive features exist. Their main contributions are from Ge s and p states, and they have some features corresponding to the calculated band dispersions. For example, the parabolic feature centered at the T point with its bottom at $E_{\text{B}}\sim 3$ eV closely corresponds to the calculated band dispersions. On the other hand, weakly dispersive features exist in the vicinity of $E_{\text{F}}$, whose main contributions are from U $5f$ states. To see these weakly dispersive features in detail, an enlargement of the ARPES spectra and the simulation of the ARPES spectra based on the band-structure calculation are shown in Fig. 26(b). These spectra have been divided by the Fermi–Dirac function broadened by the instrumental energy resolution to show the structure in the vicinity of $E_{\text{F}}$. Although some weakly dispersive bands with large contributions from U $5f$ states appear in the vicinity of $E_{\text{F}}$ in the band-structure calculation, their correspondences to the experimental ARPES spectra are not clear. The calculation predicts the existence of small electron pockets formed by band 41 around the Γ point. The experimental spectra are rather broad, and the agreement between the spectra and the calculation is unsatisfactory. The three-dimensional shape of the calculated Fermi surface of UGe₂ is shown in Fig. 26(c). It has a very complex shape, suggesting that the electronic structure of UGe₂ has a three-dimensional nature.

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Figure 26. (Color online) Experimental and calculated band structures of UGe₂, URhGe, and UCoGe. (a) Comparison of the ARPES spectra of UGe₂ measured along the T–Y–T high-symmetry line. The sample temperature was 120 K, and the sample was is in the paramagnetic phase. (b) The enlargement of the ARPES spectra of UGe₂ and the simulation of the ARPES spectra based on the band-structure calculation. The spectra are divided by the Fermi–Dirac function broadened by the instrumental energy resolution. (c) Three-dimensional shape of calculated Fermi surface of UGe₂. (d–f) Same as (a–c) but for URhGe measured along the S–Y–S high-symmetry line. The sample temperature was 20 K and the sample was in the paramagnetic phase. (g–i) Same as (a–c) but for UCoGe measured along the S–X–S high-symmetry line. The sample temperature was 20 K and the sample was in the paramagnetic phase. (a–c, g–i) Reprinted figures 2, 3, and 4 with permission from Ref. 66. © 2015 American Physical Society. (d–f) Reprinted figures 2, 4, and 5 with permission from Ref. 68. © 2014 American Physical Society.

Figures 26(d)–26(f) show the corresponding diagrams for URhGe. The spectra were measured along the S–Y–S high-symmetry line. The situation is very similar to the case of UGe₂ but there are better correspondences than in the case of UGe₂. As in the case of UGe₂, strongly dispersive bands exist on the high-binding-energy side and weakly dispersive features exist in the vicinity of $E_{\text{F}}$. One large difference from UGe₂ is that contributions from Rh $4d$ states exist at $E_{\text{B}}\gtrsim 2$ eV. The Rh $4d$ bands are well separated from $E_{\text{F}}$, suggesting that they do not make significant contributions to $E_{\text{F}}$. There is overall agreement between the experimental ARPES spectra and the band-structure calculation as shown in Fig. 26(d). Figure 26(e) shows a comparison of the ARPES spectra and the simulation in the vicinity of $E_{\text{F}}$. The dashed lines in the experimental spectra are the approximate positions of bands deduced from the analysis of MDC-normalized spectra. It is shown that the experimental spectra have certain similarities to those obtained by the calculation. In particular, an electron-like Fermi surface exists around the S point, which corresponds to bands 71 and 72 in the band-structure calculation. This Fermi surface corresponds to the pillar-like Fermi surface at the corner of the Brillouin zone as shown in Fig. 26(f).

Figures 26(g)–26(i) show the corresponding diagrams for UCoGe. The ARPES spectra were measured along the S–X–S high-symmetry line. The essential structure of the spectra is very similar to that of URhGe. There are contributions from the U $5f$ quasi-particle bands in the vicinity of $E_{\text{F}}$, and strongly dispersive bands on the high-binding-energy side, which are essentially the contributions from Co $3d$ states. The Co $3d$ bands are much closer to $E_{\text{F}}$ than the Rh $4d$ bands in URhGe, and they make finite contributions to $E_{\text{F}}$. The Co $2p$–$3d$ resonant photoemission measurement of UCoGe suggested that Co $3d$ states are strongly hybridized with U $5f$ states.⁶⁶⁾ The spectra in the vicinity of $E_{\text{F}}$ shown in Fig. 26(h) suggest that although there are certain similarities between the results of the experiment and the calculation, the agreement is not straightforward. Since the crystal structures of these compounds have a low-symmetry nature, the degeneracy of bands is removed, and very complicated band structures are expected.

