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Born effective charges and vibrational spectra in superconducting and bad conducting metals

Nature Physics volume 20, pages 88–94 (2024)Cite this article

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Abstract

Interactions mediated by electron–phonon coupling are responsible for important cooperative phenomena in metals such as superconductivity and charge density waves. The same interaction mechanisms produce strong collision rates in the normal phase of correlated metals, causing sizable reductions in d.c. conductivity and reflectivity. As a consequence, low-energy excitations like phonons, which are crucial for materials characterization, become visible in the optical infrared spectra. A quantitative assessment of vibrational resonances requires the evaluation of dynamical Born effective charges, which quantify the coupling between macroscopic electric fields and lattice deformations. We show that the Born effective charges of metals crucially depend on the collision regime of conducting electrons. In particular, we describe—within a first-principles framework—the impact of electron scattering on the infrared vibrational resonances, from the undamped, collisionless regime to the overdamped, collision-dominated limit. Our approach enables the interpretation of vibrational reflectance measurements of both superconducting and bad metals, as we illustrate for the case of strongly electron–phonon-coupled superhydride H3S.

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Fig. 1: Reflectivity spectrum for H3S in the cubic \(Im\bar{3}m\) phase at 150 GPa.
Fig. 2: Electronic and vibrational optical conductivity.
Fig. 3: BECs in the undamped, damped and overdamped regimes.

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Data availability

The Eliashberg spectral function and phonon frequencies, displacements and lifetimes including anharmonic effects37 have been provided by R. Bianco. Experimental data from ref. 17 have been provided by E. J. Nicol. All other data are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

Code availability

The QUANTUM ESPRESSO code used in this research is open source and available at www.quantum-espresso.org. We used an in-house customized version of QUANTUM ESPRESSO with the additional computational capabilities discussed in this work, which is available from G.M. upon reasonable request. Wannier interpolation and solution of Migdal–Eliashberg equations have been performed using the open-source software EPIq53.

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Acknowledgements

We thank L. Baldassarre, L. Benfatto, R. Bianco, J. Lorenzana, E. Nicol and M. Ortolani for useful discussions and R. Bianco and E. Nicol for providing us with supporting information from refs. 17,37. We acknowledge financial support from the European Union ERC-SYN MORE-TEM no. 951215 (F. Mauri and G.M.), ERC DELIGHT no. 101052708 (M.C.), Graphene Flagship Core3 no. 881603 (F. Mauri and F. Macheda) and CINECA award under ISCRA Initiative grant nos. HP10BV0TBS (P.B.) and HP10CSQ37Y (G.M.) for access to computational resources. L.B. acknowledges the Fellowship from the EPFL QSE Center ‘Many-body neural simulations of quantum materials’ (grant no. 10060). Views and opinions expressed are, however, those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

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Authors and Affiliations

  1. Dipartimento di Fisica, Università di Roma La Sapienza, Roma, Italy

    Guglielmo Marchese, Paolo Barone & Francesco Mauri

  2. Istituto Italiano di Tecnologia, Graphene Labs, Genova, Italy

    Francesco Macheda, Matteo Calandra & Francesco Mauri

  3. Theory and Simulation of Materials (THEOS), and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

    Luca Binci

  4. Department of Physics, University of Trento, Povo, Italy

    Matteo Calandra

  5. Consiglio Nazionale delle Ricerche, Institute for Superconducting and Other Innovative Materials and Devices (CNR-SPIN), Area della Ricerca di Tor Vergata, Rome, Italy

    Paolo Barone

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Contributions

F. Mauri conceived the work and supervised the project with P.B. All authors contributed to the code developments in QUANTUM ESPRESSO and EPIq53. G.M. performed all the calculations with the support of F. Macheda and L.B. G.M., F. Macheda, P.B. and F. Mauri wrote the paper with contributions from M.C. and L.B.

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Correspondence to Francesco Mauri.

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Extended data

Extended Data Fig. 1 Undamped BECs in a wider frequency range.

The independent components of the undamped dynamical BECs are plotted on their typical variation scale, which is larger than the vibrational frequency range of Fig. 3. In particular, we can notice a strong variation for ℏω > 1 eV due to the onset of interband transitions. This onset is coherent with the reflectivity drop observed at the same energy (Fig. 1a). The real and imaginary parts are shown in panel (a) and (b), respectively.

Source data

Extended Data Fig. 2 Reflectivity spectra at different impurity scattering rates.

Reflectivity spectra at different impurity scattering rates (a) Superconducting and (b) normal phase reflectivity spectra, where the self-energy is evaluated with the procedure explained in the Supplementary Section VI for three different impurity scattering rates ηimp. In the left panel the offset is 0 for ηimp = 28 meV, 0.01 for ηimp = 135 meV and 0.02 for ηimp = 200 meV. The impurity scattering rate affects differently the electronic baseline and the vibrational resonances. In the superconducting phase higher impurity scattering rates correspond to sharper drops of the reflectivity for ℏω ~ 2Δ, while in the normal phase they correspond to vertical shifts of the electronic baseline. The intensity of the vibrational resonances is enhanced by an increase of impurity scattering rates in both cases, while their shape is affected in a less trivial way due to the interplay between electronic and lattice responses, as manifested by the different effects upon the two resonances centered around 84 and 148 meV.

Source data

Extended Data Table 1 Numerical validation of the static BECs sum rule
Full size table
Extended Data Table 2 Comparison of Born effective charges
Full size table

Supplementary information

Supplementary Information

Supplementary Figs. 1–5, Tables 1–3 and Sections I–VIII.

Source data

Source Data Fig. 1

A text file containing the x and y values for each curve plotted. Different curves are separated by a new line. ‘#’ denotes comments.

Source Data Fig. 2

A text file containing the x and y values for each curve plotted. Different curves are separated by a new line. ‘#’ denotes comments.

Source Data Fig. 3

A text file containing the x and y values for each curve plotted. Different curves are separated by a new line. ‘#’ denotes comments.

Source Data Extended Data Fig. 1

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Source Data Extended Data Fig. 2

A text file containing the x and y values for each curve plotted. Different curves are separated by a new line. ‘#’ denotes comments.

Source Data Extended Data Table 1

Table in TEX format.

Source Data Extended Data Table 2

Table in TEX format.

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Marchese, G., Macheda, F., Binci, L. et al. Born effective charges and vibrational spectra in superconducting and bad conducting metals. Nat. Phys. 20, 88–94 (2024). https://doi.org/10.1038/s41567-023-02203-3

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