## Neutrons peek into f-electron bands

Neutron scattering can be used to tease out the details of collective magnetic excitations that yield well-defined peaks in the data. In principle, it could also be used to look into single-electron band excitations, but collecting enough data to capture broad distributions of intensity is tricky. Goremychkin *et al.* used neutron spectrometers that could efficiently capture a large amount of data by rotating the sample, a crystal of the intermediatevalence compound CePd_{3} (see the Perspective by Georges). The measured dynamical magnetic susceptibility, in combination with detailed ab initio calculations, showed the formation of coherent f-electron bands at low temperatures.

## Abstract

In common with many strongly correlated electron systems, intermediate valence compounds are believed to display a crossover from a high-temperature regime of incoherently fluctuating local moments to a low-temperature regime of coherent hybridized bands. We show that inelastic neutron scattering measurements of the dynamic magnetic susceptibility of CePd_{3} provides a benchmark for ab initio calculations based on dynamical mean field theory. The magnetic response is strongly momentum dependent thanks to the formation of coherent f-electron bands at low temperature, with an amplitude that is strongly enhanced by local particle-hole interactions. The agreement between experiment and theory shows that we have a robust first-principles understanding of the temperature dependence of f-electron coherence.

The Anderson impurity model, used to describe magnetic impurities in metals, formulates the interaction of localized f-electron orbitals with more delocalized d-electron bands through an onsite hybridization, whose strength is typically represented by a single energy scale, the Kondo temperature, *T*_{K} (*1*). It has been successfully applied to intermediate valence compounds, such as CePd_{3} and CeSn_{3}, even though the f electrons in these materials are not on isolated impurities but sit on a periodic lattice (*2*). Nevertheless, various deviations from the expected behavior of single impurities at temperatures well below *T*_{K} have long been interpreted as evidence of the formation of coherent f-electron bands, with strongly renormalized quasiparticle masses (*3*), an interpretation that was supported by phenomenological theory (*4*, *5*).

The concept of a crossover from coherent quasiparticles at low temperature to incoherent electronic fluctuations at high temperature is important in strongly correlated electron systems, whether in the context of heavy fermions (*6*, *7*), intermediate valence compounds (*8*), high-temperature superconductors (*9*), or “bad” metals close to a Mott transition (*10*). Although there is an extensive body of theoretical work predicting a gradual loss of quasiparticle spectral weight at the Fermi energy with increasing temperature, it has only recently become possible to perform realistic calculations by combining density functional theory with dynamical mean field theory (DFT+DMFT) in order to include both strong local correlations and itinerant band structures on an equal footing (*11*, *12*). Angle-resolved photoemission spectroscopy has provided experimental evidence of a loss of quasiparticle coherence with increasing temperature in heavy fermions (*13*, *14*), but quantitative comparison with DFT+DMFT is limited by the need to correct for matrix elements and the difficulty of combining high-energy resolution with broad momentum coverage. Furthermore, many of the relevant one-electron states are unoccupied, particularly in cerium compounds, and so inaccessible to photoemission measurements.

An alternative spectroscopic probe of quasiparticle coherence is inelastic neutron scattering. The neutron cross section, or scattering law, S(**Q**,ω), is proportional to the dynamic magnetic susceptibility, χ′′(**Q**,ω), of band electrons (*15*). For noninteracting electrons, this is derived from the Lindhard susceptibility, whose imaginary part is proportional to the joint density of states of the one-electron bands.(1)The resulting scattering intensity would be enhanced at momentum transfers, **Q**, and energy transfers, ω, that connect regions of high densities of state in the single-electron bands, *E*_{k}, whose states are occupied with probability *f*_{k}. Neutrons therefore probe both occupied and unoccupied states within the same measurements.

The advent of pulsed neutron sources, which have an enhanced flux of high-energy neutrons, stimulated interest more than 30 years ago in the possibility of using inelastic neutron scattering to study one-electron band structures (*16*, *17*), but the earliest estimates of the neutron cross section for weakly correlated electron bands were discouraging. Predicted signals were in the range 10^{–4} to 10^{–3} barns/steradians/eV, spread over wave vectors covering the entire Brillouin zone and energies up to the band width (*16*). Broad distributions of intensity have been challenging to measure at pulsed neutron time-of-flight spectrometers, where measurements were, until recently, made using a fixed sample geometry. Consequently, high-energy spectrometers have mostly been used to measure coherent excitations, such as spin waves in the copper oxide and iron-based superconductors (*18*, *19*). These also represent the magnetic excitations of band electrons, but they are easier to measure because strong interatomic exchange interactions generate poles in the dynamic susceptibility that yield well-defined peaks in the cross section.

