Heavily boron-doped diamond exhibits a nonadiabatic Kohn anomaly: adiabatic density-functional perturbation theory fails to reproduce its measured high-energy phonon dispersions, while a many-body treatment of dynamical (beyond Born-Oppenheimer) electron screening succeeds, and neglecting these effects at a carrier density of 1.4×1021 1.4\times10^{21}\,cm−3^{-3} overestimates the electron-phonon coupling strength λ\lambda by about 20% and the superconducting transition temperature TcT_c by up to 50% (Caruso et al., Phys. Rev. Lett. 119, 017001 (2017)). That paper closes by calling for a systematic, material-by-material investigation of nonadiabatic effects and Kohn anomalies in the phonon dispersions of three-dimensional heavily doped semiconductors and superconducting oxides, and for nonadiabatic effects to be included in future TcT_c predictions. Subsequent first-principles work established large nonadiabatic effects in two-dimensional materials and in MgB2_2, quantified nonadiabatic renormalizations of λ\lambda and TcT_c for MgB2_2 and hole-doped diamond, and settled methodological questions about screened versus bare electron-phonon vertices — but no systematic nonadiabatic study of three-dimensional heavily doped semiconductors beyond diamond, nor of superconducting oxides such as doped SrTiO3_3 or the cuprates, exists, and dynamical screening of the electron-phonon matrix element remains explicitly beyond current first-principles implementations. The problem is open.

Background

In first-principles lattice dynamics, phonons are usually computed with density-functional perturbation theory (DFPT), the linear-response form of density-functional theory in which the electronic density is differentiated with respect to atomic displacements. Standard DFPT works within the Born-Oppenheimer (adiabatic) approximation: electrons are assumed to adjust instantaneously to nuclear motion, so the screening of the ion displacement by the electrons is static. A Kohn anomaly is a pronounced kink or softening in a phonon dispersion branch at wavevectors that connect points on the Fermi surface (Kohn, 1959); such anomalies appear in any metal, and heavily doped semiconductors become metallic once degenerate. Nonadiabatic effects arise when this instantaneous-screening picture fails: for long-wavelength optical phonons in polar or small-gap systems, the phonon frequency can be comparable to the characteristic electronic excitation energies, and the phonon then probes the dynamical (frequency-dependent) dielectric response of the electrons rather than its static limit. In many-body language the phonon acquires a Fan-Migdal self-energy — a correction to the phonon propagator from electron-hole pair creation — whose real part shifts frequencies and whose imaginary part gives phonon linewidths. Early experimental hints of this physics date back to the doping-dependent Raman shifts of Si and Ge (Cerdeira and Cardona, 1972) and to the electron-phonon mass enhancement in the perovskite oxide semiconductor SrTiO3_3 (van Mechelen et al., 2008). The pivotal result motivating this problem came from a joint theory-experiment study of superconducting boron-doped diamond (Tc=2.8 T_c = 2.8\,K at 1.4×1021 1.4\times10^{21}\,cm−3^{-3} boron): high-resolution inelastic x-ray scattering (IXS, the measurement of phonon energies by spectrally analyzing x-rays scattered from the sample) showed that adiabatic DFPT misplaces the high-energy optical phonons near the Brillouin-zone center by about 10%, while a many-body field-theoretic treatment of nonadiabatic electron screening reproduces the measured dispersions, revealing a nonadiabatic Kohn anomaly and a breakdown of the Born-Oppenheimer approximation in a three-dimensional covalent crystal (Caruso et al., 2017). The consequences propagate into superconductivity: in Migdal-Eliashberg theory — the standard framework in which superconductivity arises from electrons exchanging phonons — the electron-phonon coupling strength λ\lambda and the Eliashberg spectral function α2F(ω)\alpha^2F(\omega) (the phonon density of states weighted by the coupling) determine TcT_c through the McMillan/Allen-Dynes formula. Since λ\lambda scales roughly as the inverse square of the phonon frequencies, the ~10% adiabatic error in the diamond LO phonon overestimates λ\lambda by ~20% and TcT_c by up to 50% (Caruso et al., 2017). Since 2017 the field has established the phenomenon widely but unevenly across dimensions. Large nonadiabatic renormalizations, including phonon-frequency splitting and dramatic reshaping of long-wavelength optical modes, were computed for doped monolayer MoS2_2 (Garcia-Goiricelaya et al., 2020), and adiabaticity breakdown was tied to Lifshitz transitions (topological changes of the Fermi surface) in doped MoS2_2 and WS2_2 monolayers; analogous nonadiabatic physics was analyzed in graphene, in MgB2_2 Raman spectra (Novko, 2018), and in one-dimensional metals (Marazzi et al., 2025). For the superconducting side, Girotto and Novko (2023) computed nonadiabatic corrections to phonons, λ\lambda, α2F(ω)\alpha^2F(\omega) and TcT_c from first principles for bulk MgB2_2 and hole-doped diamond plus several doped monolayers, finding that nonadiabatic renormalization measurably improves agreement with measured TcT_c (e.g. it lowers the overestimated MgB2_2 TcT_c by 8-12%); they simultaneously state that dynamical screening of the electron-phonon matrix element g(q,ω)g(\mathbf{q},\omega) itself — as opposed to the phonon self-energy — remains out of reach of current many-body-perturbation-plus-DFPT implementations and is a crucial but very demanding extension. On methodology, Berges et al. (2023) assessed whether the phonon self-energy should use screened or bare electron-phonon vertices, finding that the standard adiabatic double-screening approximation is accurate for phonon self-energy corrections, and Paleari and Marini (2021, 2023) developed overscreening-free phonon self-energies; Requist et al. (2019) formulated an exact-factorization density-functional theory that treats electron-phonon coupling nonadiabatically from the outset. Very recent work extends nonadiabatic Eliashberg treatments with vertex corrections to high-pressure hydride superconductors and introduces material-specific diagnostics for when beyond-adiabatic theory is needed (Mishra et al., 2026), and higher-order nonadiabatic corrections to the phonon self-energy (including finite electron lifetimes) have been worked out for MgB2_2 (Girotto Erhardt and Poncé, 2026). What is missing is exactly what the source paper called for: apart from diamond itself and the non-oxide MgB2_2, no three-dimensional heavily doped semiconductor (e.g. doped Si, Ge, SiC, GaN) and no superconducting oxide (e.g. doped SrTiO3_3 or the cuprates) has been subjected to a systematic first-principles nonadiabatic analysis of its phonon dispersions and Kohn anomalies with spectroscopic validation. Separately, no available framework yet includes dynamical screening of the electron-phonon vertex in TcT_c prediction — an extension identified as crucial but very demanding by subsequent work (Girotto and Novko, 2023). This omission matters because superconducting oxides such as dilutely doped SrTiO3_3 sit precisely in the antiadiabatic regime where the Fermi energy is comparable to or smaller than the relevant soft optical-phonon energies, where the adiabatic hierarchy is inverted.

