⇦Chinese Physics Letters, 2021, Vol. 38, No. 10, Article code 107402 The 4$f$-Hybridization Strength in Ce$_m$$M$$_n$In$_{3m+2n}$ Heavy-Fermion Compounds Studied by Angle-Resolved Photoemission Spectroscopy Jiao-Jiao Song (宋姣姣)1, Yang Luo (罗洋)1, Chen Zhang (章晨)1, Qi-Yi Wu (吴旗仪)1, Tomasz Durakiewicz2, Yasmine Sassa3,4, Oscar Tjernberg5, Martin Månsson5, Magnus H. Berntsen5, Yin-Zou Zhao (赵尹陬)1, Hao Liu (刘豪)1, Shuang-Xing Zhu (朱双兴)1, Zi-Teng Liu (刘子腾)1, Fan-Ying Wu (邬钒颖)1, Shu-Yu Liu (刘姝妤)1, Eric D. Bauer6, Ján Rusz3, Peter M. Oppeneer3, Ya-Hua Yuan (袁亚华)1, Yu-Xia Duan (段玉霞)1, and Jian-Qiao Meng (孟建桥)1* Affiliations 1School of Physics and Electronics, Central South University, Changsha 410083, China 2Institute of Physics, Maria Curie Sklodowska University, 20-031 Lublin, Poland 3Department of Physics and Astronomy, Uppsala University, Box 516, S-75120 Uppsala, Sweden 4Department of Physics, Chalmers University of Technology, 41296 Göteborg, Sweden 5Department of Applied Physics, KTH Royal Institute of Technology, AlbaNova Universitetscentrum, 106 91 Stockholm, Sweden 6Condensed Matter and Magnet Science Group, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Received 20 August 2021; accepted 10 September 2021; published online 26 September 2021 Supported by the National Natural Science Foundation of China (Grant Nos. 12074436 and 11574402), and the Innovation-Driven Plan in Central South University (Grant No. 2016CXS032). J.R. and P.M.O. acknowledge support through the Swedish Research Council (VR) and the Swedish National Infrastructure for Computing (SNIC), for computing time on computer cluster Triolith at the NSC center Linköping (supported by VR Grant No. 2018-05973). Y.S. acknowledges the support from the Swedish Research Council (VR) through a Starting Grant (No. Dnr. 2017-05078). O.T. acknowledges support from the Swedish Research Council (VR) and the Knut and Alice Wallenberg foundation. M.M. is partly supported by a Marie Sklodowska-Curie Action, International Career Grant through the European Commission and Swedish Research Council (VR) (Grant No. INCA-2014-6426), as well as a VR neutron project (Grant No. BIFROST, Dnr. 2016-06955). Further support was also granted by the Carl Tryggers Foundation for Scientific Research (Grant Nos. CTS-16:324 and CTS-17:325). Work at Los Alamos was performed under the auspices of the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering.
*Corresponding author. Email: jqmeng@csu.edu.cn Citation Text: Song J J, Luo Y, Zhang C, Wu Q Y, and Durakiewicz T et al. 2021 Chin. Phys. Lett. 38 107402    Abstract We systemically investigate the nature of Ce 4$f$ electrons in structurally layered heavy-fermion compounds Ce$_m$$M$$_n$In$_{3m+2n}$ (with $M$ = Co, Rh, Ir, and Pt, $m=1$, 2, $n=0$–2), at low temperature using on-resonance angle-resolved photoemission spectroscopy. Three heavy quasiparticle bands $f^0$, $f^1_{7/2}$ and $f^1_{5/2}$, are observed in all compounds, whereas their intensities and energy locations vary greatly with materials. The strong $f^0$ states imply that the localized electron behavior dominates the Ce 4$f$ states. The Ce 4$f$ electrons are partially hybridized with the conduction electrons, making them have the dual nature of localization and itinerancy. Our quantitative comparison reveals that the $f^1_{5/2}$–$f^0$ intensity ratio is more suitable to reflect the $4f$-state hybridization strength. DOI:10.1088/0256-307X/38/10/107402 © 2021 Chinese Physics Society Article Text The heavy-fermion (HF) system is an excellent platform for studying the behavior of correlated electrons in unconventional superconductors, quantum criticality and non-Fermi liquids, antiferromagnetism.[1–3] Previous studies on HF superconductors reveal that both dimensionality[4–6] and hybridization[7] are closely related to superconductivity. Tuning dimensionality and hybridization is the key to revealing the microscopic mechanism of unconventional superconductivity in heavy fermions. It is found that the lattice parameters ratio $c/a$ of Ce$M$In$_5$ ($M$ = Co, Rh, Ir) has a roughly linear relationship with its superconducting transition temperature $T_{\rm c}$.[8] Pu-based 115 materials $T_{\rm c}$ is also linear in $c/a$.[9] Uniaxial pressure measurement on CeIrIn$_5$ revealed that $T_{\rm c}$ changed linearly with both $a$-axis and $c$-axis pressure.[10] The $c/a$ ratio may be the control parameter for the scaling of the superconducting dome width and $T^{\max}_{\rm c}$. The value of $T_{\rm c}$ in magnetically mediated superconductors is believed to be dependent on dimensionality.