Lifshitz transition from valence fluctuations in YbAl3

In mixed-valent Kondo lattice systems, such as YbAl3, interactions between localized and delocalized electrons can lead to fluctuations between two different valence configurations with changing temperature or pressure. The impact of this change on the momentum-space electronic structure is essential for understanding their emergent properties, but has remained enigmatic. Here, by employing a combination of molecular beam epitaxy and in situ angle-resolved photoemission spectroscopy we show that valence fluctuations can lead to dramatic changes in the Fermi surface topology, even resulting in a Lifshitz transition. As the temperature is lowered, a small electron pocket in YbAl3 becomes completely unoccupied while the low-energy ytterbium (Yb) 4f states become increasingly itinerant, acquiring additional spectral weight, longer lifetimes, and well-defined dispersions. Our work presents a unified picture of how local valence fluctuations connect to momentum-space concepts such as band filling and Fermi surface topology in mixed valence systems.

K ondo lattice systems host a wide variety of quantum states such as antiferromagnetism 1 , heavy Fermi liquids 2 , hidden order 3 , and unconventional superconductivity 4 , which can often be controlled by modest perturbations using magnetic field or pressure, thereby providing access to quantum phase transitions [5][6][7] . These states generally emerge from a complex manybody state that is formed by enhanced Kondo coupling between the local rare-earth moments and the band-like conduction electrons at low temperatures. In mixed valence systems [8][9][10] , this coupling also results in a change of the rare-earth valence, which can be determined by core-level spectroscopies that probe the local chemical environment (r-space) [11][12][13] , but the implications for the momentum-space (k-space) electronic structure remain poorly understood. To gain insight into the emergent properties of these systems, it is crucial to understand how delocalized carriers and the low-energy momentum-space electronic structure emerge from these local interactions.
Here, we choose YbAl 3 as a simple prototypical mixed valence system with two nearly degenerate ytterbium (Yb) valence configurations, Yb 2+ (4f 14 ) and Yb 3+ (4f 13 ). The average Yb valence, ν f , decreases with temperature, changing by ∼Δν f = −0.05 from 300 K to below T * ≈ 34-40 K 11,12,[14][15][16] , when it becomes a heavy Fermi liquid, attributed to the enhanced Kondo screening at low temperatures 17 . We selected YbAl 3 due to its relatively large change in valence as well as its large energy scales, with a reported single ion Kondo temperature T K ≈ 670 K 17,18 , which should make these changes observable in momentum space. The lack of a welldefined, pristine surface in cleaved YbAl 3 single crystals 19 , however, has previously prevented momentum-resolved measurements of its electronic structure. We have circumvented this problem by synthesizing epitaxial thin films of YbAl 3 and its conventional metal analog LuAl 3 by molecular beam epitaxy (MBE) 20 and have combined it with in situ angle-resolved photoemission spectroscopy (ARPES) to directly measure their electronic structure as a function of temperature. Our measurements reveal a strong temperature-dependent change in both the real and momentumspace electronic structure of YbAl 3 . The local Yb valence decreases as the temperature is lowered, accompanied by a large shift in the chemical potential which leads to a Lifshitz transition of a small electron pocket at Γ, along with the emergence of renormalized heavy quasiparticles near the Fermi energy (E F ). We establish a direct one-to-one correspondence between these observed changes, which we believe to be generic to all mixed valence systems.

Results
Synthesis and electronic structure. Both YbAl 3 and LuAl 3 crystallize in a cubic Pm3m structure where Yb or Lu atoms occupy the vertices of the unit cell while Al atoms occupy the face centers, as illustrated in the inset of Fig. 1b. LuAl 3 has fully occupied 4f orbitals with zero net moment and a lattice constant (4.19 Å) closely matched to YbAl 3 (4.20 Å). Thus, LuAl 3 serves as an ideal reference compound to understand the light, Al-derived band-like conduction electron states, which are also common to YbAl 3 . Epitaxial thin films of both LuAl 3 and YbAl 3 with (001) out-of-plane orientation were synthesized by co-evaporation on MgO (001) substrates (4.21 Å) at temperatures of 200-350°C and a chamber base pressure below 2 × 10 −9 Torr. For all films, a 1.2 nm thick aluminum (4.05 Å) buffer layer was deposited at 500°C, which allowed the growth of continuous, smooth films of LuAl 3 / YbAl 3 on top. In these studies, we investigated a 30 nm thick LuAl 3 film and a 20 nm thick YbAl 3 film (the YbAl 3 was synthesized on top of a 20 nm thick LuAl 3 buffer layer on top of the Al buffer, which improved the quality of the YbAl 3 layers). All films were sufficiently thick so that any photoemission intensity from the buffer layers or substrate and thickness-dependent finite size effects can be ignored. Additional details about the synthesis can be found in "Methods" section as well in ref. 20 .