All these results are summarized as follows. Although some characteristic features were explained by the band-structure calculation, the states in the vicinity of $E_{\text{F}}$ have very complicated structures, and the agreement is not as good as for the cases of heavy fermions compounds such as URu₂Si₂ and U$M_{2}$Al₃. This suggests that the shapes of the Fermi surfaces of these compounds are qualitatively different from the calculated shapes, which might be caused by the electron correlation effect in the complicated band structures of the low-symmetry crystals. Further detailed discussion is given in Refs. 66 and 68.

6.8.3 Ferromagnetic transition in URhGe

We next discuss the changes in the electronic structure due to the ferromagnetic transition. Figures 27(a) and 27(b) show the ARPES spectra of URhGe measured along the X–Γ–X line in the paramagnetic phase (20 K) and the ferromagnetic phase (6 K), respectively. Both spectra are normalized by the intensities of the Rh $4d$ bands located at $E_{\text{B}} >1.5$ eV. There are small but substantial temperature dependences in the ARPES spectra. In the vicinity of $E_{\text{F}}$, narrow quasi-particle bands exist, and the feature has small differences between the paramagnetic and ferromagnetic phases. Although their details were not resolved in the present spectra, the changes are presumably due to the splitting of bands into majority-spin and minority-spin bands in the ferromagnetic phase, as has been observed in UGe₂¹³⁷⁾ and UTe¹³⁸⁾ since U $5f$ electrons also have an itinerant nature in URhGe. In addition to this change, the intensities around $E_{\text{B}}\sim 1$ eV at the Γ point are lower in the ferromagnetic phase. This change might originate from the incoherent part of the spectrum due to the correlation effect of U $5f$ states. Riseborough suggested that the incoherent spin excitation produces an incoherent peak in the off-$E_{\text{F}}$ region in a weak ferromagnet near a quantum critical point.¹³⁹⁾

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Figure 27. (Color online) Temperature dependence of the ARPES spectra of URhGe. (a) ARPES spectra measured at 20 K (paramagnetic phase) and (b) 6 K (ferromagnetic phase). Data replotted from Ref. 68.

6.9 UIr

UIr is also a ferromagnetic superconductor for which the Curie temperature of $T_{\text{C}}=46$ K.⁷⁰⁾ It undergoes a superconducting transition at $T_{\text{SC}} = 0.14$ K under a pressure of 2.7–2.8 GPa.⁷¹⁾ Figure 28 shows the crystal structure of UIr and its Brillouin zone. UIr has a monoclinic crystal structure with lattice parameters of $a=5.604$ Å, $b=10.55$ Å, $c=5.573$ Å, and $\beta=99.12$°⁷⁰⁾ as shown in Fig. 28(a). It does not have an inversion symmetry, and there are four uranium and iridium sites in a unit cell. Its electronic structure is expected to be very complicated owing to this low-symmetry nature of the crystal structure. Its monoclinic Brillouin zone is indicated in Fig. 28(b). It should be noted that the $c^{\ast}$-axis (Γ–Z-direction) is parallel to the $k_{z}$-axis while the $a^{\ast}$-axis (Γ–B-direction) is not parallel to the $k_{x}$-axis. It was reported that UIr is an itinerant ferromagnet with Ising-like anisotropy.¹⁴⁰⁾ dHvA experiments have been performed on UIr, and many branches with heavy cyclotron effective masses were observed,¹⁴¹⁾ suggesting that heavy quasi-particle bands exist in this compound.

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Figure 28. (Color online) Crystal structure and Brillouin zone of UIr. (a) Monoclinic crystal structure of UIr, which does not have any plane symmetries. There are four inequivalent uranium and iridium atomic sites. The a-axis is assumed to be parallel to the x-axis in this figure. (b) Monoclinic Brillouin zone of UIr. The $c^{\ast}$-axis (Γ–Z-direction) is parallel to the $k_{z}$-axis while the $a^{\ast}$-axis (Γ–B-direction) is not parallel to the $k_{x}$-axis.

6.9.1 Valence band spectrum of UIr

Figure 29 shows the valence band spectrum of UIr and the calculated pDOS.⁷²⁾ The sample temperature was 60 K, and the sample was in the paramagnetic phase. The calculated pDOS are multiplied by the Fermi–Dirac function and are broadened with the experimental energy resolution. The experimental spectra have a sharp peak at $E_{\text{F}}$. On the high-binding-energy side ($E_{\text{B}}\gtrsim 1$ eV), complicated peak structures exist. Comparison with the calculated pDOS suggests that the features at $E_{\text{F}}$ and $E_{\text{B}}\gtrsim 1$ eV are contributions mainly derived from U $5f$ and Ir $5d$ states, respectively. The experimental spectrum agrees well with the calculated pDOS, although the structures of the Ir $5d$ states have somewhat different shapes.