[4–6] Superconducting properties can be enhanced by lowering dimensionality. It is suggested that the difference in hybridization sets the overall temperature scale for each family, but dimensionality governs the behavior within each family. To explore the character of $f$ electrons is essential to understanding the complex physics in HF compounds. Notably, $f$-electrons participate in the formation of Fermi surface through hybridization with conduction electrons. The localized or itinerant properties of $f$-electrons, i.e., hybridization between conduction and $f$ electrons ($c$–$f$), will significantly affect the band structure near the Fermi energy $E_{\rm F}$. After $c$–$f$ electron hybridization, the complex itinerant heavy electron state is formed, which leads to the delocalization of $f$ electron. Therefore, $f$ electron posses the dual nature of local and itinerant even at very low temperatures, which makes the system complex and difficult to understand.[11,12] HF compounds Ce$_m M_n$In$_{3m+2n}$ ($M$ = Co, Rh, Ir, and Pt) are very suitable for investigating dimensionality and hybridization, and a lot of studies have been carried out on such a system. Theoretical calculation suggested that Ce $4f$ electron spectral weight is mainly located above $E_{\rm F}$, and only a tiny fraction appears below $E_{\rm F}$.[13] In recent years, with the improvement of energy and momentum resolution of angle-resolved photoemission spectroscopy (ARPES), it is possible to study the fine structure of heavy quasiparticle bands, such as $c$–$f$ hybridization and the crystal electric field splitting of the spin-orbital split state $f_{5/2}^1$ and $f_{7/2}^1$.[13–23] The $f_{5/2}^1$ peak observed near $E_{\rm F}$ is actually the tail of the Kondo resonance peak, and $f_{7/2}^1$ is the satellite peak. According to the single impurity Anderson model (SIAM),[24] generally, the more localized the Ce 4$f$ is, the stronger the $f^0$ peak is, and the weaker the $c$–$f$ hybridization is.[25,26] In this study, we carried out high-resolution on-resonance ARPES experiments on HF compounds Ce$_m M_n$In$_{3m+2n}$, with $m=1$, 2, $n=0$–2, and discuss the localization or itinerancy of Ce 4$f$ in these compounds and the relationship between the dimensionality and the $c$–$f$ hybridization strength. It is found that compared with the $f^1$–$f^0$ intensity ratio, the $f^1_{5/2}$–$f^0$ intensity ratio can better reflect $c$–$f$ hybridization strength.
Table 1. Physical properties of cerium compounds studied in the present study. $T_{\rm N}$: Néel temperature.
Compounds Structure $\gamma$ [mJ/(mol-Ce$\cdot$K$^2$)] Order $T_{\rm c}$ (K) $T_{\rm N}$ (K) $a$ (Å) $c$ (Å) $c/a$ $d_{{\rm Ce}-M}$ (Å)
CeIn$_3$ Cubic 120[27] AF 0.2 @2.6 GPa[4] 10.3[28] 4.6876 4.6876[29] 1
Ce$_2$IrIn$_8$ Tetragonal 700[30] 4.6897 12.195[31] 2.600 3.73
Ce$_2$RhIn$_8$ Tetragonal $\sim$400[30] AF 2 @2.3 GPa[32] 2.8[30] 4.667 12.247[31] 2.624 3.74
CeCoIn$_5$ Tetragonal 290[33] 2.3[33] 4.601 7.540[34] 1.639 3.77
CeIrIn$_5$ Tetragonal 720[35] 0.4[35] 4.668 7.515[35] 1.610 3.76
CeRhIn$_5$ Tetragonal $\sim$400[30] AF 0.1[36] 3.8[37] 4.652 7.542[30] 1.621 3.77
CePt$_2$In$_7$ Tetragonal 328[38] AF 2.1 @3.5 GPa[39] 5.2[38] 4.6093 21.627[40] 4.692 4.98
Summary of Studied Compounds. The HF compounds Ce$_m M_n$In$_{3m+2n}$ possess a relatively higher $T_{\rm c}$ than other HF materials. Its crystal structure is composed of $m$-layers of CeIn$_3$ separated by $n$-layers of $M$In$_2$ along the $c$ axis of the tetragonal unit cell. This family has rich phase diagrams and is a good platform to investigate the effect of dimensionality, hybridization, magnetism, superconductivity, etc. The basic physical properties of HF compounds investigated in the present study are listed in Table 1. From top to bottom, CeIn$_3$, Ce$_2 M$In$_8$, Ce$M$In$_5$ and Ce$_2$PtIn$_7$, the electronic structure becomes more two-dimensional, and the superconducting transition temperature $T_{\rm c}$ is enhanced. The last column $d$ is the Ce–$M$ bond distance, which is the distance between Ce and its nearest neighbors $M$ atoms. It is worth noting that Ce$_2 M$In$_8$ has only one nearest and four next-nearest neighbors $M$ atoms, while Ce$M$In$_5$ has two nearest and eight next-nearest neighbor $M$ atoms. In other words, the hybridization strength of Ce$M$In$_5$ will be significantly stronger than that of Ce$_2 M$In$_8$. The table also gives the value of $c/a$, which is often used to describe the structural dimension. CeIn$_3$ is the parent and fundamental unit of this family and crystallizes in the infinite-layer (cubic, 3D) structure ($Pm\bar{3}m$).[29] CeIn$_3$ is an antiferromagnet at ambient pressure with predominantly localized moments, and Sommerfeld coefficient $\gamma$ is 120 mJ$\cdot$mol$^{-1}$$\cdot$K$^{-2}$, undergoes a superconducting transition at $T_{\rm c} = 0.2$ K under pressure of 2.6 GPa.