Visualizing Heavy Fermion Formation and their Unconventional Superconductivity in f -Electron Materials

In solids containing elements with f -orbitals, the interaction between f -electron spins and those of itinerant electrons leads to the development of low-energy fermionic excitations with a heavy effective mass. These excitations are fundamental to the appearance of unconventional superconductivity observed in actinide- and lanthanide-based compounds. We use spectroscopic mapping with the scanning tunneling microscope to detect the emergence of heavy excitations with lowering of temperature in Ce- and U-based heavy fermion compounds. We demonstrate the sensitivity of the tunneling process to the composite nature of these heavy quasiparticles, which arises from quantum entanglement of itinerant conduction and f -electrons. Scattering and interference of the composite quasiparticles is used in the Ce-based compounds to resolve their energy-momentum structure and to extract their mass enhancement, which develops with decreasing temperature. Finally, by extending these techniques to much lower temperatures, we investigate how superconductivity, with a nodal d -wave character, develops within a strongly correlated band of composite excitations.


Introduction
A remarkable variety of collective electronic phenomena have been discovered in compounds with partially filled f-orbitals where electronic correlations are dramatically enhanced. 1,2) In these compounds the entanglement of the rather localized f-electrons with the surrounding itinerant electrons starts at relatively high temperature leading to the development of low-energy composite quasiparticles with a heavy effective mass. Tuning the hybridization between f-orbitals and itinerant electrons can destabilize the heavy Fermi liquid state at low temperatures towards an antiferromagnetically ordered ground state. [3][4][5][6][7][8] In proximity to such a quantum phase transition, between itinerancy and localization of f-electrons, many heavy fermion systems exhibit magnetism and unconventional superconductivity at low temperatures [ Fig. 1(a)]. 9) Thermodynamic and transport studies have long provided evidence for heavy quasiparticles, their unconventional superconductivity, and non-Fermi liquid behavior in a variety of Kondo lattice systems. 1,2,[9][10][11][12] However, the emergence of a coherent band of heavy quasiparticles near the Fermi energy, as a result of the hybridization of the localized f-electrons with conduction electrons [ Fig. 1(b)], remains not well understood. [12][13][14][15] Part of the challenge has been the inability of spectroscopic measurements to probe the development of heavy quasiparticles with lowering of temperature and to characterize their properties with highenergy resolution. Recently, various theoretical approaches, including several numerical studies, remarkably reproduce the generic composite band structure of Fig. 1(b). [16][17][18][19][20] Theoretical modeling has also shown that tunneling spectroscopy can be a powerful probe of this composite nature of heavy fermions. [21][22][23][24] Depending on the relative tunneling amplitudes to the light conduction (t c ) or to the heavy f-like (t f ) components of the composite quasiparticles, and due to their interference, tunneling spectroscopy can be sensitive to different features of the hybridized band structure [Figs. 1(c) and 1(d); see detail below]. Such precise measurements of heavy fermion formation are not only required for understanding the nature of these electronic excitations close to quantum phase transitions 25) but are critical to identifying the source of unconventional superconductivity near such transitions, which continues to be at the forefront of unsolved problems in all of physics.
