Coexisting Kondo hybridization and itinerant f-electron ferromagnetism in UGe2

Kondo hybridization in partially filled f-electron systems conveys significant amount of electronic states sharply near the Fermi energy leading to various instabilities from superconductivity to exotic electronic orders. UGe2 is a 5f heavy fermion system, where the Kondo hybridization is interrupted by the formation of two ferromagnetic phases below a 2nd order transition Tc ~ 52 K and a crossover transition Tx ~ 32 K. These two ferromagnetic phases are concomitantly related to a spin-triplet superconductivity that only emerges and persists inside the magnetically ordered phase at high pressure. The origin of the two ferromagnetic phases and how they form within a Kondo-lattice remain ambiguous. Using scanning tunneling microscopy and spectroscopy, we probe the spatial electronic states in the UGe2 as a function of temperature. We find a Kondo resonance and sharp 5f-electron states near the chemical potential that form at high temperatures above Tc in accordance with our density functional theory (DFT) + Gutzwiller calculations. As temperature is lowered below Tc, the resonance narrows and eventually splits below Tx dumping itinerant f-electron spectral weight right at the Fermi energy. Our findings suggest a Stoner mechanism forming the highly polarized ferromagnetic phase below Tx that itself sets the stage for the emergence of unconventional superconductivity at high pressure.

Over the past decade, interest in exploiting ferromagnetic superconductivity with non-trivial topology dominated the field of quantum matter due to their robust functionalities in quantum information [1][2][3] . Yet, quantum materials that exhibit natural coexistence of ferromagnetism and superconductivity remain rare. To date, only a handful of such synthesized single crystalline materials exist, with the majority being uranium-based heavy fermion compounds including UGe2 4 , URhGe 5-7 , UCoGe 8 and the recently discovered UTe2 9 . In these compounds, the 5felectrons play a critical role in the emergence of the exotic superconductivity 10 , making it particularly crucial to understand their controversial normal state behavior.
f-electrons in heavy fermion compounds exhibit dual characteristics of being itinerant and localized, driving an electronic competition between magnetism, very often of antiferromagnetic character, and heavy Fermi liquid behavior with quenched magnetic moments [11][12][13][14][15] . Recent experimental and computational work demonstrated the dual nature of 5f electrons in USb2 16,17 , an antiferromagnetic heavy fermion system, through orbital selectivity, providing a natural explanation of how localized magnetism and itinerant heavy fermions of the same uranium 5f electrons coexist. The emergence of f-electron ferromagnetism and its interplay with Kondo coherence remains much less explored.
The ferromagnetic heavy fermion UGe2 displays an interesting phase diagram 5,18,19 . At ambient pressure, a second order paramagnetic-to-ferromagnetic phase transition at a relatively high Tc ~ 52 K 18 is followed by a crossover meta-magnetic transition from a weakly polarized ferromagnetic state, FM1, to a strongly polarized ferromagnetic state FM2 20 at Tx ~ 32 K. Transport measurements show the emergence of the Kondo-lattice effect at TK ~ 110 K, well above Tc 19 . How the Kondo effect is impacted by the emergent ferromagnetism at and below Tc remains a question to be answered. Below Tc, specific heat measurements display a broad hump centered around Tx 19,21 which, along with magnetization measurements 19,22 and neutron scattering 23 shows the presence of itinerant and localized subset of the uranium 5f electrons. In the same temperature range, Hall effect studies reveal a rapid increase of charge carriers below Tx suggestive of some sort of Fermi surface reconstruction 24 . This reconstruction is argued to be caused by the sudden delocalization of the uranium 5f electrons. The microscopic origins of Tc and Tx are particularly important for the mechanism of emergent exotic superconductivity in UGe2. With the application of hydrostatic pressure, the pressure-dependent FM2 transition line Tx(P) decreases and terminates at the maximum of the emergent superconducting dome at Px ∼ 1.2 GPa, suggesting its fluctuations and destruction are directly related to the mechanism of superconductivity 7 . Furthermore, the superconducting dome only persists inside the FM1 phase, where both phases simultaneously disappear at the exact same pressure Pc ∼ 1.5 GPa indicating an intimate relation between the ferromagnetism and superconductivity 25,26 . Therefore, the emergence of itinerant 5f-electrons through the Kondo effect in UGe2 and their evolution across Tc and Tx forms the low temperature normal state near the Fermi energy (EF) out of which ferromagnetic superconductivity develops.