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Figure 29. (Color online) Valence band spectrum measured of UIr at $h\nu=800$ eV and the calculated U $5f$ and Ir $5d$ pDOS. The sample temperature was 60 K and the sample was in the paramagnetic phase. The calculated pDOS are multiplied by the Fermi–Dirac function and are broadened with the experimental energy resolution.

6.9.2 Band structure of UIr

Figure 30 summarizes the ARPES spectra of UIr. The photon energy was $h\nu=425$ eV and the momentum normal to the sample surface was $k_{\bot}\sim 36.4(\pi/b)$. Since the ARPES cut does not pass through any high-symmetry lines, we define the points as the $\Gamma'$ and D′ points, as shown in the Fig. 30(a). Its electronic structure is expected to be quasi-two-dimensional owing to the very compressed nature of the Brillouin zone along the $b^{\ast}$-axis (Γ–Y direction), and the ARPES spectra along this cut should be similar to those along the Γ–D–Γ high-symmetry line. The sample temperature was 60 K and the sample was in the paramagnetic phase.

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Figure 30. (Color online) ARPES spectra of UIr and result of band-structure calculation. (a) Position of ARPES cut. The spectra trace the line with $k_{\bot}\sim 36.4 (\pi/b)$. The points near the Γ and D points are defined as the $\Gamma'$ and D′ points. (b) Experimental ARPES spectra measured along the $\Gamma'$–D′–$\Gamma'$ high-symmetry line measured at $h\nu=425$ eV. The sample temperature was 60 K and the sample was in the paramagnetic phase. (c) Results of band-structure calculation. The color coding represents the contributions from U $5f$ and Ir $5d$ states. About 140 bands exist in this energy range. (d) Simulation of ARPES spectra based on the band-structure calculation.

Figure 30(b) show the ARPES spectra measured along the $\Gamma'$–D′–$\Gamma'$ line. The spectra have very complicated peak structures. In the vicinity of $E_{\text{F}}$, sharp peak structures exist, and their intensities have some momentum dependence. On the high-binding-energy side, there are strongly dispersive features. Figure 30(c) exhibits the calculated energy bands. The color coding represents the contributions from U $5f$ and Ir $5d$ states. There are about 140 bands in this energy region, which is due to the very low symmetry and multiple-site nature of the crystal structure. It is very difficult to compare the experimental ARPES spectra with the calculated dispersions. Figure 30(d) shows a simulation of ARPES spectra based on the band-structure calculation. Although its comparison is still difficult, the features in the simulation have some similarity to the experimental spectra. The features originating from quasi-particle bands in the vicinity of $E_{\text{F}}$ have some correspondences between the experiment and the calculation. On the high-binding-energy side, broad and strongly dispersive features also have some similarities between the experiment and the calculation. Therefore, some correspondences exist between the ARPES spectra and the result of the band-structure calculation, but a detailed comparison between them is difficult at the present energy resolution. Meanwhile, it is worth noting that there is some overall agreement between the experimental ARPES spectra and the results of the band-structure calculation. The situation is very similar to the cases of UGe₂, URhGe, and UCoGe. Since there are many nearly flat bands in the vicinity of $E_{\text{F}}$, the shapes of the Fermi surfaces are very difficult to predict even from the band-structure calculation.

7. Core-Level Spectra of U-Based Compounds

Core-level spectroscopy is a very powerful experimental technique for studying the local electronic structures of materials. In particular, the core-level spectra of strongly correlated materials often exhibit complex peak structures, which originate from various screening channels in the final state of the photoionization process. They have been utilized to probe various physical parameters of local atomic sites. For example, the core-level spectra of cerium-based compounds generally exhibit three-peak structures, and they have been analyzed on the basis of the Gunnarsson and Schönhammer (GS) model,¹⁴²⁾ which assumes the SIAM with $4f^{0}$, $4f^{1}$, and $4f^{2}$ configurations. The physical parameters of a local cerium site such as the hybridization $V_{fc}$, the Coulomb interaction between f-electrons $U_{ff}$, and the bare energy of the f-electron state $\epsilon_{f}$ can be obtained by analyzing the core-level spectra.

The core-level spectra of uranium-based compounds also show a wide variety of satellite structures.¹⁴³^,¹⁴⁴⁾ On the other hand, the microscopic origin of these satellite structures is not well understood. This is due to the more complex initial and final states in the case of uranium-based compounds. For example, more than three f-electron configurations must be included in the final states owing to the more hybridized nature of $5f$ states than $4f$ states. Furthermore, the number of f-electrons is expected to be $n_{f} = 2{\text{--}}3$ in uranium-based compounds, and the multiplet effect, which originates from the different configurations of f-electrons within the f-shell, plays an essential role in the spectral profiles. The GS model limits the number of final states by assuming a small hybridization, and it further ignores the multiplet effects. Therefore, a straightforward application of the GS model to the uranium-based compounds generally does not work. Although theoretical approaches are still difficult, the experimentally obtained core-level spectra show some systematic behaviors that depend on the electronic structure.³⁷^,¹⁴⁴⁾ These results suggest that the GS model is a promising starting point to understand their basic structures. Here, we studied the core-level spectra of 19 uranium-based compounds and attempted to systematically understand their core-level spectra.