Although measuring the electronic structure of weakly correlated electrons was not thought to be technically feasible with neutrons, strong intensity enhancements were predicted in strongly correlated electron systems—in particular, intermediate valence compounds (*16*). There has been increasing evidence over the past decade that compounds such as CePd_{3} (*20*), YbAl_{3} (*21*), and CeInSn_{2} (*22*) show variations in χ′′(**Q**,ω) that could result from band excitations. So far, these interpretations have been based on conceptual models of f*-*d hybridized bands (*4*), because it was not possible to do a quantitative comparison within the experimental limitations of fixed-geometry neutron measurements. However, recently, band calculations (*23*) were shown to have qualitative consistency with earlier neutron scattering data (*20*).

We can now go beyond qualitative comparisons because of the advent of a new generation of inelastic neutron scattering spectrometers with large position-sensitive detectors (*24*, *25*) that allow efficient measurements of four-dimensional (4D) S(**Q**,ω) in single crystals by rotating the sample during the data collection. This can be accomplished either by measuring at discrete steps of the rotation angle (*26*) or by collecting the data continuously as the sample rotates (*27*). Both methods produce equivalent results that overcome the limitations of fixed-geometry measurements by measuring entire volumes of (**Q**,ω)-space rather than a sparse set of hypersurfaces through that volume. Because the experimental data can be placed on an absolute scale by normalizing the intensity to a vanadium standard, it is possible to produce a direct comparison of experiment and theory.

With this instrumental capability, we have now performed a detailed quantitative comparison of neutron scattering measurements with a full calculation of S(**Q**,ω) using DFT+DMFT. As expected, the calculations show broad distributions of intensity with diffuse maxima at high-symmetry points that shift within the Brillouin zone as a function of energy transfer. These match the measured distributions of the dynamic susceptibility at low temperature with absolute cross sections that are within 20% of the theoretical predictions. Peaks in the dynamic susceptibility at fixed momentum transfer are associated with values of **Q** and ω that connect relatively flat regions of the coherent quasiparticle bands. Although the **Q** dependence therefore results from the dispersion of coherent quasiparticles, the intensity is strongly amplified by local particle-hole interactions that are accurately predicted by two-particle vertex corrections within DFT+DMFT theory. At high temperature, the **Q** dependence is suppressed by the loss of quasiparticle spectral weight at low energies, in agreement with experiment, showing that the calculations realistically model the crossover to a regime of incoherent spin fluctuations.

## Ab initio theory for the dynamic susceptibility of CePd_{3}

The dynamic susceptibility of CePd_{3} was derived from a calculation of the one-electron Green’s function using DFT+DMFT. This approach allows the incorporation of local correlations—i.e., Hund’s rule and spin-orbit coupling as well as on-site Kondo screening—into realistic band structures based on DFT. Figure 1 shows the spectral function A(**k**,ω), which includes the one-particle vertex correction—i.e., the electron self-energy. At 100 K, the calculations show strongly renormalized but well-defined quasiparticle excitations within f-electron bands that are hybridized with the more dispersive d bands. There are two small Fermi surface pockets centered at the Γ points—i.e., **Q** = (000)—and R points—i.e., ). Because of the strong spin-orbit coupling, the f bands have contributions from both and angular momentum states, with the former spread over ~100 meV around the Fermi level and the latter at a few hundred meV above the Fermi energy. The spin-orbit coupling is responsible for the weak incoherent spectral weight visible in Fig. 1A about 200 meV below the Fermi energy. At 400 K (Fig. 1B), there is a substantial broadening of the quasiparticle excitations, with a consequent reduction of their spectral weight, particularly close to the Fermi energy. We will discuss this later when presenting the high-temperature neutron scattering results.

The dynamic magnetic susceptibility, χ′′(**Q**,ω), is computed from the polarization bubble of the fully interacting DFT+DMFT one-particle Green’s function, whose spectral weight is shown in Fig. 1, by incorporating particle-hole interactions through two-particle irreducible vertex corrections, , which are assumed to be local in the same basis in which the DMFT self-energy is local (*28*, *29*). The calculations were performed at 100 K because of slow convergence times at lower temperatures, but Fig. 1 shows that the temperature is sufficiently low for the quasiparticles to be coherent. Further details are given in (*30*).

The calculations indicate scattering throughout the Brillouin zone, with broad maxima at high-symmetry points in **Q**, which shift with energy transfer. Figure 2 shows **Q** = [*H*,*K*] scattering planes with *L* = 1 and , where H, K, and L are reciprocal lattice coordinates, at energy transfers of 35 meV and 55 meV. The calculations (Fig. 2, A to C) are displayed in an extended zone scheme with corrections for the f-electron magnetic form factor. These show that there are maxima in the intensity at the Γ and R points at ω = 35 meV but at the M (0) and X (00) points at ω = 55 meV. These maxima are not connected by dispersive modes. Instead, the dynamic susceptibility consists of columns of intensity peaked at ~35 and ~55 meV in different regions of the Brillouin zone. This is illustrated in Fig. 3, which shows slices in the *L*-ω plane centered at the X and M points above 50 meV, and fig. S4B, which shows the shift to lower energy at the Γ and R points (*30*).