[4] In previous ARPES experiments, a weak heavy quasiparticle band with an energy dispersion of 4 meV was observed, indicating a weak $c$–$f$ hybridization.[17] Ce$_2$IrIn$_8$ is a tetragonal compound. The observation of a spin-glass state,[41] a relatively high Sommerfeld coefficient ($\sim$700 mJ$\cdot$mol$^{-1}$$\cdot$K$^{-2}$),[30] and the absence of long-range magnetic order indicate that the Ce 4$f$ electrons in Ce$_2$IrIn$_8$ have itinerant character. ARPES is observed with a flat band near the $E_{\rm F}$ around the $\varGamma$ point, which shows no large difference in Fermi momentum.[18] Ce$_2$RhIn$_8$ with a tetragonal crystal structure is an antiferromagnetic superconductor with two magnetic transitions at $T_{\rm N} = 2.8$ and $T_{\rm m} = 1.65$ K.[32,42] ARPES measurements suggested that the Ce 4$f$ electrons are essentially localized.[26,43,44] CeCoIn$_5$ is a well-studied compound. Its low energy behavior is similar to that of the underdoped cuprates, with a relatively high $T_{\rm c} \approx 2.3$ K at ambient pressure. In the beginning, $f$ electrons were considered to be itinerant at low temperature because the experimental results of optical conductivity,[45] de Haas-van Alphen (dHvA),[46,47] and scanning tunneling microscope (STM)[48,49] were in good agreement with the theoretical calculation based on fully itinerant $f$ electrons.[7,46] Earlier ARPES experiments suggested that $f$ electrons were itinerant up to 105 K,[22] while subsequently there appeared evidence to support the dominance of localized $f$ electrons at low temperature,[50,51] which showed an itinerant nature due to partial hybridizations.[51] Recent ARPES experiments suggested that the localized-to-itinerant transition occurs at surprisingly high temperatures, and $f$ electrons are largely localized even at the lowest temperature.[19] CeIrIn$_5$ is a superconductor with $T_{\rm c} = 0.4$ K at ambient pressure.[35] Early dHvA experiments suggested that $f$ electron delocalizes and involves forming Fermi surface at low temperature, which means that $f$ electrons have itinerant character.[46,47,52] Earlier ARPES experiments suggested that $f$ electrons were nearly localized.[25] Later, on-resonance ARPES experiments have observed heavy quasiparticle peaks, suggesting that although 4$f$ electrons are mainly localized, a small itinerant component is responsible for superconductivity.[53] Recently, ARPES measurements found that the localized $f$ electrons evolve into the HF state starting from a temperature much higher than the coherence temperature.[20] CeRhIn$_5$, unlike CeCoIn$_5$ and CeIrIn$_5$, is an antiferromagnetic compound with Néel temperature $T_{\rm N} = 3.8$ K at ambient pressure.[37] In contrast to CeCoIn$_5$ and CeIrIn$_5$, dHvA experiments suggested that the Ce 4$f$ electrons are localized in CeRhIn$_5$.[46,47,52] The theoretical calculation also gives the picture of localized $f$ states.[54] Different ARPES research groups found seemingly conflicting results. Some believed that 4$f$ electrons were nearly localized in the paramagnetic state,[25] while others suggested that the 4$f$ electrons participate in band formation.[55] Recently, ARPES experiments found band-dependent $c$–$f$ hybridization.[21] CePt$_2$In$_7$ with body-centered tetragonal crystal structure is an antiferromagnetic superconductor with $T_{\rm c} = 2.1$ K near 3.5 GPa.[39] Quantum oscillation[56] and ARPES[13,23,57] measurements suggested that partially 4$f$-electrons contribute to FS formation. Experimental Details. High-resolution ARPES experiments were performed on SIS X09LA beamline at the Swiss Light Source, using a VG-SCIENTA R4000 photoelectron spectrometer. The characteristics of Ce 4$f$ electrons were obtained by on-resonance ARPES at a low temperature of 10 K, which is a practical approach to explore the 4$f$ electron states. All compounds were measured with 122 eV photons, except CeIrIn$_5$ and CePt$_2$In$_7$ compounds measured with 123 eV photons. The energy and momentum resolution was set to $\sim $25–35 meV and 0.2$^{\circ}$, respectively. All the samples were single crystals, and cleaved in situ, and measured in an ultra-high vacuum. All measurements were carried out under very similar conditions. Since the thermocouple does not directly contact the sample, the actual temperature of the sample is slightly higher than the measured temperature. Thus, all samples, including CeIn$_3$, are in the paramagnetic phase.
cpl-38-10-107402-fig1.png
Fig. 1. Photoemission images for HF compounds Ce$_m M_n$In$_{3m+2n}$. (a1)–(a7) On-resonance ARPES spectra for CeIn$_3$, Ce$_2$IrIn$_8$, Ce$_2$RhIn$_8$, CeCoIn$_5$, CeIrIn$_5$, CeRhIn$_5$, and CePt$_2$In$_7$, respectively. (b1)–(b7) The second-derivative images with respect to energy corresponding to (a1)–(a7), respectively.