7.1 Basic structure of core-level spectra

We consider the core-level spectra of the typical itinerant compound UB₂ and the localized compound UPd₃ to understand the basic structure of the core-level spectra of uranium compounds. Figure 31(a) displays their U $4f_{7/2}$ core-level spectra. They have very different spectral profiles, indicating that they reflect the different natures of U $5f$ states in these compounds. The core-level spectrum of UB₂ has a relatively simple spectral line shape with a single peak. The negative second derivative of the spectrum indicates that the peak position is about $E_{\text{B}}\sim 377$ eV. On the other hand, the main line of UPd₃ has a rather symmetric shape and is located at around $E_{\text{B}}\sim 379$ eV. Furthermore, complicated satellite structures exist on the high-binding-energy side as well as a shoulder structure at around $E_{\text{B}}\sim 377$ eV.

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Figure 31. (Color online) U $4f_{7/2}$ core-level spectra of typical uranium compounds and a simplified schematic of the photoemission process. (a) U $4f_{7/2}$ core-level spectra of itinerant compound UB₂ and localized compound UPd₃ and their negative second derivatives. (b) Simple schematic of the core-level photoemission process based on the single-impurity Anderson model. $|5f^{2}\rangle$ and $|5f^{3}\underline{L}\rangle$ initial states and $|5f^{2}\underline{c}\rangle$, $|5f^{3}\underline{Lc}\rangle$, and $|5f^{4}\underline{L^{2}c}\rangle$ final states are considered in the first approximation. The electron configurations in their ground states (U $5f^{3}$ for UB₂ and U $5f^{2}$ for UPd₃) suggest that the main lines of UB₂ and UPd₃ correspond to the dominant contributions from the U $5f^{4}$ and U $5f^{3}$ final states, respectively.

Here, we discuss the origin of these peaks. Since the local electronic structure has an important role in the core-level photoemission process, we focus on the local uranium site and the neighboring ligand states. The core-level spectrum of La-substituted UPd₂Al₃ is identical to that of the parent compound, suggesting that the SIAM is essentially applicable to uranium compounds.¹⁴³⁾ Therefore, as the first approximation, we consider these spectral profiles based on an analogy of the GS model. Figure 31(b) shows a simple schematic of the core-level photoemission process of uranium-based compounds. In uranium compounds, the local electronic structure of the uranium site in the ground state should be expressed by a linear combination of the U $5f^{2}$ and U $5f^{3}$ configurations. On the other hand, in the final state of the core-level photoemission process, the $5f$-levels are lowered by the attractive potential of the core hole ($U_{fc}$), which is typically about 4 eV.¹⁴⁵⁾ Then, the potential from the core hole might or might not be screened by the transfer of ligand electrons onto the lowered f-level. Since the screening process depends on the electronic configuration in the ground state as well as the strength of the hybridization, the core-level spectra of f-based compounds should split into multiple peaks.

In the case of UB₂, U $5f$ electrons have an itinerant character, and the number of f-electrons estimated from the band-structure calculation is $n_{f}\sim 2.82$. Therefore, the local electronic configuration of the uranium site in the ground state can be considered as nearly close to the U $5f^{3}$ configuration. In the final state, U $5f$ levels are lowered by $U_{fc}$, and they might be screened by the transfer of electrons from ligand states through the strong hybridization. Then, the U $5f^{4}$ final state becomes dominant in the core-level spectrum owing to the strong $U_{fc}$. Therefore, the peak at $E_{\text{B}}\sim 377$ eV can be regarded as the U $5f^{4}$ final state. On the other hand, the spectrum of UPd₃ has a very complicated structure. The negative second derivative suggests that the U $4f_{7/2}$ peak consists of three peaks located at 384, 379, and 377 eV. The U $5f$ electrons in UPd₃ have localized character, and the local U $5f$ configuration is U $5f^{2}$. The dominant final state should be the U $5f^{3}$ configuration, and the peak at around 379 eV can be ascribed to the final state with the U $5f^{3}$ configuration. The peak at around 377 eV can be assigned to the final state with the U $5f^{4}$ configuration since the position coincides with the main line of UB₂. The peak at around 384 eV is ascribed to the final state with the U $5f^{2}$ configuration since it has a stronger intensity in UPd₃ than in UB₂. Accordingly, the consideration based on the ground state properties suggests that these peaks mainly originate from the U $5f^{2}$, U $5f^{3}$, and U $5f^{4}$ components. Therefore, by comparing the core-level spectra of uranium-based compounds with these typical compounds, the valence state of uranium atoms, namely, the number of f-electrons in the ground state, can be deduced. Note that strong hybridization between U $5f$ and ligand states in uranium compounds leads to a more mixed nature of the final-state configurations ($f^{n}$) in each peak, and these nominal assignments are approximate ones.