## Inelastic neutron scattering from CePd_{3}

Using a large single crystal of CePd_{3}, with a mass of 17.72 g, we have performed measurements of 4D S(**Q**,ω) by rotating the sample at fixed incident energies on the time-of-flight spectrometer Merlin and the wide Angular-Range Chopper Spectrometer (ARCS), at the ISIS Pulsed Neutron Facility and Spallation Neutron Source, respectively (*24*, *25*). These possess large banks of position-sensitive detectors that allow the scattered neutrons to be counted as a function of polar and azimuthal angle, with respect to the incident beam. When combined with the sample rotation angle and the neutron time-of-flight, this four-coordinate scattering geometry can be readily transformed into 3D reciprocal space coordinates, **Q**, and a fourth energy coordinate, ω. The transformed data fill large volumes of (**Q**,ω), allowing arbitrary cuts to be made at constant energy or momentum transfer (Fig. 2, D to F, and Fig. 3, A and B). Correction for the temperature factor and calibration to a vanadium standard allows the dynamic magnetic susceptibility to be directly compared with the DFT+DMFT calculations. There is an overall 20% scale uncertainty in the cross section.

Figure 4 shows the measured and calculated energy dependence of the scattering at four points in the Brillouin zone. The agreement between the experiment and theory shows that the DFT+DMFT calculations accurately reproduce the energy scale for the magnetic fluctuations. Below 30 meV, the measured spectra are dominated by nonmagnetic phonon scattering, which is not included in the calculations.

Figure 2, D to F, shows the **Q** dependence of the dynamic magnetic susceptibility derived from experiment, confirming the theoretically predicted shift in the maxima between the Γ and R points at 35 meV to the M and X points at 55 meV (Fig. 2, A to C). The magnetic scattering is superposed on a **Q-**dependent background from the sample environment, which increases monotonically with momentum transfer, shifting the maxima away from **Q** = 0.

To provide a more quantitative comparison, we show constant energy cuts along a number of high-symmetry directions in Fig. 5, where the data are plotted against the theoretical calculations on an absolute scale. The neutron spectra were fitted to the calculated dynamic susceptibility, adjusted by a single scale factor, and an instrumental background, produced by scattering off the sample environment, which is well described by a quadratic function in **Q**. The scale factor derived from the fits indicates that the accuracy of the absolute normalization is ~20%, consistent with uncertainties due to the absorption of the irregularly shaped sample (*30*). In Fig. 5, the fitted background has been subtracted from the data to show just the magnetic scattering. The unsubtracted spectra, along with additional details about the background estimates, are given in (*30*). The quantitative agreement confirms the qualitative consistency of experiment and theory evident in Figs. 2 and 3.

Finally, we compare the calculations made at higher temperatures to the experimental data. Figure 6 shows that the **Q** dependence of the magnetic scattering is almost entirely suppressed at room temperature, which is well above the coherence temperature inferred from transport measurements. This is also predicted by the theoretical calculations. An inspection of the spectral functions at 100 and 400 K in Fig. 1 shows that this results from the substantial reduction in spectral weight of coherent quasiparticles and confirms the importance of this coherence in generating the observed **Q** variations in the dynamic susceptibility at low temperature.

## Discussion and conclusion

Our results demonstrate that it is now possible to determine the electronic structure of intermediate valence materials with considerable accuracy by incorporating local correlations into band structures through the combination of density functional theory and dynamical mean field theory. The agreement of the calculated and measured neutron cross sections is complete throughout the Brillouin zone—extending over a broad range of energy transfers from ~20 to 65 meV—and accurately describes the shift in the scattering maximum from around 35 meV at the Γ and R points to around 55 meV at the M and X points. Although magnetic neutron scattering is a well-established probe of collective magnetic excitations, such as spin waves, that produce sharp dispersion surfaces in **Q**-ω space, our experiment shows it is now possible to measure and theoretically account for the dynamic magnetic susceptibility arising from correlated electron bands.

