Three-dimensional bulk electronic structure of the Kondo lattice CeIn3 revealed by photoemission.

We show the three-dimensional electronic structure of the Kondo lattice CeIn3 using soft x-ray angle resolved photoemission spectroscopy in the paramagnetic state. For the first time, we have directly observed the three-dimensional topology of the Fermi surface of CeIn3 by photoemission. The Fermi surface has a complicated hole pocket centred at the Γ-Z line and an elliptical electron pocket centred at the R point of the Brillouin zone. Polarization and photon-energy dependent photoemission results both indicate the nearly localized nature of the 4f electrons in CeIn3, consistent with the theoretical prediction by means of the combination of density functional theory and single-site dynamical mean-field theory. Those results illustrate that the f electrons of CeIn3, which is the parent material of CeMIn5 compounds, are closer to the localized description than the layered CeMIn5 compounds.

Heavy fermion (HF) compounds CeMIn5 (M=Co, Rh, Ir) have attracted much attention in the last decade because of their novel properties123456. For example, the 4f electrons of CeCoIn5 go through a transition from the localized state to the itinerant state and begin to participate in the modification of the Fermi surface (FS) at low temperatures178, resulting in an abnormal enhancement of the electron mass. Under certain conditions, those heavy f electrons condense into Cooper pairs, e.g., CeCoIn5 exhibits the highest superconductivity (SC) temperature recorded in Ce-based HF compounds23910. However, there are still many unresolved questions regarding this system. First, there is not an explicit definition of the crossover line from the localized f electrons to the itinerant f electrons state. Second, the SC of CeCoIn5 cannot be explained by the BCS theory121112. As layered compound, the structure of CeMIn5 is comprised of alternating layers of CeIn3 and MIn2. The three-dimensional (3D) component CeIn3 in CeMIn5 contributes all the f electrons and can be viewed as adding an effective positive pressure on the CeIn3 crystal at ambient pressure1314. Further investigation of the electronic structure of CeIn3 is of particular importance for understanding the nature of CeMIn5. However, unlike the two-dimensional (2D) CeMIn5 compounds78, the electronic structure of CeIn3 is 3D. Consequently, the study of CeIn3 also provides an opportunity to study the influence of the layered structure on the properties of f electrons in this system.

To investigate the properties of f electrons in CeIn3, many experiments have been previously performed. Transport15, optical conductivity spectra16, and inelastic neutron scattering171819 results all indicate the existence of HF at low temperature, even inside the antiferromagnetic (AFM) phase below 10 K. Quantum oscillation measurements2021 reveal that the HF states occupy only a small portion of the FS. Polycrystal photoemission measurements22, angular correlation of the electron-positron annihilation radiation23, and de Haas-van Alphen (dHvA)24 measurements all reveal the localized nature of f electrons at ambient pressure. Photoemission spectroscopy is a powerful tool to detect the reconstruction of the electronic structure and is often used to judge the nature of f electrons25262728. Photoemission spectroscopy can even be used to detect the SC energy gap in HF systems29. However, angle-resolved photoemission spectroscopy (ARPES) studies of CeIn3 have not been previously performed due to the difficulty of cleaving the sample and the complicated 3D electronic structure of CeIn3.

In this work, the electronic structure of CeIn3 in the paramagnetic state is characterized using the soft x-ray ARPES technique for the first time. The 3D topology of the FS of CeIn3 is observed. The FS has a complicated hole pocket centred at the Γ-Z line and an elliptical electron pocket centred at the R point of the Brillouin zone (BZ). Photoemission results indicate the nearly localized nature of the 4f electrons in CeIn3, consistent with the theoretical predictions of the localized assumption. These results can help us to understand the behaviours of f electrons in CeIn3 and the derived CeMIn5 systems.