In Fig. 1, we show Fermi surface maps and the electronic structure from LuAl 3 and YbAl 3 thin films from the Fermi energy (E F ) to a binding energy of 10.5 eV. For LuAl 3 , only the Lu 4f 13 final states are observed, with the J = 7/2 and 5/2 core levels at binding energies of 6.7 and 8.2 eV, respectively. Highly dispersive Al-derived bands can be observed in both LuAl 3 (Fig. 1c) and YbAl 3 (Fig. 1f), which extend from about 6 eV binding energy to near E F . By matching the experimentally determined dispersion of these bands as well the Fermi surface contours measured with both He Iα (21.2 eV) and He IIα (40.8 eV) photons to density functional theory (DFT) calculations, we are able to determine our out-of-plane momenta (k z ) values for both YbAl 3 and LuAl 3 . Due to the lack of strong correlations in LuAl 3 (its 4f shell is entirely filled), DFT calculations should accurately describe its electronic structure and indeed we find good agreement between both the DFTcalculated band dispersions as well as Fermi surface contours to the experimentally determined dispersions and Fermi surface from ARPES, assuming an inner potential of V 0 − ϕ = 13.66 eV (Fig. 1a, c and Supplementary Fig. 1). We observe broadly dispersive, primarily Al-derived bands in YbAl 3 (Fig. 1f) analogous to those observed in LuAl 3 , and also found excellent correspondence between the measured electronic structure in LuAl 3 and YbAl 3 over the entire Brillouin zone ( Supplementary  Fig. 3) indicating that we are probing a similar k z as in LuAl 3 , as one might expect given their highly similar electronic and crystal structures. Using the value of V 0 − ϕ, we determine that for hν = 21.2 eV, we are probing near the zone center, Γ, k z = 0 ± 0.1π/c. More details about the k z determination can be found in Supplementary Note 1.
A two dimensional slice at k z = Γ of the three-dimensional Fermi surface of LuAl 3 accesses a multiply connected Fermi surface sheet consisting of electron-like pockets centered at (0, 0) and (π, π), consistent with our ARPES data, shown in Fig. 1a-c. On the other hand, in YbAl 3 , we clearly observe both the Yb 4f 13 and 4f 12 final states around 0-2 eV and 6-10.5 eV binding energy, respectively, consistent with its mixed valence character. The near-E F electronic structure in YbAl 3 is, however, significantly modified by a shift in its chemical potential due to the differing average Lu and Yb valence and the interaction between the broad, dispersive bands and the renormalized Yb 4f states. In the Fermi surface map of YbAl 3 (Fig. 1d), large Fermi surface sheets are prominent and centered at zone edges (π, π).
Evolution of the electron pocket at Γ. Having discussed the basic electronic structure, we now turn towards its temperature dependence in YbAl 3 . In Fig. 2a, we show a series of ARPES spectra obtained along (0, 0) to (0, π) at k z ≈ Γ between 255 and 21 K, which establish a clear temperature-dependent shift of the chemical potential Δμ with the 4f-derived states moving closer to E F as the temperature is lowered, consistent with earlier angleintegrated measurements 14 . The most dramatic effect of Δμ is on a small parabolic electron pocket centered at Γ. At 255 K, the electron pocket can be clearly observed with its band bottom at 40 ± 5 meV binding energy and a k F of 0.20 ± 0.01π/a. As the temperature is lowered, the electron pocket is lifted in energy and becomes entirely unoccupied around 21 K. Since the pocket is centered at Γ, its lifting above E F would then coincide with a Lifshitz transition. To within experimental resolution, the dispersion or effective mass of the electron pocket does not change apart from a rigid shift due to Δμ. Furthermore, while the Yb 4f 13 final states also shifted in energy, the Yb 4f 12 states did not shift appreciably with temperature indicating that the Δμ shift arises from an alteration in band filling due to the emergence of a Kondo screened many-body state.