Results and Discussion. Figures 1(a1)–1(a7) show the spectra taken from freshly cleaved HF compounds Ce$_m M_n$In$_{3m+2n}$. Because ARPES spectra is a surface sensitive technique with 122/123 eV photon energies, this results in a large Ce 4$f$ spectra weight distribution at deeper binding energies. Figures 1(b1)–1(b7) display the dispersion reproduced by second-derivative with respect to the energy to sharpen the band structures while maintaining the main band structure. Three flat bands, $f^0$, $f^1_{5/2}$, and $f^1_{7/2}$ final states, can be resolved. The high-intensity flat band observed in all samples corresponds to the $f^0$ state prevalent in the Ce-based HFs material.[13–21,25] It can be observed that the location, intensity, and width of $f^0$ state vary greatly with materials. Weak heavy quasiparticle bands close to $E_{\rm F}$ are observed, corresponding to $f^1_{5/2}$ and $f^1_{7/2}$ final states.[13–21] The intensities of $f^1_{5/2}$ and $f^1_{7/2}$ final states also vary significantly with materials. The $f^1_{5/2}$ final state, the tails of Kondo resonant peaks, can be easily distinguished from the raw images of CeIn$_3$ and Ce$M$In$_5$, while those of Ce$_2$IrIn$_8$, Ce$_2$RhIn$_8$ and CePt$_2$In$_7$ show only limited spectral weight. The $f^1_{7/2}$ final state, i.e., the spin-orbit coupling sideband of the $f^1_{5/2}$ state, located at around $-300$ meV, can be observed in the raw data of all samples. As seen in Fig. 1(a4), the $f^1$ state of CeCoIn$_5$ is heavily overlapped with the Co 3$d$ state, giving the fact that the photoionization cross-section of the Co 3$d$ level is of one order higher than those of the Rh 4$d$ and Ir 5$d$ at the Ce 4$d$–$4f$ excitation threshold.
cpl-38-10-107402-fig2.png
Fig. 2. Comparison of angle-integrated PES spectra of CeIn$_3$, Ce$_2$IrIn$_8$, Ce$_2$RhIn$_8$, CeCoIn$_5$, CeIrIn$_5$, CeRhIn$_5$, and CePt$_2$In$_7$.
Figure 2 displays the energy distribution curves (EDCs) of Ce$_m M_n$In$_{3m+2n}$, which is integrated over the momentum range shown in Fig. 1. It is found that our ARPES results are consistent with the previous results.[13,17–21,26] The $f^0$, $f^1_{7/2}$ and $f^1_{5/2}$ final states mentioned above were observed in EDCs. These spectral features were widely observed in Ce-based HFs. Several notable features can be seen from the figures. The relative intensities of the three final states change significantly with the material. Only in CeIn$_3$ and CeIrIn$_5$ can sharp peaks of $f^1_{5/2}$ and $f^1_{7/2}$ be observed. For others, the intensity of $f^1_{5/2}$ is considerably weaker than that of $f^1_{7/2}$. The $f^0$ final state intensity is very strong for the whole family members, except CeCoIn$_5$, suggesting that localized electrons dominate the Ce 4$f$ states. The $f^0$ intensities of CePt$_2$In$_7$ and CeCoIn$_5$ were the strongest and the weakest in the family, respectively. According to the SIAM, they may correspond to the most localized (weakest hybridized) and the most itinerant (most hybridized), respectively. The deduction for CePt$_2$In$_7$ seems to be consistent with the fact that it has the largest $d$ value in this family.[39] However, the inference for CeCoIn$_5$ is inconsistent with previous experimental and calculation results, which suggested that the $c$–$f$ hybridization weakens in the order of CeIrIn$_5$, CeCoIn$_5$, and CeRhIn$_5$.[7,21] It seems thus not a good idea to use the $f^0$ strength alone to reflect the hybridization strength and itinerancy of $f$ electrons. A more suitable parameter is needed to quantitatively reflect the hybridization strength.
cpl-38-10-107402-fig3.png
Fig. 3. (a) and (b) Angle-integrated PES spectra (open circles) and the corresponding fitting results (solid lines and dot-dashed lines). Filled green, purple, and blue profiles represent the fitting results of $f^0$, $f^1_{7/2}$, and $f^1_{5/2}$ final states, respectively. Other dot-dashed lines originate from conduction bands and background. (c) The $f^1$–$f^0$ and $f^1_{5/2}$–$f^0$ intensity ratios of Ce$_m M_n$In$_{3m+2n}$.
The $4d$–$4f$ on-resonance spectra in Ce-based HF compounds can be well understood by the SIAM,[24–26] which considers that the ground state is a linear combination of the $f^0$ and $f^1_{5/2}$ states. Previous experiments suggested that the stronger the $c$–$f$ hybridization, the stronger the $f^1$ peaks.[25,26] To compare hybridization strength quantitatively, the $f^1$–$f^0$ intensity ratios [[$I$($f^1_{7/2}$)+$I$($f^1_{5/2}$)]/$I$($f^0$)] were calculated, which has been considered to reflect the $c$–$f$ hybridization strength.[25] The angle-integrated PES spectra were well fitted with multiple Lorentzian peaks.[16] Figures 3(a) and 3(b) display the typical fitted results to CeIn$_3$ and CeCoIn$_5$ spectra. The filled shadow curves represent the heavy $f$ responses. As evidenced by red, yellow, and cyan dot-dashed lines, the spectra include a large contribution of conduction bands. Figure 3(c) display the calculated $f^1$–$f^0$ intensity ratio, together with $f^1_{5/2}$–$f^0$ intensity ratio $I$($f^1_{5/2}$)/$I$($f^0$). It can be seen that both intensity ratios $f^1$–$f^0$ and $f^1_{5/2}$–$f^0$, vary greatly with the materials. From CeIn$_3$ to CePt$_2$In$_7$, the ratios do not change monotonically with $c/a$ or $d$. Interestingly, CeIn$_3$, which has a 3D structure, and CePt$_2$In$_7$, which is towards the two-dimension limit in the Ce$_m M_n$In$_{3m+2n}$ family, both have very low intensity ratio and possess the weakest hybridization/itinerant. The ratios of four antiferromagnets (CeIn$_3$, Ce$_2$RhIn$_8$, CeRhIn$_5$, and CePt$_2$In$_7$) are very small. The strong localized nature of $f$ electron may be responsible for the antiferromagnetic transition at low temperatures.[26] The $f^1$–$f^0$ value of Ce$M$In$_5$ is the largest of this family, suggesting that Ce$M$In$_5$ may have the strongest $c$–$f$ hybridization strength in the Ce$_m M_n$In$_{3m+2n}$ family.
cpl-38-10-107402-fig4.png
Fig. 4. Energy positions of $f^0$ and $f^1_{7/2}$ final states.