Here we review our recent advances in the application of STM techniques to study the formation of heavy fermions and their superconductivity. [26][27][28] To provide a controlled study of the formation of heavy fermion excitations within a Kondo lattice system and visualize the emergence of heavy electron superconductivity, we carried out studies on the Ce 1 M 1 In 5 (with M ¼ Co, Rh) material system. These socalled 115 compounds offer the possibility to tune the interaction between the Ce's f-orbitals and the itinerant spd conduction electrons using isovalent substitutions at the transition metal site within the same tetragonal crystal structure. Consequently, the ground state of this system can be tuned (in stoichiometric compounds) between antiferromagnetism, as in CeRhIn 5 (T N ¼ 3:5 K), to superconductivity, as observed in CeCoIn 5 (T c ¼ 2:3 K) and CeIrIn 5 (T c ¼ 0:4 K). 9) Transport studies show a drop in the electrical resistivity in CeCoIn 5 around T Ã ¼ 50 K, which has been interpreted as evidence for the development of a coherent heavy quasiparticle band, followed by a linear resistivity at lower temperature (above T c ) 29)a behavior that has been associated with the proximity to the QCP. Quantum oscillations and thermodynamic measurements find a heavy effective mass (10{50m 0 , where m 0 is the bare electron mass) for CeCoIn 5 , while in the same temperature range the felectrons in CeRhIn 5 are effectively decoupled from the conduction electrons. 30,31) We demonstrate the sensitivity of the tunneling process to the composite nature of these heavy quasiparticles in CeCoIn 5 , which arises from quantum entanglement of itinerant conduction and f-electrons. We contrast this observation in CeCoIn 5 with the exotic heavy fermion compound URu 2 Si 2 , which also shows similar composite heavy fermion behavior at high temperatures but undergoes an enigmatic second order phase transition at T HO ¼ 17:5 K to a "hidden order" state. Using spectroscopic imaging of quasiparticle interference in Ce-115 compounds, we visualize the energy-momentum structure of these composite heavy fermion excitations which develops below T Ã near the Fermi energy. Upon further lowering of temperature, we find the spectrum of these heavy excitations to be strongly modified just prior to the onset of superconductivity by a suppression of the spectral weight near E F , reminiscent of the pseudogap state in the cuprates. 32,33) Finally, we demonstrate how nodal superconductivity develops within this strongly correlated band of composite excitations.

Cleaved surfaces and topographs of CeCoIn 5
Figures 2(a) and 2(b) show STM topographs of a single crystal of CeCoIn 5 doped with 0.15% of Hg-CeCo-(In 0.9985 Hg 0.0015 ) 5 for reasons which will be addressed below. All high temperature measurements (T ! 20 K) were performed on CeCo(In 0.9985 Hg 0.0015 ) 5 . For simplicity however, from here on we will refer to it as CeCoIn 5 . The samples were cleaved in situ in our variable temperature ultra-high vacuum STM. The cleaving process results in exposing multiple surfaces terminated with different chemical compositions. The crystal symmetry necessarily requires multiple surfaces for cleaved samples, as no two equivalent consecutive layers occur within the unit cell. Therefore breaking of any single chemical bond will result in different layer terminations on the two sides of the cleaved sample. Experiments on multiple cleaved samples have mostly revealed three different surfaces, two of which are atomically ordered (termed surfaces A and B in Fig. 2) with a periodicity corresponding to the lattice constant of the bulk crystal structure, while the third surface (termed surface C, Fig. 2) is reconstructed. Comparison of the relative heights of the sub-unit cell steps between the different layers [Figs. 2(c) and 2(d)] to the crystal structure determined from scattering experiments 34) enables us to identify the chemical composition of each exposed surface [ Fig. 2(d)]. Experiments on the isostructural CeRhIn 5 reveal similar results (not shown here), where cleaving exposes the corresponding multiple layers.

Composite nature of heavy fermion excitations in
CeCoIn 5 Spectroscopic measurements of CeCoIn 5 show the sensitivity of the tunneling process to the composite nature of the hybridized heavy fermion states. As shown in Fig. 3(a), tunneling spectra on surface A (identified as the Ce-In layer) The composite nature of the heavy fermion excitations manifests itself by displaying different spectroscopic characteristics for tunneling into the different atomic layers. Figure 3(b) shows spectra measured on surface B (identified as Co) of CeCoIn 5 that looks remarkably different than those measured on surface A [ Fig. 3(a)]. In the same temperature range where spectra on surface A [ Fig. 3(a)] develop a depletion of spectral weight near the Fermi energy, surface B shows a sharp enhancement of spectral weight within the same 30-40 meV energy window [ Fig. 3(b)]. With further lowering of temperature, the enhanced tunneling on surface B evolves into a double-peak structure. As a control experiment, measurements on the corresponding surface in CeRhIn 5 , once again, display no sharp features in the same temperature and energy windows [ Fig. 3 A model calculation for tunneling to composite heavy excitations can reproduce the different spectroscopic lineshapes on the two different surfaces. Following recent theoretical efforts, 22,23) we compute the spectroscopic properties of a model band structure in which a single hole-like itinerant band of spd-like electrons E c k ðk x ; k y Þ hybridizes with a narrow band of f-like electrons E f k ðk x ; k y Þ, with Here, t and ® represent the nearest neighbor hopping of the conduction electrons and the chemical potential, respectively, and 0 , 1 , and " f 0 represent the nearest and next-nearest site spin correlations, and the position of the heavy band with respect to the Fermi energy, respectively. The hybridization of these two bands with a hybridization amplitude v yields the generic heavy fermion band structure of Fig. 1(b). The differential conductance dI=dV, which represents the tunneling to the hybridized band structure, can then be calculated by ½t ImĜðk; !