7.2 Core-level spectra of various uranium compounds

Figure 32 summarizes the U $4f_{7/2}$ core-level spectra of uranium compounds and their second derivatives. These spectra are categorized into three groups. The first group shown in Fig. 32(a) consists of the itinerant U $5f$ compounds UB₂, UFeGa₅, UPtGa₅, UN, USb₂, UAl₃, and UGa₃. The second group shown in Fig. 32(b) consists of the heavy-fermion superconductors and related compounds UGe₂, UIr, UCoAl, UCoGe, URhGe, UNi₂Al₃, and URu₂Si₂. The third group shown in Fig. 32(c) consists of compounds with noteworthy spectral shapes (UIn₃, UPd₂Al₃, and UPt₃) and localized U $5f$ compounds (UPd₃ and UGa₂). The core-level spectra of these compounds essentially have a common feature consisting of a main line at $E_{\text{B}}\sim 377{\text{--}}378$ eV and a satellite structure $E_{\text{B}}\sim 380{\text{--}}387$ eV. The shapes of the main lines as well as the satellite vary widely among the compounds.

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Figure 32. (Color online) U $4f_{7/2}$ core-level spectra of 19 uranium compounds and their negative second derivatives. The dotted vertical lines ($E_{\text{B}}=376.6{\text{--}}377.3$ eV) represent the range of peak positions of the itinerant uranium compounds. (a) Core-level spectra of simple itinerant uranium compounds and their second derivatives. The small splitting of the spectrum of USb₂ might be due to the presence of two different uranium sites in the crystal structure. (b) Same as (a) but for heavy-fermion superconductors and related compounds. The main lines of these compounds are located in the range of $E_{\text{B}}=376.6{\text{--}}377.3$ eV ($\Delta E_{\text{B}} = 0.7$ eV). The shoulder peak observed in the spectrum of UIr might be due to the existence of four different uranium sites in the crystal structure. (c) Same as (a) but for localized or nearly localized uranium compounds. Their main lines are broad (UIn₃, UPd₂Al₃, and UGa₃) or show clear splitting (UPd₃ and UPt₃). Data of UB₂, UFeGa₅, UPtGa₅, UGe₂, UCoGe, URhGe, UNi₂Al₃. UPd₂Al₃, URu₂Si₂, UPt₃, and UPd₃ replotted from Ref. 37.

7.3 Itinerant compounds

First, we discuss the core-level spectra of the itinerant compounds shown in Fig. 32(a). These spectra have rather simple peak structures, having a sharp and asymmetric peak in the energy range of $E_{\text{B}}=376.6{\text{--}}377.3$ eV ($\Delta E_{\text{B}} = 0.7$ eV). This peak is ascribed to the U $5f^{4}$ final state, which originates from the strongly hybridized U $5f^{3}$ ground state. Intensities of other peaks are very weak, suggesting that the local electronic configuration in the ground state should be dominated by U $5f^{3}$. Meanwhile, the main line of USb₂ shows a splitting of 0.35 eV. This is due to the existence of two different chemical sites of uranium atoms in its crystal structure. There are finite intensities from satellite structures in these compounds although their intensities are generally suppressed. Therefore, there are influences from the multiple final-state configurations even in these itinerant compounds.

7.4 Heavy-fermion superconductors and related compounds

We next discuss the spectra of the heavy-fermion superconductors and related compounds shown in Fig. 32(b). These compounds have rather large specific heat coefficients, suggesting that the electron correlation effect plays an important role. The essential peak structures are very similar to those for the itinerant compounds shown in Fig. 32(a) except for the case of UIr, for which there is a hump structure on the high-binding-energy side of the main line. This is presumably due to the existence of four uranium sites with different chemical states in its crystal structure. The intensities of the satellite structure located at around $E_{\text{B}}\gtrsim 380$ eV are somewhat enhanced compared with those for the itinerant cases. In particular, the satellite intensities of UCoGe, URhGe, UNi₂Al₃, and URu₂Si₂ are considerably stronger than those of itinerant compounds, suggesting that the hybridization is not strong as for the simple itinerant compounds. Nevertheless, their main lines have rather simple structures, as in the case of the itinerant compounds, and therefore their electronic structures should be close to the itinerant U $5f^{3}$ configuration in the ground state. The nearly U $5f^{3}$ configuration in the ground state of the heavy-fermion compounds is consistent with other experimental results. For example, an electron energy loss spectroscopy experiment on URu₂Si₂ suggested a U $5f$ electron count of 2.6–2.8.¹⁴⁶⁾ An X-ray magnetic circular dichroism study on UCoGe also suggested that a U $5f$ electron count of about 3.¹⁴⁷⁾