Theoretical understanding of the nature of the ferromagnetic state below Tx is controversial. One mechanism comes from the phenomenological ideas following the rigid-band Stoner approach, where two sufficiently sharp and narrowly separated density-of-state peaks located near the Fermi energy form the majority and minority spin bands 27 . Another idea involves charge and spin density waves emerging below Tx 18 . In either case, direct experimental signature of a double peak structure or density waves have not been observed to date.
A sharp resonance in the density of states can naturally arise in heavy fermion compounds, whose energy relative to EF depends on the valence of the f-electrons in the material system 28,29 . Scanning tunneling microscopy (STM) has the spatial and energy resolution to probe the sharp resonance and it's possible splitting 30 . Yet, due to the lack of a natural cleaving plane in UGe2, which is crucial to obtain clean surfaces for STM, such an experiment have so far not been carried out. Here we use STM to probe the local electronic states and their temperature evolution near EF in single crystal UGe2. We find multiple peaks in the density of states located near the chemical potential above T c . The finding is in qualitative agreement with our DFT+ Gutzwiller calculations presented here, attributing their origin to the different 5f-electronic orbital characters. With lowering of temperature, the two peaks located nearest to EF strengthen and narrow. Below Tc and particularly near Tx ~ 32 K, the peaks split, forming additional kinks that further develop and evolve with temperature suggesting a Stoner mechanism of the ferromagnetic order. Our finding indicates a significant degree of itinerant character of the f-electrons through Kondo hybridization and the itinerant nature of the ferromagnetism involving the same f-electrons. At the lowest temperature, a sharp f-electronic density of states is formed at EF setting the stage for the emergence of ferromagnetic spin-polarized superconductivity at higher pressure. Figure 1a, b shows STM topographs of the (010) surface of single crystal UGe2, in-situ cleaved in our ultra-high vacuum, variable temperature STM. Cleaving exposes alternating terraces of two chemically different surfaces, termed A and B in Fig.1c. While surface A displays spatial uniformity with no atomic corrugation indicating the extended nature of the electronic states, surface B undergoes a surface reconstruction, whose quasiperiodic structure differs between different cleaves, as seen in Fig.1a, b. The asymmetry in the step height (A→B > B→A, with A↔A = B↔B ≡ b-axis unit cell) between the different terraces allows us to compare the results to the crystal structure. Assuming only a single chemical bond breaking during the cleaving process leads us to identify surfaces A and B as uranium and germanium terminated, respectively (Fig.1c). Such an assumption is justified by the fact that only two surfaces have been observed based on ten different sample cleaves. The Ge surface with two Ge atoms per ac-plane unit cell, as compared to a single U or Ge atom per ac-plane for all other layers (see Fig.1), is also more likely to undergo surface relaxation and buckling. Nevertheless, the surface assignment here does not change the conclusions reached below. STM spectra probed on the two surfaces just above Tc reveal two asymmetric low-energy peaks in the density of states in close proximity to the chemical potential, whose intensities are different on the two surfaces (Fig.1d). An additional broader high-energy peak near ~100 meV is also observed. Note that the intensity of the peaks on surface-B are spatially non-uniform and is further elaborated below. At low temperatures (8 K), the low energy peaks undergo a splitting with the emergence of a kink/shoulder (Fig.1e). The high-energy peak remains mostly unchanged.
Peaks in the density of states near EF is the hallmark of itinerant f-electron systems and have been seen in previous STM experiments in various 4f and 5f heavy fermions 31 . They are the results of the Kondo hybridization of localized f-orbitals with conduction electrons. To corroborate this observation, we compute the electronic structure of UGe2 by employing the generalized gradient approximation to density functional theory (DFT) in combination with the Gutzwiller approximation (DFT+Gutzwiller) 32 . This method captures electronic correlations beyond the single-particle picture of DFT and has been successfully applied to other f-electron systems such as UO2 33 . The local Coulomb interaction strength and the Hund's coupling constant are U = 6.0 eV and J = 0.57 eV, respectively, for the correlated U 5f-orbital. The DFT+Gutzwiller calculations were performed at T = 0 K within the paramagnetic phase. Within DFT there is a significant mixing between U 5f5/2 and 5f7/2 states, so their spin-orbit splitting is not apparent. X-ray photoemission spectroscopy 34 and x-ray magnetic circular dichroism 35 measurements indicate that there is a clear spin-orbit splitting between U 5f5/2 and 5f7/2. Furthermore, the 5f5/2 level lies below the 5f7/2 level and the magnitude of the splitting is around 1.1 eV. This feature is well captured by the DFT+Gutzwiller calculations, which gives a spin-orbit splitting of ~1.5 eV. Our calculations indeed show multiple uranium 5f-electron peaks with different orbital character to reside near EF (Fig.1f). More specifically, three major peaks located at energies of -18meV, +35meV and +66meV that have characters of U (J = 5/2, mJ = ±1/2, ±5/2, ±1/2), respectively, are qualitatively consistent with the high temperature experimental data observed on surface A and/or B at ~ -20meV, ~ +25meV, and ~ +100 meV (Fig.1d, e).