However, in the subfamily Ce$M$In$_5$, the $f^1$–$f^0$ intensity ratio decreases in the order of CeCoIn$_5$, CeIrIn$_5$, and CeRhIn$_5$. According to the SIAM, the $c$–$f$ hybridization weakens in the same order, which is inconsistent with the previous experimental and theoretical results.[7,21] Noticeably, the $f^1_{5/2}$–$f^0$ intensity ratio is in line with previous results,[7,21] which seems to reflect the hybridization strength of Ce$M$In$_5$ better. Both intensity ratios could thus be employed. Significantly, our results suggest that the $f^1_{5/2}$–$f^0$ intensity ratio is more suitable than the $f^1$–$f^0$ to quantify the hybridization strength. In fact, this is an intuitive result. Previous studies have shown that compared with the $f^1_{5/2}$ state, the $f^1_{7/2}$ state is not sensitive to temperature.[19–23,58] This means that the $f^1_{7/2}$ state is not a good parameter for determining Kondo temperature $T_{\rm K}$, and the $f^1_{5/2}$ state is a more suitable parameter. With the development of ARPES, we may even further consider using the low-lying crystal electric field excitation of $f^1_{5/2}$ state to quantitatively determine $T_{\rm K}$. We note that the change of atomic distance $d$ in subfamily Ce$M$In$_5$ is very small, while the hybridization strength changes significantly. Similarly, the change of $c/a$ in Ce$M$In$_5$ is not consistent with the change of hybridization strength. However, the $c/a$ ratio can roughly distinguish the hybridization strength among different subfamilies. If the number of adjacent $M$ atoms is taken into account, $d$ can also be used to distinguish the hybridization strength among different subfamilies. Finally, Fig. 4 summarizes the energy location of $f_{7/2}^1$ and $f^0$ final state of Ce$_m M_n$In$_{3m+2n}$. Since the $f_{5/2}^1$ state close to $E_{\rm F}$ is the tail of Kondo resonant peak, it is greatly influenced by the resolution-convoluted Fermi–Dirac distribution. The $f_{5/2}^1$ will not be discussed here. The energy locations of $f_{7/2}^1$ and $f^0$ final states vary significantly with the material. They have similar behaviors with the change of materials, but not at the same magnitude. The variation of $f^0$ position is nearly an order of magnitude larger than that of $f_{7/2}^1$. For example, the positions of $f^1_{7/2}$ and $f^0$ peaks are $-0.26$ and $-2.08$ eV at CeIn$_3$ and $-0.32$ and $-2.53$ eV at CePt$_2$In$_7$. The $f^1_{7/2}$ and $f^0$ states are shifted down by 0.06 and 0.45 eV, respectively. In summary, we have studied the electronic structures of HF compounds Ce$_m M_n$In$_{3m+2n}$ by ARPES and observed the heavy quasiparticle bands $f_{5/2}^1$, $4f_{7/2}^1$ and $f^0$ in all compounds. It is found that the strength of all three peaks varies greatly with compositions of materials. The hybridization strength of the subfamily Ce$M$In$_5$ is the strongest in the Ce$_m M_n$In$_{3m+2n}$ family. Compared with the $f^1$–$f^0$ intensity ratio, the $f^1_{5/2}$–$f^0$ intensity ratio is a better indicator of hybridization strength. The $c/a$ ratio and Ce–$M$ band distance $d$, considering the number of adjacent $M$ atoms, set the overall hybridization strength within each subfamily. References Superconductivity in the Presence of Strong Pauli Paramagnetism: Ce Cu 2 Si 2 Commensurate antiferromagnetism in CePt 2 In 7 , a nearly two-dimensional heavy fermion systemSuperconductivity and Quantum Criticality in C e C o I n 5 Magnetically mediated superconductivity in heavy fermion compoundsMagnetically mediated superconductivity in quasi-two and three dimensionsMagnetically mediated superconductivity: Crossover from cubic to tetragonal latticeDynamical mean-field theory within the full-potential methods: Electronic structure of CeIrIn 5 , CeCoIn 5 , and CeRhIn 5 Multiple phase transitions in Ce(Rh,Ir,Co)In5Structural Tuning of Unconventional Superconductivity in P u M G a 5 ( M = C o , R h )Anisotropic Dependence of Superconductivity on Uniaxial Pressure in CeIrIn 5 Observing the heavy fermions in CeCoIn5 by angle-resolved photoemissionQuasi-two-dimensional Fermi surfaces of