7.5 Compounds with complex main-line structures

Figure 32(c) shows the U $4f_{7/2}$ core-level spectra of UIn₃, UPd₂Al₃, UPt₃, UPd₃, and UGa₂. The core-level spectra of these compounds have very different profiles from those of the itinerant and heavy-fermion compounds. Their main lines consist of two components or the satellite structures have strongly enhanced intensities. The main line of UIn₃ has a rather symmetric shape, and its second derivative clearly shows that it has a double peak structure. It is very different from the spectra of the isostructural compounds UAl₃ and UGa₃. This might originate from the more localized nature of the U $5f$ states in UIn₃ than those in UAl₃ and UGa₃. The lattice constant of UIn₃ (4.601 Å)⁴⁹⁾ is about 8% larger than those of UAl₃ (4.264 Å) and UGa₃ (4.260 Å),⁴⁷⁾ suggesting that the U $5f$-ligand hybridization is greatly suppressed in UIn₃ compared with in UAl₃ and UGa₃. A similar double peak structure was observed in the core-level spectrum of UPd₂Al₃. The satellite intensity is strongly enhanced in this compound, which also indicates that its electronic structure does not comprise simple itinerant states. The overall spectral line shape has similarities to both the itinerant and localized compounds. Furthermore, the core-level spectrum of UPt₃ is more similar to that of the localized compound UPd₃ than to those of the itinerant compounds. The main line has a clear double peak structure, and the intensity of the satellite located at around 384 eV is greatly enhanced. Therefore, the electronic structure of UPt₃ is very close to that of UPd₃, although there are heavy quasi-particle bands in the vicinity of $E_{\text{F}}$.³⁷⁾ On the other hand, the spectrum of another localized compound UGa₂ has a very different shape from that of UPd₃. It has a broad main line and an enhanced satellite peak at around $E_{\text{B}}\sim 384$ eV. This suggests that the electronic structure of UGa₂ is very different from that of UPd₃, and the microscopic mechanism of the localization of U $5f$ electrons should also have a different origin.

7.6 Satellite structure

Here, we discuss the behavior of the satellite in the core-level spectra. It has been claimed that the intensity of the satellite is more enhanced in localized compounds than in itinerant compounds.¹⁴⁸⁾ In general, such a trend can be observed in these spectra. For example, the satellite intensities are more enhanced in the localized compounds shown in Fig. 32(c) than in the itinerant compounds shown in Fig. 32(a). On the other hand, there are exceptions. For instance, the satellite intensities of USb₂ or UGa₂ are comparable to that of the nearly localized compound UPt₃. Therefore, there is a general trend in the relationship between the satellite intensity and the itinerant or localized nature of U $5f$ states, but it is not a strict relationship. At present, we do not understand the microscopic mechanisms of the satellite intensity, but the calculation based on the SIAM reported in Ref. 145 with a realistic ligand density of states suggested that the shape of the ligand bands has a strong effect on the satellite intensity.¹⁴⁹⁾ Therefore, a realistic model calculation is required to understand the behaviors of satellite structures. There are some theoretical approaches to understanding their spectral shapes,¹⁴⁵^,¹⁵⁰⁾ and further development would provide new approaches to revealing the electronic structures of $5f$ materials.

7.7 Summary of core-level spectra

From the above, the core-level spectra of the uranium compounds suggest that most heavy fermion superconductors have nearly U $5f^{3}$ configurations with considerable hybridization with ligand states in the ground state, except for UPd₂Al₃ and UPt₃. In UPd₂Al₃, there is a finite contribution from the U $5f^{3}$ final state, suggesting that the hybridization is not as strong as in other heavy-fermion superconductors. This should lead to the more correlated nature of U $5f$ states in this compound. The spectrum of UPt₃ is very similar to that of UPd₃, suggesting that its U $5f$ electrons are nearly localized. The spectrum of UGa₂ is very different from those of itinerant compounds and UPd₃, indicating that the microscopic mechanism of the localization is different from that of UPd₃. These results indicate that the fine structure of the main line is very sensitive to the localized or itinerant nature of U $5f$ states in uranium-based compounds, and it can be used as a measure of local U $5f$ electronic configurations.

8. Summary and Outlook

The band structures and Fermi surfaces of uranium compounds as well as the local electronic structures of uranium sites were discussed in terms of the results of SX-ARPES and core-level spectroscopy experiments. The conclusions drawn from these experiments are summarized as follows.