The calculations reveal the important role of local particle-hole interactions in renormalizing the dynamic magnetic susceptibility. In Fig. 6, B and C (and figs. S3 and S4), we compare the calculated magnetic response both with and without the two-particle vertex correction, , which represents the interactions between the electron and the hole excited by the neutron. The correction has two effects: First, it smooths out some of the fine structure in the energy dependence of the spectra while broadly preserving both the **Q** variation and the overall energy scale; and second, it produces a strong enhancement of the intensity that is both energy and temperature dependent, for example, by a factor of ~6.5 at ω = 60 meV at 100 K. This shows that the **Q** dependence of the scattering is predominantly determined by the one-electron joint density of states, as expected for band transitions, whereas the overall intensity is amplified by the strong electron correlations.

In (*20*), we showed that the Anderson impurity model (AIM) is successful in explaining a number of important properties [the magnetic susceptibility χ(*T*), the 4f occupation number *n*_{f}(*T*), the 4f contribution to the specific heat *C*_{4f}(*T*), and the **Q**-averaged dynamic susceptibility χ′′(ω)] of intermediate valence compounds such as CePd_{3}, even though the cerium atoms are not impurities but sit on a periodic lattice. We speculated that the reason the (incoherent) impurity model works so well for these periodic systems is that the strong inelastic Kondo scattering of the electronic quasiparticles by valence/spin fluctuations broadens the spectral functions. Such broadening can be seen even at low temperature for states away from the Fermi energy in Fig. 1A. The properties χ(*T*), *n*_{f}(*T*), and *C*_{4f}(*T*), are primarily sensitive to χ′′(ω), which represents local 4f moment fluctuations. The DFT+DMFT calculations show that the vertex corrections caused by the inelastic scattering result in spectra that, when averaged in **Q**, are very similar to the AIM result of (*20*), thus helping explain why the impurity model works as well as it does. These inelastic processes are what drives the rapid loss of coherence with temperature shown in Fig. 1B. The effect on the dynamic susceptibility, shown in Fig. 6, is that the spectrum of CePd_{3} is nearly **Q** independent at room temperature and has the quasielastic spectral shape expected for an incoherent Anderson impurity system (*20*).

It should be possible to extend this work to materials with even stronger electronic correlations, such as heavy fermions, but the reduced coherence temperature and the increased complexity of potential many-body states will make it more computationally intensive. DFT+DMFT calculations of the magnetic susceptibility converge more slowly at low temperature and in the presence of multiple f-electron energy levels caused by crystal fields or magnetic interactions. In the case of CePd_{3}, the f*-*d hybridization is strong enough to suppress the crystal field splittings but weak enough that a temperature of 100 K was sufficiently low to reveal the onset of coherence. Nevertheless, the DFT+DMFT method has been successfully applied to the single-particle excitations of heavy fermions such as CeCoIn_{5} (*6*), so calculations of the two-particle spectra would be challenging but should be technically feasible.

The results of this comparison between theory and experiment provide insight into the nature of the correlations in intermediate valence systems. The magnetic fluctuations show a much richer structure than was implied by earlier “toy” models of hybridized bands, and there is a complex interplay between coherent and incoherent contributions to the electronic spectra that is reflected in the evolution of the dynamic magnetic susceptibility with temperature. The transition from coherent f-electron bands to local moment physics, so long postulated in heavy fermion and intermediate valence systems, is confirmed by combining the latest advances in inelastic neutron scattering and ab initio theories of correlated electron systems.

## Supplementary Materials

www.sciencemag.org/content/359/6372/186/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S8

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**The research at the Joint Institute for Nuclear Research was supported by the Russian Foundation for Basic Research project 16-02-01086. The research at Argonne National Laboratory and Los Alamos National Laboratory was supported by the Materials Sciences and Engineering Division, Office of Basic Energy Sciences, U.S. Department of Energy. The research at Oak Ridge National Laboratory’s Spallation Neutron Source was supported by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy. Neutron experiments were performed at the Spallation Neutron Source, Oak Ridge National Laboratory, and the ISIS Pulsed Neutron Source, Rutherford Appleton Laboratory. We gratefully acknowledge the computing resources provided on Blues, a high-performance computing cluster operated by the Laboratory Computing Resource Center at Argonne National Laboratory. We are also grateful for useful discussions with P. Riseborough. Files containing the data sets used in this paper are available for download from http://dx.doi.org/doi:10.18126/M2C914. Four-dimensional inelastic neutron scattering data from ARCS are stored in HDF5 files, conforming to the NeXus standard (www.nexusformat.org), which can be viewed using the open-source application, NeXpy (http://nexpy.github.io/nexpy/). The data can be compared with DFT+DMFT calculations, produced using the Wien2K+DMFT package (

*31*), which are also stored in NeXus files. Inelastic neutron scattering data from MERLIN are available as RAR archives, containing files produced by the Horace suite of MATLAB programs (http://horace.isis.rl.ac.uk/), which can be used to extract cuts and slices through the 4D data.