Directly Characterizing the Relative Strength and Momentum Dependence of Electron-Phonon Coupling Using Resonant Inelastic X-Ray Scattering

The coupling between lattice and charge degrees of freedom in condensed matter materials is ubiquitous and can often result in interesting properties and ordered phases, including conventional superconductivity, charge-density wave order, and metal-insulator transitions. Angle-resolved photoemission spectroscopy and both neutron and nonresonant x-ray scattering serve as effective probes for determining the behavior of appropriate, individual degrees of freedom—the electronic structure and lattice excitation, or phonon dispersion, respectively. However, each provides less direct information about the mutual coupling between the degrees of freedom, usually through self-energy effects, which tend to renormalize and broaden spectral features precisely where the coupling is strong, impacting one’s ability to quantitatively characterize the coupling. Here, we demonstrate that resonant inelastic x-ray scattering, or RIXS, can be an effective tool to directly determine the relative strength and momentum dependence of the electron-phonon coupling in condensed matter systems. Using a diagrammatic approach for an eight-band model of copper oxides, we study the contributions from the lowest-order diagrams to the full RIXS intensity for a realistic scattering geometry, accounting for matrix element effects in the scattering cross section, as well as the momentum dependence of the electron-phonon coupling vertex. A detailed examination of these maps offers a unique perspective into the characteristics of electron-phonon coupling, which complements both neutron and nonresonant x-ray scattering, as well as Raman and infrared conductivity.

Popular Summary

Coupling between lattice and charge degrees of freedom in condensed matter materials is ubiquitous. Such coupling—deformation, piezoelectric, or electrostatic, for example—can often result in interesting properties and ordered phases, including conventional superconductivity, charge-density wave order, and metal-insulator transitions. The application of resonant inelastic x-ray scattering—in which high-energy x rays are scattered off of a material to determine its electronic structure—has blossomed in recent years as a key technique to probe fundamental excitations in solids. Here, we demonstrate that resonant inelastic x-ray scattering can be an effective tool to directly determine how electrons and lattice excitations known as phonons couple in solid-state materials and also the relative strength and momentum dependence of the coupling.

We focus on copper oxides in situations where a weakly interacting core electron excited into the conduction band. Our investigations are additionally limited to situations in which a single-phonon process dominates. We study how phonons are resonantly created by impinging x rays. We additionally measure how a material’s electrons are, in turn, coupled to various branches of these lattice excitations. We show that a detailed examination of resonant inelastic x-ray scattering offers a unique perspective into the characteristics of electron-phonon coupling, which complements and builds upon the results of both neutron and nonresonant x-ray scattering, as well as Raman and infrared conductivity.

We expect that our findings will advance studies of electron-phonon coupling and pave the way for more detailed investigations of emergent phenomena like superconductivity.

Figure 1

Electron-hole contribution to RIXS (bare diagram). The dotted lines denote the Cu $2p$ hole, and solid lines denote the $3d$ conduction electrons. Notation for the vertices has been suppressed.

Figure 2

Leading-order one-phonon contribution to RIXS. The dotted lines denote the Cu $2p$ hole, solid lines denote the $3d$ conduction electrons, and the circular line denotes the phonon. Notation for the vertices has been suppressed.

Figure 3

Bare contribution to RIXS from particle-hole scattering. Momentum transfer is given in units of $\pi /a$.

Figure 4

Longitudinal acoustic phonon contribution to RIXS. Momentum transfer is given in units of $\pi /a$.

Figure 5

$B1$ phonon contribution to RIXS. Momentum transfer is given in units of $\pi /a$.

Figure 6

$A1$ phonon contribution to RIXS. Momentum transfer is given in units of $\pi /a$.

Figure 7

Cu-O bond-stretching (breathing) phonon contribution to RIXS. Momentum transfer is given in units of $\pi /a$.

Figure 8

Apical oxygen phonon contribution to RIXS. Momentum transfer is given in units of $\pi /a$.

Figure 9

Sum of all contributions from phonons to RIXS. Momentum transfer is given in units of $\pi /a$.

Figure 10

Sum of all contributions. Momentum transfer is given in units of $\pi /a$.