Results

Fermi surface mapping

The topology of the FS of CeIn3 is presented in Fig. 1. Figure 1(a) shows the slice of the FS in the k-k plane, observed by hν-dependent ARPES. Although the FS contours contain complicated features, we can obtain the symmetry of the electronic structure along kz direction. The slices of the FS observed by 542 eV and 595 eV photon energies correspond to the centre and boundary of the BZ, respectively. Besides, a complicated band structure d centred at the Γ-Z line and an elliptical shaped pocket a centred at R point are observed. To obtain a better understanding of the FS, two selected photon energies are adopted to characterize the FS contours in k-k plane. Figure 1(b,c) are the k-k maps at k~0 (hν = 542 eV) and k~π (hν = 595 eV), respectively. At the centre of the BZ in Fig. 1(b), the square-like pocket d3 at the M point and the double-ring band d2 centred at the Γ point are observed. The spectral weight of d2 is quite strong in the Γ-X direction. At the boundary of the BZ in Fig. 1(c), a square structure d1 around the Z point and an elliptical structure a centred at the R point are displayed.

Figure 1

3D FS map of CeIn3.

(a) FS map in the k-k plane at the Fermi energy integrated over a window of [EF − 100 meV, EF + 100 meV]. Photon energies are varied from 480 eV to 650 eV at a step of 4 eV. 2D FS maps in the k-k plane at the Fermi energy: (b) integrated over a window of [EF − 80 meV, EF + 80 meV] for a photon energy of 542 eV (k~0) and (c) integrated over a window of [EF − 130 meV, EF + 130 meV] for a photon energy of 595 eV (k~π). (d) Bulk BZ of CeIn3 and the high symmetry points. The green, yellow, and purple planes correspond to the FS slices in (a–c), respectively. (e) LEED pattern of the obtained CeIn3 (001) surface. The bright spots in the square lattice reflect the pristine 1 × 1 surface.

The topology of the FS is highly 3D in nature and agrees with the dHvA experiment and the full-potential linear augmented plane wave calculations results24, which exhibit a complicated structure (denoted as d) centred at the Γ-Z line and an ellipsoid-like structure around the R point. Figure 1(e) shows the low energy electron diffraction (LEED) pattern of the obtained surface. A clean 1 × 1 pattern of the (001) surface is observed.

Valence band structure

An important goal in the study of a HF system is to understand the properties of f electrons. Previously, the polarized ARPES technique has been conducted to resolve the multi-orbit nature of the 3d states in iron-based superconductors and has proven to be an effective method to distinguish the different orbits of 3d electrons30. The HF compound YbRh2Si2 has also been studied using different light polarizations3132. Some states in YbRh2Si2 are proven to possess dichroic effects, indicating that electronic states with the same symmetry can be identified by their similar response to a change of light polarization. Therefore, to investigate the possible multi-orbit properties of the f electrons in the Ce-based HF system, we have performed polarization-dependent ARPES measurements in CeIn3.

The experimental valence band structures of CeIn3 are displayed in Fig. 2. The electron-like band 1, band 2 centred at the Γ point and band 3 around the M point can be clearly observed in Fig. 2(c1), forming the double-ring structure d2 and the hole pocket d3 observed in Fig. 1(b). Band 4 is located at the bottom of band 3. Band 5 extends to high energy and nearly connects with band 1 around the Γ point at approximately 1 eV binding energy (BE). Three bands (bands 6, 7 and 8) are observed along the Γ-X direction in Fig. 2(d1). The electron-like band 6 and hole-like band 7 are adjacent at the Fermi level. Band 8 is located at approximately 1.2 eV BE. Except for the band structures discussed above, two nearly non-dispersive bands located at EF and 300 meV BE are observed in Fig. 2(c1,d1). They have much weaker intensity weight than the other conduction bands and can be observed more clearly in Fig. S1 in the SI. The origin of the non-dispersive bands could be Kondo effect or spin-polaron effect3334, which both lead to the many-body resonance in the electron density of states near EF. However, the temperature dependent resistivity of CeIn3 can be described by a function ρ~-lnT beyond the coherent temperature1535, indicating Kondo physics takes effect. Therefore, the two non-dispersive bands correspond to the state and its spin orbit coupling (SOC) sideband 82225. The two flat bands are observed more clearly, as shown in Fig. 2(c2,d2), where significant changes occur when light is changed to p-polarized light. Bands 2, 4, and 5 along Γ-M and bands 6 and 7 along Γ-X disappear. Bands 1 and 3 and the two non-dispersive f bands remain. The suppression of the intensity of the conduction states is presumably due to the effect of the relative spatial orientations of the electronic states with the polarization vector of the light. The phenomena, by using p-polarized light, might be applied to other Ce-based HF compound, even with a low photoemission cross-section, as the suppression of non-f states makes f states in ARPES intensity plots clearer and purer.