Þt ij : Where the matrixtðt c ; t f Þ controls the ratio of tunneling to the c-and f-bands andĜ defines the full Green's function describing the hybridization between the c-and f-electron bands. 22,23) The results of our calculations [Figs. 3(c) and   2 Si 2 To demonstrate the generic behavior of the tunneling sensitivity to the composite nature of heavy fermion excitations, we show in Fig. 4 similar measurements carried out on another heavy fermion compound. URu 2 Si 2 displays heavy electron formation below T Ã % 80 K, as probed by transport measurements. 35) However, this compound has long puzzled scientists due to its enigmatic phase transition to a hidden order state at T HO ¼ 17:5 K, whose order parameter and connection to the heavy excitations has since remained a mystery. 10,36) In Figs. 4(a) and 4(b) we show STM topographs of the cleaved surfaces of URu 2 Si 2 , which similar to CeCoIn 5 expose multiple atomic layers within the unit cell. (Here we show the two relevant surfaces, AA and BA. A third surface, which undergoes surface reconstruction, is also observed. 26 3(b)]. The spectra on surface AA of URu 2 Si 2 display an asymmetric Fano lineshape, a signature of quantum interference between tunneling to a discrete (t f ) and continuum (t c ) electronic states. 37,38) Similar measurements on surface BA display a double peak structure, indicating an enhanced co-tunneling to the discrete f-like states on this surface. As in CeCoIn 5 , the double-peak structure, extended over µ40 meV, reflects the high density of states originating from the flat dispersions of the two heavy fermion bands [Figs. 1(b) and 1(d)], which are already formed above T HO . These similarities in the two, rather different, material systems demonstrate the generic behavior of the tunneling process to the composite nature of heavy fermion excitations. Extending the measurements in URu 2 Si 2 below T HO further reveals that the hidden order occurs on the lower heavy fermion band with an energy scale of µ4 meV around the Fermi energyan order of magnitude smaller than the hybridization energy scale of µ40 meV.

Visualizing quasiparticle mass enhancement in
CeCoIn 5 To directly probe the energy-momentum structure of heavy quasiparticles we consider the Ce-115 material systems again. Spectroscopic mapping with the STM enables us to visualize scattering and interference of these quasiparticle excitations from impurities or structural defects. Elastic scattering of quasiparticles from these imperfections gives rise to standing waves in the conductance maps at wavelengths corresponding to 2=q, where q ¼ k f À k i is the momentum transfer between initial (k i ) and final (k f ) states at the same energy. We expect that q's with the strongest intensity connect regions of high density of states on the contours of constant energy and hence provide energy-momentum information of the quasiparticle excitations. We characterize the scattering q's using discrete Fourier transforms (DFTs) of STM conductance maps measured at different energies. Figure 5(a) shows examples of energy-resolved STM conductance maps on surface A of CeCoIn 5 measured at 20 K displaying signatures of scattering and interference of quasiparticles from defects and step edges. These conductance maps show clear changes of the wavelength of the modulations as a function of energy. Perhaps the most noticeable are the changes around each random defect (0.15% of Hg dopants, which is doped intentionally to introduce quasiparticle scattering centers. Their presence, however, at this low concentration does not change the thermodynamic properties). Figure 5(b) shows DFTs of such maps that display sharp non-dispersive Bragg peaks [at the corners, (AE2=a; 0), (0; AE2=a)] corresponding to the atomic lattice, as well as other features (concentric square-like shapes) that disperse with energy, collapse [ Fig. 5(b); 0 meV], and disappear [ Fig. 5(b); 9 meV] near the Fermi energy. We have carried out such measurements both at low temperatures [20 K, Fig. 5(b)], where the spectrum shows signatures of hybridization between conduction electrons and f-orbitals, as well as at high temperatures [70 K, Fig. 5(c)] where such features are considerably weakened [e.g., Fig. 5(c); 2 meV, 10 meV]. As a control experiment, we have also carried out the same measurements on the corresponding surface of CeRhIn 5 [ Fig. 5(d)], for which signatures of heavy electron behavior   are absent [e.g., Fig. 3(a)] in the same temperature window (20 K). Whereas these measurements on surface A display QPI patterns dominated by the lighter part of the composite heavy fermion bands that weaken near E F at low temperatures, measurements on surface B of CeCoIn 5 show a strongly dispersing QPI signal that is present only near E F , in unison with related signatures in the tunneling spectra (Fig. 6).