8.1 Itinerant compounds

In itinerant uranium compounds, there are strongly dispersive bands with large contributions from U $5f$ states, which form Fermi surfaces. Their band structures and Fermi surfaces are quantitatively described by the band-structure calculation treating all U $5f$ electrons as itinerant. Minor deviations in the band structure and Fermi surface from the calculation might originate from small but finite contributions from the electron correlation effect, which renormalizes quasi-particle U $5f$ bands. Furthermore, their core-level spectra are very close to those of simple metallic compounds, suggesting that the local electronic structures of their uranium sites have a well-hybridized $5f^{3}$ configuration.

8.2 Localized compounds

In the localized compound UPd₃, the U $5f$ states are located on the high-binding-energy side, and do not contribute to the Fermi energy. Meanwhile, the localized U $5f$ states have finite energy dispersions, indicating that they have a finite hybridization with ligand states. The main-line structure of its core-level spectrum is very different from those of itinerant compounds, demonstrating that it is a sensitive probe of the electronic structures of uranium compounds. The core-level spectrum of UGa₂ is very different from that of UPd₃, suggesting that the microscopic mechanism of the localization is different from that of UPd₃.

8.3 Heavy-fermion superconductors and related compounds

In most heavy-fermion superconductors such as UNi₂Al₃, and URu₂Si₂, U $5f$ electrons form quasi-particle bands in the vicinity of $E_{\text{F}}$. Their overall band structures are qualitatively explained by the band-structure calculations. Meanwhile, UPd₂Al₃ and UPt₃ exhibit somewhat different behaviors. For UPd₂Al₃, the overall band structure is explained by the band-structure calculation, suggesting that the itinerant description is still an appropriate starting point. On the other hand, its core-level spectrum shows similarities to those of both itinerant and localized compounds, indicating that a strong electron correlation effect exist. For UPt₃, although there is a sharp peak in the vicinity of $E_{\text{F}}$ originating from the U $5f$ quasi-particle bands, its core-level spectrum is very similar to that of the localized compound UPd₃. This suggests that the electronic structure of UPt₃ is nearly localized, although there is an itinerant nature in the ground state.

8.4 Ferromagnetic superconductors

In ferromagnetic superconductors such as UGe₂, URhGe, UCoGe, and UIr, there are some good correspondences between the experimental ARPES spectra and the results of the band-structure calculation at a sub-electronvolt energy scale. On the other hand, the states in the vicinity of $E_{\text{F}}$ have complex structures, and the agreement is not as good as for itinerant compounds and other heavy-fermion superconductors. This suggests that the shapes of the Fermi surfaces of these compounds are qualitatively different from the result of the calculations, which is possibly due to the electron correlation effect in the complicated band structures of the low-symmetry crystals. Their core-level spectra are essentially very similar to those of itinerant compounds, suggesting that their local electronic structures are well-hybridized nearly U $5f^{3}$ configuration.

8.5 Outlook and future prospects

As we have shown in the present paper, SX-PES experiments on uranium compounds have revealed the basic electronic structures of uranium compounds. These results can be used as a basis to understand their unique physical properties. To further understand their electronic structures, it is important to combine the data from PES experiments with various incident photon energies since they have different characteristics. In addition, advances in theoretical approaches are required to describe the SX-ARPES spectra as well as core-level spectra. Many-body calculations based on realistic electronic structures, such as by dynamical mean field theory with the LDA calculation, are required to comprehend the electronic structure of metallic $5f$ compounds. In particular, the application of such realistic models for the core-level spectra¹⁵¹^,¹⁵²⁾ is desirable to obtain important physical parameters such as valence numbers.

Acknowledgments

We would like to thank I. Kawasaki, T. Ohkochi, and A. Yasui for conducting part of the present research during their careers at Japan Atomic Energy Agency. We would also like to thank J. W. Allen, D. Aoki, N. Aso, A. Chainani, J. D. Denlinger, T. Durakiewicz, L. Havela, J. J. Joyce, S. Kambe, H. Kusunose, G. H. Lander, K. Miyake, J. Mydosh, Y. Nanba, K. Okada, A. F. Santandar-Syro, D. D. Sarma, N. K. Sato, Z.-X. Shen, K. Shimada, J. G. Tobin, P. Riseborough, D. Vyalikh, and G. Zwicknagl for stimulating discussions and comments on the reported studies. The experiment was performed under proposal Nos. 2005B3821, 2006A3807, 2006B3808, 2007A3833, 2008B3821, 2008B3824, 2009A3821, 2009A3824, 2010A3820, 2010B3824, 2011A3821, 2012B3820, 2012B3821, 2013A3820, 2014A3820, 2014B3820, 2015A3820, and 2015B3820 at SPring-8 BL23SU. The present work was financially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan, under Contact Nos. 21740271 and 26400374; Grants-in-Aid for Scientific Research on Innovative Areas “Heavy Electrons” (Nos. 20102002 and 20102003) from the Ministry of Education, Culture, Sports, Science and Technology, Japan; and the Shorei Kenkyu program of Hyogo Science and Technology Association.