We note that the small electron pocket at Γ is not reported in previous de Haas-van Alphen (dHvA) studies of YbAl 3 , which can be explained by the fact that the electron pocket is only occupied at higher temperatures (T > 20 K), whereas dHvA measurements are conducted at low temperatures (T ≈ 20 mK-1.5 K) 18,21 . Nevertheless, DFT calculations suggest the presence of a quasi-spherical electron pocket at Γ, but whose size strongly depends on the binding energy of the Yb 4f states ( Supplementary Fig. 4), and thus the value of U used in the calculations. Furthermore, our observation of the temperaturedependent chemical potential shift may explain the need to artificially shift the chemical potential in previous lowtemperature quantum oscillation experiments of YbAl 3 which compared their results to band structure calculations 21 . The large Fermi surface sheets centered at (π, π) (see Fig. 1d and Supplementary Fig. 2) measured by ARPES would give an oscillation frequency of ≈1.0 ± 0.2 × 10 8 Oe, which is comparable but somewhat larger than the largest reported quantum oscillation frequency along the (100) direction, 6.51 × 10 7 Oe. This discrepancy might be due to the fact that the quantum oscillations are measuring a closed Fermi surface contour at a different k z than our ARPES measurements at k z = 0, which might correspond to an open contour. We also observed another Fermi surface sheet centered at the M point (π, π) with 40.8 eV photon energy which corresponds to an oscillation frequency of 3.5 × 10 7 ± 1 × 10 7 , roughly consistent with the β pocket reported in the dHvA measurements (4.55 × 10 7 Oe).
In Fig. 2b, we show a series of angle-integrated wide energy valence band in situ x-ray photoemission spectra (XPS) showing a dramatic temperature-dependent change in relative intensity of the 4f 13 and 4f 12 final states. As the temperature is lowered, the relative intensity of the 4f 13 final states increases, while that of the 4f 12 final states decreases indicating a reduction of the effective Yb valence in YbAl 3 at lower temperatures, which is found to be Δν f ≈ 0.05 from room temperature to below ≈ 45 K, in agreement with previously reported results from bulk samples 11,12,[14][15][16] .
Relation between real-space and momentum-space electronic structure. In Fig. 2c, we make a quantitative comparison between the observed change in the temperature-dependent band filling and the estimated change in the Yb valence from core-level spectroscopy, both in our thin films and previous measurements on YbAl 3 single crystals. The change in average Yb valence in our thin films has been estimated by resonant x-ray emission spectroscopy (RXES) and XPS, details for which can be found in "Methods" section and in Supplementary Note 2. Assuming a spherical geometry 4 3 πk 3 F À Á due to its location at k = (0, 0, 0) and the cubic symmetry, we plot the change in Luttinger volume of the electron pocket Δν Lutt , vs. the estimated change in Yb valence Δν f , from core-level spectroscopy. Without any adjustable parameters or scaling factors, we discover a precise, one-to-one correspondence between Δν Lutt from the electron pocket and Δν f as a function of temperature. This provides direct microscopic evidence that in YbAl 3 , the Kondo screening of the 4f moments by the conduction electrons that results in the emergence of composite heavy fermion quasiparticles leads to a Lifshitz transition of the Fermi surface, which is also reflected in the reduction of the average Yb valence, and should be generic to other mixed valence systems.
A qualitative model of the temperature-dependent changes in both real and momentum-space is presented in Fig. 2d. As the temperature is lowered, the filling of the small electron pocket is gradually reduced as those electrons are transferred into the Kondo screening cloud at the Yb site leading to the formation of renormalized Kondo screened many-body states 22 and a reduction of the effective Yb valence as measured by XPS and RXES studies. This model would explain the direct one-to-one correspondence between the measured changes in both the Yb valence and the Luttinger volume of the electron pocket as a function of temperature.
We should note that previous studies of other mixed valence systems, such as YbRh 2 Si 2 have not reported temperaturedependent changes in the band structure or Fermi surface topology 23 , although this may have been because of the much larger Δν f in YbAl 3 (0.05 vs. 0.01) in the accessed temperature range of the experiments, as well as its larger energy scales (T K = 670 K vs. 25 K) 23 .