Problem Statement

Carry out the systematic, material-by-material investigation of nonadiabatic effects and Kohn anomalies in the phonon dispersions of three-dimensional heavily doped semiconductors and superconducting oxides that Caruso et al. (Phys. Rev. Lett. 119, 017001 (2017)) called for, together with its superconducting consequence. Concretely: (i) compute from first principles, with a nonadiabatic (frequency-dependent) many-body treatment of the electron-phonon interaction, the adiabatic-versus-nonadiabatic phonon dispersions and linewidths — including the Kohn-anomaly regions — for representative heavily doped three-dimensional semiconductors beyond diamond (e.g. degenerately doped Si, Ge, SiC, GaN and related covalent/compound semiconductors) and for representative superconducting oxides (e.g. dilutely doped SrTiO3_3 and cuprate superconductors), validating the nonadiabatic predictions against spectroscopic measurements (inelastic x-ray scattering, Raman, or inelastic neutron scattering) where available; (ii) establish from these calculations the material parameters — carrier density, band structure and effective masses, dielectric screening strength, phonon frequencies, and dimensionality of the Fermi surface — that control the magnitude and sign of the breakdown of the adiabatic (Born-Oppenheimer) approximation in three-dimensional crystals; and (iii) quantify how nonadiabatic phonon renormalization propagates into the electron-phonon coupling strength λ\lambda, the Eliashberg spectral function α2F(ω)\alpha^2F(\omega), and predicted superconducting transition temperatures TcT_c for these materials, in line with the source's call that nonadiabatic effects be considered in future studies of TcT_c. The common scientific objective uniting these deliverables is a validated, quantitative map of where and why the Born-Oppenheimer approximation fails for phonons in three-dimensional doped crystals, and a correspondingly corrected, predictive account of phonon-mediated superconductivity in the material classes named by the source. The material classes are those named in the source paper and are open-ended; a complete answer applies a consistent, validated nonadiabatic methodology across representative members of both classes and includes at least two heavily doped three-dimensional semiconductors beyond diamond and at least one superconducting oxide, each treated with adiabatic-versus-nonadiabatic phonon dispersions and linewidths and, where spectroscopic data exist, a comparison against them.