the heavy-fermion superconductor Ce 2 PdIn 8 Crystal electric field splitting and f -electron hybridization in heavy-fermion CePt 2 In 7 ARPES view on surface and bulk hybridization phenomena in the antiferromagnetic Kondo lattice CeRh2Si2Electronic structure and 4 f -electron character in Ce 2 PdIn 8 studied by angle-resolved photoemission spectroscopyAngle-resolved photoemission spectroscopy view on the nature of Ce 4 f electrons in the antiferromagnetic Kondo lattice Ce Pd 5 Al 2 Direct observation of heavy quasiparticles in the Kondo-lattice compound CeI n 3 Hybridization Effects Revealed by Angle-Resolved Photoemission Spectroscopy in Heavy-Fermion Ce 2 IrIn 8Direct observation of how the heavy-fermion state develops in CeCoIn 5 Tracing crystal-field splittings in the rare-earth-based intermetallic CeIrIn 5 Band Dependent Interlayer f -Electron Hybridization in CeRhIn 5 Hybridization effects in Ce Co In 5 observed by angle-resolved photoemissionThree-dimensional and temperature-dependent electronic structure of the heavy-fermion compound CePt 2 In 7 studied by angle-resolved photoemission spectroscopyElectron spectroscopies for Ce compounds in the impurity modelNearly localized nature of f electrons in Ce T In 5 ( T = Rh , Ir)Angle-resolved and resonant photoemission spectroscopy on heavy-fermion superconductors Ce 2 Co In 8 and Ce 2 Rh In 8 Low-temperature specific heat of RIn3 (R = La-Ho)Magnetic ordering in the presence of fast spin fluctuations: A neutron scattering study of Ce In 3 Magnetic Susceptibilities of Rare‐Earth–Indium Compounds: RIn 3Superconductivity and magnetism in a new class of heavy-fermion materialsSingle-Crystal Growth of Ln 2 MIn 8 (Ln = La, Ce; M = Rh, Ir): Implications for the Heavy-Fermion Ground StateMagnetism and superconductivity in Ce 2 RhIn 8 Heavy-fermion superconductivity in CeCoIn 5 at 2.3 KA new heavy-fermion superconductor CeIrIn 5 : A relative of the cuprates?Ambient-pressure bulk superconductivity deep in the magnetic state of Ce Rh In 5 Pressure-Induced Superconductivity in Quasi-2D CeRhIn 5 Single crystal study of the heavy-fermion antiferromagnet CePt 2 In 7Pressure-induced superconducting state and effective mass enhancement near the antiferromagnetic quantum critical point of CePt 2 In 7 The crystal structure of the new indide CePt2In7 from powder dataRandom spin freezing in Ce 2 M In 8 ( M = Co , Rh , Ir ) heavy-fermion materialsThermal expansion and magnetovolume effects in the heavy-fermion system Ce 2 RhIn 8 Band structure and Fermi surface of heavy Fermion compounds Ce2TIn8 studied by angle-resolved photoemission spectroscopyElectronic structure of Ce 2 RhIn 8 : A two-dimensional heavy-fermion system studied by angle-resolved photoemission spectroscopyOptical conductivity of the heavy fermion superconductor CeCoIn 5 Fermi Surface, Magnetic and Superconducting Properties of LaRhIn 5 and CeTIn 5 (T: Co, Rh and Ir)Calculated de Haas–van Alphen quantities of Ce M In 5 ( M = Co , Rh, and Ir) compoundsImaging Cooper pairing of heavy fermions in CeCoIn5Visualizing nodal heavy fermion superconductivity in CeCoIn5Electronic structure of CeCoIn 5 from angle-resolved photoemission spectroscopyBand structures of 4 f and 5 f materials studied by angle-resolved photoelectron spectroscopy 4 f -Electron Localization in C e x L a 1 − x M I n 5 with M = C o , Rh, or IrDirect observation of a quasiparticle band in Ce Ir In 5 : An angle-resolved photoemission spectroscopy studyProbing the electronic structure of pure and doped Ce M In 5 ( M = Co , Rh , Ir ) crystals with nuclear quadrupolar resonanceThe electronic structure of CeRhIn5 and LaRhIn5 from ARPESFermi surface of CePt 2 In 7 : A two-dimensional analog of CeIn 3 Electronic structure of heavy fermion system CePt 2 In 7 from angle-resolved photoemission spectroscopyLarge Fermi surface expansion through anisotropic mixing of conduction and f electrons in the semimetallic Kondo lattice CeBi
[1] Steglich F, Aarts J, Bredl C D, Lieke W, Meschede D, Franz W, and Chäfer H 1979 Phys. Rev. Lett. 43 1892