Understanding details of the QPI in Figs. 5 and 6 requires detailed modeling of the complex band structure of the 115 compounds, which from previous theoretical calculations, quantum oscillation, and angle resolved photoemission spectroscopy measurements is known to consist of multiple three-dimensional bands. 39,40) These previous measurements and calculations have also shown that the so called ¡ and ¢ bands are the most relevant near E F . Our QPI measurements show features that are consistent with 2k F scattering originating from the ¡ and ¢ bands. However, inferring a unique Fermi surface from STM measurements in a threedimensional, multi-band material without making large number of assumptions is not possible. Regardless, analyzing the features of the energy-resolved DFT maps provide direct evidence for mass enhancement of quasiparticles. . This is the direct signature of the quasiparticles acquiring heavy effective mass at low energies near the Fermi energy. Detailed analysis of the QPI bands estimates the mass enhancement near the Fermi energy to be about 20{30m 0 [Figs. 7(c) and 7(d)], a value which is close to that seen in quantum oscillation studies of CeCoIn 5 . 30,31) Contrasting low temperature QPI patterns on CeCoIn 5 to measurements on the same compound at high temperatures [70 K, Figs. 7(e) and 7(f )], where the hybridization gap is weak, or to measurements on CeRhIn 5 [20 K, Figs. 7(g) and 7(h)], where signatures of a hybridization gap are absent in the tunneling spectra, confirms that the development of this gap results in apparent splitting of the bands which are responsible for both the scattering and the heavy effective mass in the QPI measurements. Furthermore, these measurements show that the underlying band structure responsible for the scattering wavevectors away from the Fermi energy is relatively similar between CeCoIn 5 and CeRhIn 5 . Only when f-electrons of the Kondo lattice begin to strongly hybridize with conduction electrons and modify the band structure within a relatively narrow energy window (µ30 meV), we see signatures of heavy fermion excitations in QPI measurements, signaling a transition from small to large Fermi surface. 5 We now turn to low temperatures to address the emergence of superconductivity in CeCoIn 5 Fig. 8(a)]. However, instead of focusing on measurements of surface A, where the tunneling is dominated by the lighter part of the composite band, we turn to measurements of surface B, which probes the narrow bands of heavy excitations resulting in the double peak structure near E F . Lowering the temperature from 7.2 to 5.3 K, above T c , we find that this peak is modified by the onset of a pseudogaplike feature at a smaller energy scale [ Fig. 8(b)]. Further cooling shows the onset of a distinct superconducting gap below T c inside the pseudogap. Measurements in a magnetic  Fig. 8(c)]. This behavior is reminiscent of the pseudogap found in underdoped cuprates, where the superconducting gap opens inside an energy scale describing strong correlations that onset above T c . However, unlike cuprates, here we clearly distinguish between the two energy scales by performing high-resolution spectroscopy in a magnetic field large enough to fully suppress superconductivity. The spectroscopic measurements suggest that electronic or magnetic correlations alter the spectrum of heavy excitations by producing a pseudogap within which pairing takes place. These measurements also show the shape of the spectra at the lowest temperature to be most consistent with a d-wave superconducting gap, as they have a nearly linear density of states near zero energy (Fig. 8). However, measurements on all surfaces and on several samples reveal that this d-wave gap (with a magnitude of 535 AE 35 µV, consistent with that extracted from point contact data 18,19) ) is filled (40%) with low energy excitationsa feature that cannot be explained by simple thermal broadening (determined to be 245 mK). The complex multiband structure of CeCoIn 5 could involve different gaps on different Fermi surface sheets, and there is the possibility that some remain ungapped even at temperatures well below T c . 41) Another contribution to the in-gap density of states could come from surface impurities, since even non-magnetic impurities perturb a nodal superconductor. The first such signature can be found by examining the response of low-energy excitations to extended potential defects such as atomic step edges. Spectroscopic mapping with the STM upon approaching such steps shows direct evidence for the suppression of superconductivity in their immediate vicinity [Figs. 9(a) and 9(b)]. This suppression is consistent with the expected response of a nodal superconductor to non-magnetic scattering [ Fig. 9(c)], analogous to similar observations in the cuprates, 42) and in marked contrast with step-edge measurements of conventional s-wave superconductors. 28) The data in Fig. 9(d) provide a direct measure of the Bardeen-Cooper-Schriefer (BCS) coherence length 43) BCS ¼ 56 AE 10 Å, in agreement with BCS $ h " v F =Á $ 60 Å using the gap observed in Fig. 8 (0.5 meV) and the Fermi velocity extracted from Fig. 7 (1:5 Â 10 6 cm/s).