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Author Biographies

Shin-ichi Fujimori was born in 1970 and is originally from Aomori Prefecture, Japan. He obtained his B.Sc. from the Department of Physics, Hirosaki University (1993) and his M.Sc. (1995) and Ph.D. (1998) from Graduate School of Science, Tohoku University. He has been working at SPring-8, Japan Atomic Energy Agency (formerly Japan Atomic Energy Research Institute) since 1998. During this period, he was a visiting physicist of Professor Z.-X. Shen's group of Stanford University (2007–2008). He has been working on the spectroscopic study of strongly correlated materials, particularly f-electron systems.

Yukiharu Takeda was born in 1973 and is originally from Hiroshima Prefecture, Japan. He obtained his B.E., M.Sc., and D.Sc. from Hiroshima University in 1996, 1998, and 2001, respectively. He worked at Hiroshima Synchrotron Radiation Center of Hiroshima University from 2001 to 2003. Since 2003, he has been working at SPring-8, Japan Atomic Energy Agency. He has been engaged in the improvement of X-ray circular magnetic dichroism experiments and studying magnetism of condensed matter, particularly actinide compounds.

Tetsuo Okane was born in 1968 and is originally from Ibaraki Prefecture, Japan. He obtained his B.Sc., M.Sc., and D.Sc. from Tohoku University in 1990, 1992, and 1995, respectively. Since 1995, he has been working as a researcher of Japan Atomic Energy Agency (Japan Atomic Energy Research Institute until 1998), focusing on X-ray spectroscopy experiments on actinide and rare-earth compounds utilizing SPring-8.

Yuji Saitoh He obtained his D.Eng (1995) from the Graduate School of Engineering Science, Osaka University. Since 1995, he has been working at Japan Atomic Energy Agency (formerly Japan Atomic Energy Research Institute). He has been engaged in the construction and upgrading of the soft X-ray beamline SPring-8 BL23SU.

Atsushi Fujimori was born in Tokyo, Japan in 1953. He obtained his B.Sci. (1976) and D.Sci. (1981) from the University of Tokyo. He was a research scientist (1978–1988) at National Institute for Research in Inorganic Materials. He has been an associate professor (1988–1994) at the Department of Physics and a professor (1994–2007) at the Department of Complexity Science and Engineering of the University of Tokyo. Since 2007, he has been a professor at the Department of Physics, the University of Tokyo. He has worked on the study of strongly correlated electron systems using photoemission spectroscopy and core-level spectroscopy.

Hiroshi Yamagami was born in 1962 in Niigata Prefecture, Japan. He obtained his B.Sc. (1986), M.Sc. (1988), and Ph.D. (1991) from Niigata University. After that, he was an assistant professor (1991–2000) at Tohoku University. During this period, he carried out researches (1995–1997) under Professor J. Kübler at the Technical University of Darmstadt through Alexander von Humbolt fellowship. He then became an associate professor (2000–2004) and a professor (2004–) at the Department of Physics, Kyoto Sangyo University. Since 2005, he has been working as a group leader (visiting researcher) at SPring-8, Japan Atomic Energy Agency. He has been working on the first-principles band-structure calculation of actinide and rare-earth.

Yoshinori Haga He obtained his B.Sc. (1990), M.Sc. (1992), and Ph.D. (1995) from the Department of Physics, Tohoku University. He has been working at Advanced Science Research Center, Japan Atomic Energy Agency (formerly Japan Atomic Energy Research Institute) since 1995. During this period, he worked with Dr. Jacques Flouquets group in CEA Grenoble. He also had an assignment as a visiting associate professor at Tohoku University (2005–2011). He has been working on the single-crystal growth and material science of actinide and rare-earth materials.

Etsuji Yamamoto was born in Osaka Prefecture, Japan in 1964. He obtained his B.S. from Osaka University in 1988, M.S. from Kyushu University in 1990, and his Doctor's degree from Osaka University in 2000. Since 1994, he has held a research position at Japan Atomic Energy Research Institute. His research is focused on the crystal growth and low-temperature measurements of heavy-fermion compounds.

Yoshichika Ōnuki was born in Tochigi Prefecture, Japan in 1947. He obtained his B.S. and M.S. from Kyoto University in 1971 and 1973, respectively, and his Doctors degree from University of Tokyo in 1976. He has worked as a lecturer at Saitama Institute of Technology, lecturer, and associate professor, and professor at University of Tsukuba. Since 1994, he has been a professor at the Graduate School of Science, Osaka University. His research is focused on the crystal growth and low-temperature measurements of heavy-fermion compounds. He was awarded the IBM Science Award for the Crystal Growth and Fundamental Property of a Typical Heavy Fermion Compound CeCu₆ in 1989 and shared the Nishina Memorial Award with Professor A. Hasegawa for the Fermi Surface Study of Itinerant Heavy Fermions in 1992.