[2] Aproberts-Warren N, Dioguardi A P, Shockley A C, Lin C H, Crocker J, Klavins P, and Curro N J 2010 Phys. Rev. B 81 180403(R)
[3] Sidorov V A, Nicklas M, Pagliuso P G, Sarrao J L, Bang Y, Balatsky A V, and Thompson J D 2002 Phys. Rev. Lett. 89 157004
[4] Mathur N D, Grosche F M, Julian S R, Walker I R, Freye D M, Haselwimmer R K W, and Lonzarich G G 1998 Nature 394 39
[5] Monthoux P and Lonzarich G G 2001 Phys. Rev. B 63 054529
[6] Monthoux P and Lonzarich G G 2002 Phys. Rev. B 66 224504
[7] Haule K, Yee C, and Kim K 2010 Phys. Rev. B 81 195107
[8] Pagliuso P G, Movshovich R, Bianchi A D, Nicklas M, Moreno N O, Thompson J D, Hundley M F, Sarrao J L, and Fisk Z 2002 Physica B 312–313 129
[9] Bauer E D, Thompson J D, Sarrao J L, Morales L A, Wastin F, Rebizant J, Griveau J C, Javorsky P, Boulet P, Colineau E, Lander G H, and Stewart G R 2004 Phys. Rev. Lett. 93 147005
[10] Dix O M, Swartz A G, Zieve R J, Cooley J, Sayles T R, and Maple M B 2009 Phys. Rev. Lett. 102 197001
[11] Koitzsch A, Borisenko S, Inosov D, Geck J, Zabolotnyy V B, Shiozawa H, Knupfer M, Fink J, Buechner B, Bauer E D, Sarrao J L, and Follath R 2007 Physica C 460–462 666
[12] Götze K, Klotz J, Gnida D, Harima H, Aoki D, Demuer A, Elgazzar S, Wosnitza J, Kaczorowski D, and Sheikin I 2015 Phys. Rev. B 92 115141
[13] Duan Y X, Zhang C, Rusz J, Oppeneer P M, Durakiewicz T, Sassa Y, Tjernberg O, Månsson M, Berntsen M H, Wu F Y, Zhao Y, Song J J, Wu Q Y, Luo Y, Bauer E D, Thompson J D, and Meng J Q 2019 Phys. Rev. B 100 085141
[14] Patil S, Generalov A, Güttler M, Kushwaha P, Chikina A, Kummer K, Rödel T C, Santander-Syro A F, Caroca-Canales N, Geibel C, Danzenbächer S, Kucherenko Y, Laubschat C, Allen J W, and Vyalikh D V 2016 Nat. Commun. 7 11029
[15] Yao Q, Kaczorowski D, Swatek P, Gnida D, Wen C H P, Niu X H, Peng R, Xu H C, Dudin P, Kirchner S, Chen Q Y, Shen D W, and Feng D L 2019 Phys. Rev. B 99 081107(R)
[16] Yuan Y H, Duan Y X, Rusz J, Zhang C, Song J J, Wu Q Y, Sassa Y, Tjernberg O, Månsson M, Berntsen M H, Wu F Y, Liu S Y, Liu H, Zhu S X, Liu Z T, Zhao Y Z, Tobash P H, Bauer E D, Thompson J D, Oppeneer P M, Durakiewicz T, and Meng J Q 2021 Phys. Rev. B 103 125122
[17] Zhang Y, Feng W, Lou X, Yu T L, Zhu X G, Tan S Y, Yuan B K, Liu Y, Lu H Y, Xie D H, Liu Q, Zhang W, Luo X B, Huang Y B, Luo L Z, Zhang Z J, Lai X C, and Chen Q Y 2018 Phys. Rev. B 97 045128
[18] Liu H J, Xu Y J, Zhong Y G, Guan J Y, Kong L Y, Ma J Z, Huang Y B, Chen Q Y, Chen G F, Shi M, Yang Y F, and Ding H 2019 Chin. Phys. Lett. 36 097101
[19] Chen Q Y, Xu D F, Niu X H, Jiang J, Peng R, Xu H C, Wen C H P, Ding Z F, Huang K, Shu L, Zhang Y J, Lee H, Strocov V N, Shi M, Bisti F, Schmitt T, Huang Y B, Dudin P, Lai X C, Kirchner S, Yuan H Q, and Feng D L 2017 Phys. Rev. B 96 045107
[20] Chen Q Y, Wen C H P, Yao Q, Huang K, Ding Z F, Shu L, Niu X H, Zhang Y, Lai X C, Huang Y B, Zhang G B, Kirchner S, and Feng D L 2018 Phys. Rev. B 97 075149
[21] Chen Q Y, Xu D F, Niu X H, Peng R, Xu H C, Wen C H P, Liu X, Shu L, Tan S Y, Lai X C, Zhang Y J, Lee H, Strocov V N, Bisti F, Dudin P, Zhu J X, Yuan H Q, Kirchner S, and Feng D L 2018 Phys. Rev. Lett. 120 066403
[22] Koitzsch A, Borisenko S V, Inosov D, Geck J, Zabolotnyy V B, Shiozawa H, Knupfer M, Fink J, Buechner B, Bauer E D, Sarrao J L, and Follath R 2008 Phys. Rev. B 77 155128
[23] Luo Y, Zhang C, Wu Q Y, Wu F Y, Song J J, Xia W, Guo Y F, Rusz J, Oppeneer P M, Durakiewicz T, Zhao Y Z, Liu H, Zhu S X, Yuan Y H, Tang X F, He J, Tan S Y, Huang Y B, Sun Z, Liu Y, Liu H Y, Duan Y X, and Meng J Q 2020 Phys. Rev. B 101 115129
[24] Gunnarsson O and Schönhammer K 1983 Phys. Rev. B 28 4315
[25] Fujimori S, Okane T, Okamoto J, Mamiya K, Muramatsu Y, Fujimori A, Harima H, Aoki D, Ikeda S, Shishido H, Tokiwa Y, Haga Y, and Ōnuki Y 2003 Phys. Rev. B 67 144507
[26] Raj S, Iida Y, Souma S, Sato T, Takahashi T, Ding H, Ohara S, Hayakawa T, Chen G F, Sakamoto I, and Harima H 2005 Phys. Rev. B 71 224516
[27] Satoh K, Fujimaki Y, Umehara I, Itoh J, Ōnuki Y, and Kasaya M 1993 Physica B 186–188 658
[28] Lawrence J M and Shapiro S M 1980 Phys. Rev. B 22 4379
[29] Buschow K H J, de Wijn H W, and van Diepen A M 1969 J. Chem. Phys. 50 137