Evolution of the Yb 4f states. We now discuss the evolution with temperature of the 4f-derived heavy bands near E F . In Fig. 3a, we show representative energy distribution curves (EDCs) at different temperatures integrated over the momentum region indicated in Fig. 3d, together with extracted changes of the 4f binding energy, quasiparticle weight, and scattering rate as a function of temperature (Fig. 3b, c). Details of the fitting process along with extended data sets can be found in Supplementary Note 3 and Supplementary Fig. 7. We find a dramatic enhancement of the quasiparticle spectral weight of the 4f bands, consistent with previous measurements by Tjeng et al. 11 , coinciding with a precipitous drop in the scattering rate, which saturates around T * ≈ 37 K, the estimated coherence temperature of YbAl 3 20 when it becomes a Fermi liquid. The enhancement of the quasiparticle spectral weight and lifetime with decreasing temperature suggests that the screening of the 4f moments by the conduction electrons has nearly saturated around T * , and that the Lifshitz transition is coincident with this dramatic change in the 4f spectral function. This is further highlighted by the observation of a ln(T 0 /T) ≥ 255 K scaling behavior in the integrated spectral   Fig. 2 Correspondence between r-space and k-space electronic structure in YbAl 3 . a Evolution of the low-energy electronic structure with temperature. E vs. k dispersions are divided by the corresponding resolution-broadened Fermi-Dirac distribution to emphasize thermally occupied states above E F . White lines are guides to the eye showing evolution of the electron-like pocket centered at (0, 0, 0). b XPS spectra showing the temperature-dependent intensity variation of the 4f 13 and 4f 12 final states in YbAl 3 , after Shirley background subtraction 37 and normalized by the 4f 12 final state intensity. c Temperature dependence of the change in Luttinger volume, estimated from the size of the electron pocket at (0, 0, 0) and of the change in Yb valence, measured by core-level spectroscopy, revealing a precise one-to-one correspondence. Error bars reflect uncertainty in the Luttinger volume estimation due to a statistical error of one standard deviation in the extracted k F values from the fits to the Momentum distribution curves (MDCs) taken at E F . d Schematic illustrating the temperature-dependent relationship between r-space and k-space electronic structure in YbAl 3 weight, as expected from a two fluid model 22,[24][25][26] , until the onset of Fermi liquid behavior at T * , when it starts to saturate. The observation of this scaling behavior up to 255 K, the highest temperature accessed in this study, suggests that the hybridization between the local 4f moments and the conduction electrons sets in at a relatively high temperature, even though the Fermi liquid regime exists only below T * , consistent with the slow crossover scenario predicted by slave boson mean field calculations 27,28 . The saturation of the 4f quasiparticle lifetime at T * in our ARPES measurements is also consistent with earlier transport and thermodynamic measurements, which suggested that T * could be related to the formation of coherence in the 4f states 15,17,18 , which we establish spectroscopically. The shift in binding energy of the 4f states is smaller compared to the Δμ measured from the electron-like band, with the discrepancy increasing at lower temperatures, shown in Fig. 3b, indicative of enhanced hybridization between the 4f states and the conduction electrons at lower temperatures that pushes the electron pocket further towards lower binding energy.
Dispersive crystal electric field split states. In addition to the strong temperature dependence of the electronic structure, our measurements clearly show three distinct flat bands close to E F which acquire significant dispersion at certain k points. Their proximity to E F , the value of their splittings (≈0-30 meV), and their narrow bandwidths are all consistent with these being crystal electric field (CEF) split states. The dispersion of the CEF states occur when the light Al-derived bands cross the flat 4f states near E F , lifting the degeneracy of the CEF split branches, as shown in Fig. 3d (also see Supplementary Fig. 8). The observation of three distinct bands is consistent with the bulk cubic symmetry of the Yb ions, where the Yb J = 7/2 manifold should split into three crystal field levels Γ 6 , Γ 7 , and Γ 8 29 . While we cannot determine conclusively whether these states are representative of bulk vs. surface Yb atoms, the values of their splittings (0-30 meV) and bandwidths (≤ 25 meV) from ARPES and the fact that they extend from E F to a binding energy of ≈50 meV are also consistent with reports from bulk-sensitive inelastic neutron scattering which do not observe sharp CEF excitations but rather a broad continuum (≈50 meV), since those measurements would average the dispersion of the CEF states over the entire Brillouin zone [30][31][32] .

Discussion
Our work experimentally provides a unified picture of how local changes of the rare-earth valence impacts the momentum-space electronic structure in the prototypical mixed valence system, YbAl 3 . We have achieved this by combining state-of-the-art materials synthesis and advanced in situ spectroscopy, which should be readily extendable to other Kondo lattice systems or even artificial f-electron heterostructures. We have discovered that a Lifshitz transition of a small electron Fermi surface accompanies the change in average Yb valence, which had hitherto been unanticipated. This discovery underscores how the Kondo screening process can significantly alter k-space instabilities of Kondo lattice systems.