Figure 2

Polarization dependent valence band structure of CeIn3.

(a) Experimental schematic of polarization dependent ARPES. (b) The centre of the BZ of CeIn3 and the locations of the momentum cuts. (c1,d1) Photoemission intensity plots obtained using 542 eV s-polarized light along the Γ-M and Γ-X directions, respectively. (c2,d2) Photoemission intensity plots obtained using 542 eV p-polarized light along the Γ-M and Γ-X directions, respectively. The black dashed lines represent the highest intensity of the ARPES results.

Another unresolved problem in CeIn3 is determining why the f bands can be clearly observed by off-resonance photoemission spectroscopy in Fig. 2. In contrast, for many other HF systems, f bands can only be observed using on-resonance photoemission825. We have studied the photoemission cross sections of different orbits of Ce and In as a function of photon energy. Ce (4f, 5d, 6s) and In (5s, 5p) states were considered as valence states36. In the photon energy range from 500 eV to 900 eV, the total photoemission cross section is mainly contributed by 4f electrons. The contribution from Ce (5d, 6s) and In (5s, 5p) are approximately one order-of-magnitude smaller37.

Calculations

The comparison between the experimental band structures with the calculation results is an effective means to judge the properties of the f electrons. On the other hand, density functional theory merged with the single-site dynamical mean-field theory (DFT+DMFT) is probably the most powerful established method to study the electronic structures of strongly correlated materials. DFT+DMFT has been successfully applied in the studies of many HF systems, such as the temperature-dependent localized-itinerant transition in CeIrIn545, We compare the ARPES results with the DFT+DMFT band calculations of CeIn3 in Fig. 3. First, all the valence bands 1–8 in Figs 2 and 3(c,d) can be clearly duplicated in the calculation results, as are the very flat and non-dispersive f bands located at EF and the vicinity. The flat f bands appear to be constant and non-dispersive over all angles and have much weaker intensity weight than those of the other conduction bands, consistent with our experimental results in Fig. 2 and Fig. S1. This phenomenon does not agree well with the case of CeIrIn55, the f states of which have strong intensity weight and form the coherent peak on EF. Besides, the f states of CeIrIn5 show obvious dispersions around EF, induced by the strong hybridization between f states and conduction states. This indicates that the f electrons of CeIn3 have different behaviours compared with those of CeIrIn5, which have itinerant f states at low temperature. Second, a small renormalization of conduction band 7 on EF is displayed in both the calculation results in Fig. 3(a) and the MDCs in Fig. 3(d). This should be induced by the interaction between f states and the band 7. However, the interaction is so weak that the coherent peak does not form. In fact, if an additional pressure is added on CeIn3, the interaction will be greatly enhanced. Band 7 will have obvious band bending and form the coherent peak on EF together with the hybridized f band38. Through comparison of the experimental bands with the calculated results, we can conclude that the experimental band structure of CeIn3 at ambient pressure and low temperature can be described by the nearly localized model.

Figure 3

(a) The band structure calculated using the DFT+DMFT approach for CeIn3. (b) Partial density of states for the Ce-4f states of CeIn3 using DFT+DMFT. (c,d) MDCs display along Γ-M and Γ-X directions at k~0, respectively. The purple dotted lines are guides to eye, tracing the peaks of the MDCs. The orange rectangle marks the f states. The green rectangles mark the renormalization of band 7.