Heavy electron superconductivity in CeCoIn
A more spectacular demonstration of the nodal pairing character in CeCoIn 5 can be obtained from examining the spatial structure of in-gap states associated with defects on the surface of cleaved samples. The spatial structure of impurity quasi-bound states, which are mixtures of electronand hole-like states, can be a direct probe of the order parameter symmetry. Figure 10 shows an extended defect with a four-fold symmetric structure, which perturbs the low energy excitations of CeCoIn 5 by inducing an in-gap state. Probing the spatial structure of these impurity states, we not only find their expected electron-hole asymmetry, but also find that their orientation is consistent with that predicted for a d x 2 Ày 2 superconductor [Figs. 10(b)-10(e)]. 32,44) The minima (maxima) in the oscillations for hole-like (electron-like) states identify the nodes of the d-wave order to occur at 45°to the atomic axes [ Fig. 10(h)]. In fact, these features in the STM conductance maps are identical to those associated with Ni impurities in high-T c cuprates. 43,45) However, in contrast to measurements in the cuprates, we are able to determine the spatial structure that such impurities induce on the normal state by suppressing pairing at high magnetic fields. Such measurements allow us to exclude the influences of the normal state band structure of the impurity shape, or of the tunneling matrix element 43)

Conclusions
In summary, the experimental results and the model calculations presented here provide a comprehensive picture of how heavy fermion excitations in the 115 Ce-based Kondo lattice systems emerge with lowering of temperature or as a result of chemical tuning of the interaction between the Ce felectrons and the conduction electrons. The changes in the scattering properties of the quasiparticles directly signal the flattening of their energy-momentum structure and the emergence of heavy quasiparticles near the Fermi energy. Such changes are also consistent with the predicted evolution from small to large Fermi surface as the localized f-electrons hybridize with the conduction electrons. The sensitivity of the tunneling to the surface termination and the successful modeling of these data provide direct spectroscopic evidence of the composite nature of heavy fermions and offer a unique method to disentangle their components. Spectroscopic signatures in CeCoIn 5 and URu 2 Si 2 above T HO , reveal a similar hybridization energy scale (30-40 meV), which mostly effects quasiparticles above the chemical potential. Furthermore, contrasting measurements above and below T HO , show that the hidden order state in URu 2 Si 2 opens a narrow gap near E F with an energy scale much smaller than the hybridization gap. Finally, extending the measurements in CeCoIn 5 to very low temperatures reveals the appearance of a pseudogap and direct evidence for d x 2 Ày 2 superconductivity, which ties the phenomenology of the Ce-115 system to that of the high-temperature cuprate superconductors. Ali Yazdani is a professor of Physics at Princeton University whose research program focuses on the development and application of novel experimental methods to directly visualize exotic electronic phenomena in solids. Yazdani received his BA in Physics from UC Berkeley in 1989, and his Ph. D. in Physics from Stanford University in 1995. He was a postdoctoral scientist at IBM's Almaden Research Center in California, where he pioneered experiments to probe superconductivity on the atomic scale with the scanning tunneling microscope in the early 90s. After IBM, he went on to establish a research program as a faculty member at the University of Illinois at Urbana -Champaign in 1997, and then as a professor of Physics at Princeton University in 2005. Yazdani's research program spans experiments on high temperature superconductivity, magnetism in semiconductors, nanoscience, and the newly discovered topological phases of electrons. Yazdani is a fellow of the American Association for Advancement of Science and American Physical Society. J. Phys. Soc. Jpn. 83, 061008 (2014) Special Topics