Methods
Film growth and characterizaion. Single crystalline, epitaxial, atomically smooth thin films of (001) YbAl 3 and LuAl 3 were synthesized on MgO substrates in a Veeco GEN10 MBE system with a liquid nitrogen cooled cryoshroud at a base pressure better than 2 × 10 −9 Torr. Prior to growth, MgO substrates were annealed in vacuum for 20 min at 800°C and a 1-2 nm thick aluminum (Al) buffer layer was deposited at 500°C. Lu/Yb and Al were co-evaporated from Langmuir effusion cells at a rate of ≈0.4 nm/min onto a rotating substrate between 200 and 350°C with real-time reflection high-energy electron diffraction (RHEED) monitoring. Due to the co-evaporation growth, the surface termination was not deliberately controlled. After growth, the films were immediately transferred under ultra-high vacuum to an ARPES chamber for measurements. All ARPES data presented in this study were performed on 30 nm thick LuAl 3 thin films with a 1.2 nm thick Al buffer layer, or on 20 nm thick YbAl 3 thin films with 20 nm thick LuAl 3 and 1.2 nm thick Al buffer layers. The ARPES spectra did not show any thickness dependence for LuAl 3 /YbAl 3 layers that were more than 10 nm thick, the minimum thickness for this study. For further details regarding film growth and characterization see ref. 20 .
In situ ARPES and XPS. After growth, thin film samples were immediately transferred within 5 min through ultra-high vacuum into an analysis chamber consisting of a VG Scienta R4000 electron analyzer, VUV5000 helium plasma discharge lamp and a dual anode x-ray source for ARPES and XPS measurements. The base pressure of the analysis chamber was better than 5 × 10 −11 Torr. ARPES measurements were performed using He Iα (hν = 21.2 eV) and He IIα (hν = 40.8 eV) photons, while Al Kα (hν = 1486.6 eV) photons were utilized for collecting XPS data. A polycrystalline gold reference, in electrical contact with the sample was used to determine position of the Fermi level and the energy resolution.
DFT calculations. DFT calculations of the band structure and Fermi surface of LuAl 3 /YbAl 3 with were performed using full potential linearized augmented plane wave method as implemented in the Wien2k software package 33 . The exchange and correlation effects were taken into account within the generalized gradient approximation 34 . Relativistic effects and spin-orbit coupling were included. For LuAl 3 , we found that an on-site Coulomb repulsion of U = 2.08 eV 35 would give good agreement between the Lu 4f orbitals to the binding energies of the core levels measured in experiment. However, the value of U had no impact on the near-E F electronic structure in LuAl 3 . For YbAl 3 , calculations were performed both with and without application of U to the Yb 4f orbitals, which was found to have a significant impact on the near-E F electronic structure. (Supplementary Fig. 4).
RXES. RXES spectra were collected at the Cornell High Energy Synchrotron Source (CHESS) at the C1 bend magnet beamline under ring conditions of 5.3 GeV and 100 mA. Incident x-ray radiation was monochromated using a Rh mirror and a sagittal focus double Si(2 2 0) crystal monochromator. The incident energy was calibrated using a Cu foil. The x-ray emission was monochromated and focused using five spherically bent Ge(6 2 0) crystals in the Rowland geometry by using the CHESS dual array valence emission spectrometer 36 . X-rays were finally collected with a Pilatus 100 K area detector (Dectris). Use of an area detector offered significant advantages for the current experiment in terms of ease of alignment and reliable background subtraction. Two regions of interest (ROIs) were chosen, one containing more than 95% of the emission signal and another centered on the first ROI, but four times in size. The larger ROI was used to correct for the average background counts as where ROI 1 and ROI 2 are the larger and smaller ROIs, respectively. I ROI and Area ROI denotes the intensity and area corresponding to the region of interest, respectively, while I corrected is the corrected intensity after background subtraction. Measured counts were further corrected for variations in incident photon flux by normalizing with the measured incident flux using a N 2 -filled ionization chamber placed upstream of the sample stage. X-ray emission energy was calibrated measuring the K α1 and K α2 lines of a Cu foil. The overall energy resolution of the setup was determined to be better than ≈3 eV measuring quasi-elastic scattering from a polyimide sample. To minimize photodamage, a fast shutter was placed upstream of the ionization chambers that would only open during active data taking, thus minimizing x-ray dosage of the samples. No photodamage was observed even after taking more than four scans (the maximum number of scans used for measurements at a particular spot) at a single spot. The sample was mounted on a closed cycle cryostat with base temperature of 45 K. A helium filled bag covering most of the x-ray path between the sample, analyzer, and detector was placed to reduce air attenuation along the x-ray path.
Data availability. Data that support the findings of this study are available from the corresponding author upon request.