On-resonance photoemission

To clarify the explicit band structures of the f electrons, Fig. 4(a) shows on-resonance photoemission intensity plot of CeIn3. The dense and non-dispersive f bands emerge near EF and ~300 meV BE, enhanced by the 3d-4f excitation. It is well known that the f band at EF is the tail of the Kondo resonance (KR) peak, corresponding to the Ce-4 final state. The peak at 300 meV BE is the SOC sideband and corresponds to the Ce-4 final state. Although the Ce-4 band of CeIn3 is intersected by the conduction band at EF, it does not show any dispersion and does not open an energy gap to form the k-dependent bands E−, as shown in Fig. 4(c). In the Periodic Anderson Model (PAM)39, if the f electrons are itinerant and have periodicity in the lattice, it should form dispersive KR peaks crossing EF in unoccupied states, as shown in the pressure-induced itinerant state for CeIn338. However, no crossing of f electron bands and no energy gap for CeIn3 are observed at ambient pressure in Fig. 4(a).

Figure 4

On-resonance photoemission intensity plot of CeIn3.

(a) Photoemission intensity plot along the Z-R direction. The red dashed lines mark the two flat bands at −1.4 eV and −2 eV. (b) Angle-integrated photoemission spectroscopy of the ARPES result in (a). (c) Schematic of the Anderson lattice model. The dashed lines are the dispersive valence band and the non-dispersive f bands. The dotted line indicates the location of EF. For a finite hybridization, an energy gap opens, and two k-dependent branches E+ and E− form. The lower hybridized band E− crosses EF and forms the heavy particles. The blue shaded area is the angle-integrated photoemission spectroscopy curve within the hybridized model.

From another aspect, as shown in Fig. 4(b), except for the f 1 final states near EF, the non-dispersive structures at approximately 1.4 eV and 2 eV BE are also visible. These arise from pure charge excitations of the trivalent Ce ion (4f 1 → 4f 0) and are referred to the ionization peaks22. The two f bands have been observed in the Ce-termination surface in CeRh2Si240, which exhibits the nature of localized f electrons. Usually, if the hybridization strength is strong, then the f 0 peak is much weaker than the f 1 final state, as shown in the blue shaded area in Fig. 4(c). If the hybridization strength is weak, then the intensity of f 0 peak is strong2241, as demonstrated by recent calculation results using the DFT+DMFT approach by H.Y.L38. H.Y.L et al. indicate that the 4f electrons of CeIn3 will undergo a localized-itinerant transition under pressure, consistent with the dHvA results24. At ambient pressure, the 4f electrons of CeIn3 are localized, and the intensity of f 1 final state is comparable with that of f 0 peak. With the crystal volume decreasing under pressure, the intensity of the f 1 final state quickly increases, and the intensity of f 0 peak decreases. When the crystal volume of CeIn3 decreases to 65% of the volume at ambient pressure, the f 1peak is quite strong, and the f 0 peak nearly disappears. As the intensity of the f 0 peak is comparable with that of f 1 final state in CeIn3 in Fig. 4(b), the angle-integrated photoemission spectroscopy result is also consistent with the calculation result of CeIn3 under ambient pressure. In summary, the f electrons of CeIn3 are close to the description of the nearly localized model, similar to the angle-integrated photoemission spectroscopy results of f localized compounds CeRhIn5 and CeRh2Si24042.

Discussion

We have compared the ground states between CeIn3 and CeMIn5 to illustrate the different properties of the f electron. In fact, there are three possible ground states in the HF systems based on the Doniach phase diagram43 at low temperature: i) the mixed valence ground state with extremely large hybridization strength between the f electrons and the conduction electrons and 0 < n < 1, where n represents the occupation number of f electrons; ii) the magnetic ground state with weak hybridization strength and n ≈1; iii) Non-Fermi liquid (sometimes SC) ground state with strong spin fluctuations, located between the two states discussed above.