While the relative widths and weights of the spectral lineshapes are spatially uniform on surface A (see Fig.2a), they vary significantly on surface B due to the structural inhomogeneity induced by surface reconstruction, as seen in Fig.2b. For example, looking at the different peaks, one can see that their intensity can be dramatically suppressed depending on the spatial location on the surface. This also applies to the spectra in Fig.1d, where at high temperature only the negative peak is observed with almost no peak intensity on the positive side. The disappearance of the positive bias peaks is due to the particular location of the spectrum on the surface and other locations (not shown here) do reveal a finite peak at positive and negative biases. This spatial variation renders studying the detailed temperature dependent evolution unreliable on surface B. We therefore focus on surface A (U surface) to probe the temperature dependence of the spectra across the two ferromagnetic transitions. Figure 3 shows our temperature dependent spectroscopy measurements carried out on surface A. The dI/dV conductance were measured in a constant current mode, Iset, and a bias voltage applied to the sample, Vbias, with a bias modulation of Vmod. Two sets of data with different experimental settings (energy-resolution) of Iset = 150 pA, Vbias = 500 meV, Vmod = 5 meV (Figure 3a) and Iset = 1 nA, Vbias = 200 meV, Vmod = 1 meV (Figure 3b) are shown. The spectra display an asymmetric resonance analogous to that seen in other heavy fermion systems 31,[36][37][38][39][40][41] . The observed resonance is the manifestation of Kondo hybridization, delocalizing the f-electrons and merging them into the Fermi sea starting already at temperatures above Tc. As temperature is lowered below Tc, we observe the sharp kink at E1 (near the Fermi energy) starting to develop particularly below ~35 K (see insets in Fig.3a, b). At the lowest measured temperature of 8 K, a clear double peak structure can be resolved with a peak separation (E1 -E2) of ~ 16 meV.
The strong temperature broadening of the spectral lineshapes makes it difficult to pin-point the onset of the E1 kink in the raw data. In Fig.3c, d, we show the 2 nd derivative of the spectral lineshapes, which highlights the kink-structure near the Fermi energy. While the temperature evolution is weak above ~ 35 K, it becomes more pronounced at lower temperatures. To better visualize this behavior, we contrast in Fig.3e, the high resolution spectra with a model Fano lineshape. A Fano lineshape in STM spectra resembles an asymmetric resonance peak due to interference between the two tunneling paths from the tip to the heavy (resonance) and light (continuum) electronic states of a Kondo lattice and has been widely used in STM analysis of heavy fermion systems [42][43][44][45][46] . The equation below represents the Fano lineshape where E characterizes the resonance energy, Γ the resonance linewidth expressed as Half Width at Half Maximum (HWHM) and q is the tip-sample coupling, also known as the asymmetry parameter. A is related to the amplitude of the resonance. Figure 3e shows the data and the corresponding fit to a single Fano lineshape. We observe at 55K (T>Tc) that the data can be nicely modeled by a single resonance. At 35 K however, we can see that the data deviates from a single Fano lineshape particularly in the energy range of ±10 meV, where a second resonance develops and grows stronger with further cooling. We therefore use a two Fano lineshape model (one centered at E1 and another at E2) to fit the temperature dependent data. Figure 4a shows the data together with their corresponding fit to the summation of two Fano resonances. For all temperatures, the model fit shows an excellent agreement with the data. No additional background is used in the model. The extracted resonance amplitude, widths, and energies are displayed in Fig.4 b, c, d respectively.