[30] Thompson J D, Movshovich R, Fisk Z, Bouquet F, Curro N J, Fisher R A, Hammel P C, Hegger H, Hundley M F, Jaime M, Pagliuso P G, Petrovic C, Phillips N E, and Sarrao J L 2001 J. Magn. Magn. Mater. 226–230 5
[31] Macaluso R T, Sarrao J L, Moreno N O, Pagliuso P G, Thompson J D, Fronczek F R, Hundley M F, Malinowski A, and Chan J Y 2003 Chem. Mater. 15 1394
[32] Nicklas M, Sidorov V A, Borges H A, Pagliuso P G, Petrovic C, Fisk Z, Sarrao J L, and Thompson J D 2003 Phys. Rev. B 67 020506
[33] Petrovic C, Pagliuso P G, Hundley M F, Movshovich R, Sarrao J L, Thompson J D, Fisk Z, and Monthoux P 2001 J. Phys.: Condens. Matter 13 L337
[34]Kalychak Y M, Zaremba V I, Baranyak V M, Bruskov V A, and Zavalii P Y 1989 Russian Metallurgy 1 213
[35] Petrovic C, Movshovich R, Jaime M, Pagliuso P G, Hundley M F, Sarrao J L, Fisk Z, and Thompson J D 2001 Europhys. Lett. 53 354
[36] Paglione J, Ho P C, Maple M B, Tanatar M A, Taillefer L, Lee Y, and Petrovic C 2008 Phys. Rev. B 77 100505(R)
[37] Hegger H, Petrovic C, Moshopoulou E G, Hundley M F, Sarrao J L, Fisk Z, and Thompson J D 2000 Phys. Rev. Lett. 84 4986
[38] Tobash P H, Ronning F, Thompson J D, Scott B L, Moll P J W, Batlogg B, and Bauer E D 2012 J. Phys.: Condens. Matter 24 015601
[39] Bauer E D, Lee H O, Sidorov V A, Kurita N, Gofryk K, Zhu J X, Ronning F, Movshovich R, Thompson J D, and Park T 2010 Phys. Rev. B 81 180507(R)
[40] Kurenbaeva Z M, Murashova E V, Seropegin Y D, Noel H, and Tursina A I 2008 Intermetallics 16 979
[41] Morris G D, Heffner R H, Moreno N O, Pagliuso P G, Sarrao J L, Dunsiger S R, Nieuwenhuys G J, Maclaughlin D E, and Bernal O O 2004 Phys. Rev. B 69 214415
[42] Malinowski A, Hundley M F, Moreno N O, Pagliuso P G, Sarrao J L, and Thompson J D 2003 Phys. Rev. B 68 184419
[43] Souma S, Raj S, Campuzano J C, Sato T, Takahashi T, Ohara S, and Sakamoto S 2008 Physica B 403 752
[44] Jiang R, Mou D X, Liu C, Zhao X, Yao Y X, Ryu H, Petrovic C, Ho K M, and Kaminski A 2015 Phys. Rev. B 91 165101
[45] Singley E J, Basov D N, Bauer E D, and Maple M B 2002 Phys. Rev. B 65 161101(R)
[46] Shishido H, Settai R, Aoki D, Ikeda S, Nakawaki H, Nakamura N, Iizuka T, Inada Y, Sugiyama K, Takeuchi T, Kindo K, Kobayashi T C, Haga Y, Harima H, Aoki Y, Namiki T, Sato H, and Ōnuki Y 2002 J. Phys. Soc. Jpn. 71 162
[47] Elgazzar S, Opahle I, Hayn R, and Oppeneer P M 2004 Phys. Rev. B 69 214510
[48] Allan M P, Chuang T, Massee F, Xie Y, Ni N, Bud K S L, Boebinger G S, Wang Q, Dessau D S, Canfield P C, Golden M S, and Davis J C 2013 Nat. Phys. 9 468
[49] Zhou B B, Misra S, Da S N E H, Aynajian P, Baumbach R E, Thompson J D, Bauer E D, and Yazdani A 2013 Nat. Phys. 9 474
[50] Koitzsch A, Opahle I, Elgazzar S, Borisenko S V, Geck J, Zabolotnyy V B, Inosov D, Shiozawa H, Richter M, Knupfer M, Fink J, Büchner B, Bauer E D, Sarrao J L, and Follath R 2009 Phys. Rev. B 79 075104
[51] Fujimori S 2016 J. Phys.: Condens. Matter 28 153002
[52] Harrison N, Alver U, Goodrich R G, Vekhter I, Sarrao J L, Pagliuso P G, Moreno N O, Balicas L, Fisk Z, Hall D, Macaluso R T, and Chan J Y 2004 Phys. Rev. Lett. 93 186405
[53] Fujimori S, Fujimori A, Shimada K, Narimura T, Kobayashi K, Namatame H, Taniguchi M, Harima H, Shishido H, Ikeda S, Aoki D, Tokiwa Y, Haga Y, and Ōnuki Y 2006 Phys. Rev. B 73 224517
[54] Rusz J, Oppeneer P M, Curro N J, Urbano R R, Young B, Lebegue S, Pagliuso P G, Pham L D, Bauer E D, Sarrao J L, and Fisk Z 2008 Phys. Rev. B 77 245124
[55] Moore D P, Durakiewicz T, Joyce J J, Arko A J, Morales L A, Sarrao J L, Pagliuso P G, Wills J M, and Olson C G 2002 Physica B 312–313 134
[56] Altarawneh M M, Harrison N, Mcdonald R D, Balakirev F F, Mielke C H, Tobash P H, Zhu J X, Thompson J D, Ronning F, and Bauer E D 2011 Phys. Rev. B 83 081103(R)
[57] Shen B, Yu L, Liu K, Lyu S, Jia X, Bauer E D, Thompson J D, Zhang Y, Wang C, Hu C, Ding Y, Sun X, Hu Y, Liu J, Gao Q, Zhao L, Liu G, Xu Z, Chen C, Lu Z, and Zhou X J 2017 Chin. Phys. B 26 077401
[58] Li P, Wu Z Z, Wu F, Guo C Y, Liu Y, Liu H J, Sun Z, Shi M, Rodolakis F, Mcchesney J L, Cao C, Yuan H Q, Steglich F, and Liu Y 2019 Phys. Rev. B 100 155110