CeCoIn5 and CeIrIn5 become superconducting at low temperature, implying that they are located in the SC region in the Doniach phase diagram and that they have relatively strong hybridization strength. In contrast, CeIn3 has the antiferromagnetic ground state with much weaker hybridization strength. Although CeRhIn5 is also an antiferromagnetic compound below 3.8 K, the CeIn3 units in it can be viewed as adding a pressure of approximate 1.4 GPa to the CeIn3 compound at atmospheric pressure14. For CeIn3, the hybridization strength is enhanced under pressure1624. This indicates that CeRhIn5 has a larger hybridization strength than that of CeIn3. These results agree well with our ARPES results of CeIn3, regarding that the 4f electrons of CeIn3 are a nearly localized type.

Now the DFT+DMFT calculation results, our ARPES results, angular correlation of the electron-positron annihilation radiation23 and dHvA24 results all support the view that the 4f electrons of CeIn3 at ambient pressure are nearly localized. However, the optical conductivity results16, transport15 and inelastic neutron scattering171819 measurements hold the opposite view. Why are the conclusions of different references on CeIn3 totally different? First, from our DFT+DMFT calculations and experimental results in Fig. 3, the interaction between band 7 and f band really exists, although the intensity of the interaction strength is too small to form the coherent peak, implying that most of the f electrons are localized and a small portion of the f electrons tend to be itinerant, but incompletely. Second, refs 20 and 21 prove that the hybridized f holes exist at low temperature in CeIn3. However, the f holes just occupy a small portion of the FS and are not along the high symmetry directions in BZ. Such a small proportion may make some techniques hard to detect them. However, this situation will change under pressure. More f electrons begin to participate in the modification of the FS and the collective behaviors of f electrons make the system itinerant and heavy under pressure33. Based on the above discussions, we propose that most of the f electrons of CeIn3 stay localized at ambient pressure and the situation can be changed by means of adding additional pressure.

In summary, the electronic structure of CeIn3 in the paramagnetic state was characterized using the soft x-ray ARPES technique. 3D FS of CeIn3 was revealed. The FS has a complicated hole pocket d centred at the Γ-Z line and an elliptical electron pocket a centred at the R point of the BZ. The photoemission results and the calculated results all indicate a nearly localized nature of the 4f electrons in CeIn3 at ambient pressure.

Methods

High-quality single crystals of CeIn3 were grown using the self-flux method44. The fresh and smooth surfaces were obtained by performing cycles of Ar+-ion sputtering and annealing with a base pressure better than 3 × 10−10 mbar after the surfaces are polished in the atmosphere. The polarization and photon-energy dependent soft x-ray ARPES experiments were performed at the ADDRESS station of the Swiss Light Source facility. The soft x-ray ARPES spectra were obtained using a PHOIBOS-150 photoelectron analyser45. The combined energy resolution is 90 meV or better, and the angle resolution is 0.1°. The base pressure of the ultra-high vacuum system was below 5 × 10−11 mbar during the entire measurement. The samples were kept at T = 13 K in the ARPES measurements. Unless a particular explanation is given, all the data are taken using s-polarized light.

The calculation method is the density functional theory merged with the single-site dynamical mean-field theory (DFT+DMFT) that combines the first-principles aspect of DFT with the non-perturbative many-body treatment of local interaction effects in DMFT. The method used in this paper is introduced in detail in the literature38. All of the calculations were conducted at the inverse temperature β = 1000 (T = 11.6 K), which is comparable with the experimental temperature. Here we adopted U = 6.2 eV and J = 0.7 eV, where U is the Coulomb interaction strength and J the Hund’s exchange parameter.

Additional Information

How to cite this article: Zhang, Y. et al. Three-dimensional bulk electronic structure of the Kondo lattice CeIn3 revealed by photoemission. Sci. Rep. 6, 33613; doi: 10.1038/srep33613 (2016).