Looking at the amplitude of the resonances (Fig.4b), we first note that both resonances (E 1 and E 2 ) weaken with increasing temperature. While the E2 resonance amplitude remains finite and large with no apparent anomaly at Tc, the E1 resonance fades and within the experimental resolution becomes negligible above 35 K, as is reflected by the diverging error bars, which renders their values meaningless above 35 K. The extracted linewidths of the resonances also paint a similar picture. At the lowest temperature, the linewidths of the E1 and E2 resonances saturate at values of ~ 7meV and ~13meV, respectively. With increasing temperature, the E2 resonance increases and within the experimental resolution follows the conventional temperature dependence expected in Kondo lattice systems (T) = √( ) 2 + 2( ) 2 38,39 . Plotting (T) for a TK of 110 K extracted from transport measurements (blue line in Fig.4c) reveals a good agreement with the experimental data. On the other hand, the E1 linewidths grow rapidly and diverge above 35 K, where the error bars span the entire y-range and the data are therefore omitted for T > 35 K from Fig.4c. The rapid decrease of spectral weight of the E1 resonance together with its diverging linewidths with increasing temperature makes it difficult to ascertain its high-temperature evolution, particularly above 35 K. The extracted energies below 35 K show no significant temperature dependence.
We now turn to identify the origin of the observed double-peak structure at low temperatures. One possibility is the indirect Kondo hybridization gap 47 . However, this scenario can be discarded in UGe2 for two reasons. First, the observed band splitting occurs far below the Kondo lattice temperature of 110 K extracted from transport measurements. In fact, looking at the temperature dependence of the E1 linewidth, one can see that it deviates dramatically from thermal broadening and (T) (red line in Fig.4c) and diverges near 35 K. Second, contrasting our observation in UGe2 with the Kondo resonance in antiferromagnetic USb2 16 , non-magnetic UTe2 9 , and URu2Si2 38,40 above its hidden order temperature, we see that all these uranium-based heavy fermion systems display a single Fano resonance above the chemical potential. Yet, none show a splitting or an indirect Kondo hybridization gap opening at low temperature, regardless of their magnetic or Kondo-coherence temperatures. Therefore, the splitting that we observe is likely not due to an indirect hybridization gap, which should show up similarly in these other U-based systems as well.
We therefore turn to ferromagnetism as a possible origin of the band splitting (~ 16 meV) that we observe.
In itinerant ferromagnets, below their magnetic transition, the spin-majority and minority bands split due to the ferromagnetic exchange. Our observation is consistent with this Stoner mechanics of itinerant ferromagnetism. Indeed, the observation of a Kondo resonance above Tc and its further evolution below Tc provides spectroscopic evidence of the itinerant character of the f-electrons and therefore of the ferromagnetism in UGe2. This agrees with the fact that the ordered moment in the ferromagnetic phase is 1.4 B/U, much smaller than the effective paramagnetic moment of 2.7B /U 19 . Evidence of itinerant ferromagnetism and band splitting is also seen in the optical spectroscopy of UGe2 below Tc, where low energy excitations with an energy of E ~ 13.6 meV 48 have been observed. The latter is comparable to the separation of the E1-E2 resonances seen here (Fig.4d). Similarly, a Stoner gap of the order of 40 K has also been inferred from magnetic neutron diffraction 49 .
Overall, our data reveal a Kondo resonance with a characteristic Kondo-lattice temperature of 110 K, consistent with transport measurements 19 , that survives in the ferromagnetic phase. In heavy fermion systems, the magnetic ground state is generally in competition with the Kondo quenching of the magnetic moments that lead to the famous Doniach diagram. This holds true in Ce-based heavy fermions as seen in CeRhIn5 and CeCoIn5 50,51 . In UGe2 however, the two phenomena seem to coexist with no apparent competition, as seen by the largely unaffected Kondo resonance crossing Tc. A similar conclusion was reached in USb2 16,17 . Gradually below ~35 K, an additional kink/shoulder (E1 resonance) develops that coincides with the highly polarized FM2 phase that emerges as a crossover below Tx ~ 32 K. The E1 resonance shifts the f-spectral weight closer to EF, which leads to an enhanced electronic effective mass. The latter has been observed in the optical spectroscopy measurements 48 . Hall measurements are also consistent with this picture, where a rapid increase of charge carriers indicating some sort of Fermi surface reconstruction is observed below Tx 24 . Altogether, these observations indicate the splitting of the resonance that likely onsets at Tc and becomes more pronounced below the crossover temperature Tx following the rigid band Stoner model 27,49 , leading to a spin polarized f-electron band (E1) at the Fermi energy, which sets the stage for a spin polarized superconductivity to emerges at high pressure. Future spin polarized STM study on the relative peak intensities of E1 and E2 will confirm the findings here, but is beyond the scope of this study.