References

  1. F. Caruso, M. Hoesch, P. Achatz, J. Serrano, M. Krisch, E. Bustarret, F. Giustino, 'Nonadiabatic Kohn Anomaly in Heavily Boron-Doped Diamond', Phys. Rev. Lett. 119, 017001 (2017), DOI: 10.1103/PhysRevLett.119.017001, arXiv:1706.02151, https://arxiv.org/abs/1706.02151
  2. W. Kohn, 'Image of the Fermi Surface in the Vibration Spectrum of a Metal', Phys. Rev. Lett. 2, 393 (1959), DOI: 10.1103/PhysRevLett.2.393, https://doi.org/10.1103/PhysRevLett.2.393
  3. F. Giustino, 'Electron-phonon interactions from first principles', Rev. Mod. Phys. 89, 015003 (2017), DOI: 10.1103/RevModPhys.89.015003, https://doi.org/10.1103/RevModPhys.89.015003
  4. F. Cerdeira, M. Cardona, 'Effect of Carrier Concentration on the Raman Frequencies of Si and Ge', Phys. Rev. B 5, 1440 (1972), DOI: 10.1103/PhysRevB.5.1440, https://doi.org/10.1103/PhysRevB.5.1440
  5. J. L. M. van Mechelen, D. van der Marel, C. Grimaldi, A. B. Kuzmenko, N. P. Armitage, N. Reyren, H. Hagemann, I. I. Mazin, 'Electron-Phonon Interaction and Charge Carrier Mass Enhancement in SrTiO3_3', Phys. Rev. Lett. 100, 226403 (2008), DOI: 10.1103/PhysRevLett.100.226403, https://doi.org/10.1103/PhysRevLett.100.226403
  6. N. Girotto, D. Novko, 'Dynamical renormalization of electron-phonon coupling in conventional superconductors', Phys. Rev. B 107, 064310 (2023), DOI: 10.1103/PhysRevB.107.064310, arXiv:2212.08117, https://arxiv.org/abs/2212.08117
  7. J. Berges, N. Girotto, T. Wehling, N. Marzari, S. Poncé, 'Phonon Self-Energy Corrections: To Screen, or Not to Screen', Phys. Rev. X 13, 041009 (2023), DOI: 10.1103/PhysRevX.13.041009, arXiv:2212.11806, https://arxiv.org/abs/2212.11806
  8. D. Novko, 'Nonadiabatic coupling effects in MgB2 reexamined', Phys. Rev. B 98, 041112(R) (2018), DOI: 10.1103/PhysRevB.98.041112, arXiv:1806.03212, https://arxiv.org/abs/1806.03212
  9. P. Garcia-Goiricelaya, J. Lafuente-Bartolome, I. G. Gurtubay, A. Eiguren, 'Emergence of large non-adiabatic effects induced by the electron-phonon interaction on the complex vibrational quasiparticle spectrum of the doped monolayer MoS2', Phys. Rev. B 101, 054304 (2020), DOI: 10.1103/PhysRevB.101.054304, arXiv:1911.00311, https://arxiv.org/abs/1911.00311
  10. J. Krsnik, D. Novko, O. S. Barišić, 'Superconductivity in two-dimensional systems enhanced by nonadiabatic phonon-production effects', Phys. Rev. B 110, L180505 (2024), DOI: 10.1103/PhysRevB.110.L180505, arXiv:2405.12554, https://arxiv.org/abs/2405.12554
  11. E. Marazzi, S. Poncé, J.-C. Charlier, G.-M. Rignanese, 'Importance of Nonadiabatic Effects in Kohn Anomalies in 1D Metals', Phys. Rev. Lett. 135, 256402 (2025), DOI: 10.1103/PhysRevLett.135.256402, arXiv:2506.18704, https://arxiv.org/abs/2506.18704, https://doi.org/10.1103/PhysRevLett.135.256402
  12. F. Paleari, A. Marini, 'Overscreening-free electron-phonon interaction in realistic materials', arXiv:2105.09823 (2021), https://arxiv.org/abs/2105.09823 (superseded, with changed author list, by arXiv:2211.02573; the published companion is Phys. Rev. B 107, 024305 (2023) by A. Marini)
  13. A. Marini, 'Equilibrium and out-of-equilibrium realistic phonon self-energy free from overscreening', Phys. Rev. B 107, 024305 (2023), DOI: 10.1103/PhysRevB.107.024305, https://doi.org/10.1103/PhysRevB.107.024305
  14. R. Requist, C. R. Proetto, E. K. U. Gross, 'Exact factorization-based density functional theory of electron-phonon systems', Phys. Rev. B 99, 165136 (2019), DOI: 10.1103/PhysRevB.99.165136, arXiv:1901.07523, https://arxiv.org/abs/1901.07523
  15. S. B. Mishra, F. Belli, E. Zurek, E. R. Margine, 'Anharmonicity and Nonadiabaticity in Hydride Superconductors', arXiv:2608.14949 (2026), https://arxiv.org/abs/2608.14949
  16. N. Girotto Erhardt, S. Poncé, 'Higher-order nonadiabaticity governs the temperature dependence of the phonon spectrum', arXiv:2608.14882 (2026), https://arxiv.org/abs/2608.14882
  17. F. Caruso, D. Novko, 'Ultrafast dynamics of electrons and phonons: from the two-temperature model to the time-dependent Boltzmann equation', Electron. Struct. 4, 033002 (2022), DOI: 10.1080/23746149.2022.2095925, arXiv:2202.07237, https://arxiv.org/abs/2202.07237
  18. C. J. N. Coveney, N. M. Tubman, C.-E. Hsu, A. Montoya-Castillo, M. R. Filip, J. B. Neaton, Z. Li, V. Vlcek, A. M. Alvertis, 'Phonon-mediated electron attraction in SrTiO3 via the generalized Fröhlich and deformation potential mechanisms', arXiv:2501.17230 (2025), https://arxiv.org/abs/2501.17230