Abstract
Many of the applications of graphene rely on its uneven stiffness and high thermal conductivity, but the mechanical properties of graphene—and, in general, of all two-dimensional materials—are still not fully understood. Harmonic theory predicts a quadratic dispersion for the out-of-plane flexural acoustic vibrational mode, which leads to the unphysical result that long-wavelength in-plane acoustic modes decay before vibrating for one period, preventing the propagation of sound. The robustness of quadratic dispersion has been questioned by arguing that the anharmonic phonon–phonon interaction linearizes it. However, this implies a divergent bending rigidity in the long-wavelength regime. Here we show that rotational invariance protects the quadratic flexural dispersion against phonon–phonon interactions, and consequently, the bending stiffness is non-divergent irrespective of the temperature. By including non-perturbative anharmonic effects in our calculations, we find that sound propagation coexists with a quadratic dispersion. We also show that the temperature dependence of the height fluctuations of the membrane, known as ripples, is fully determined by thermal or quantum fluctuations, but without the anharmonic suppression of their amplitude previously assumed. These conclusions should hold for all two-dimensional materials.
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Data availability
Source data are provided with this paper. All other data are available from the corresponding author upon reasonable request.
Code availability
The atomistic calculations of the SCHA theory are performed with the SSCHA code. This code is open source and can be downloaded from https://sscha.eu/. The SCHA calculations in the membrane model are performed with an in-house program adapting the distributed SCHA code, which is available from the corresponding author upon reasonable request.
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Acknowledgements
We would like to thank F. Guinea for useful conversations. Financial support was provided by the Spanish Ministry of Economy and Competitiveness (FIS2016-76617-P); the Spanish Ministry of Science and Innovation (grant nos. PID2019-105488GB-I00 and PID2022-142861NA-I00); the Department of Education, Universities and Research of the Basque Government and the University of the Basque Country (IT1707-22 and IT1527-22); and the European Commission under the Graphene Flagship, Core3, grant no. 881603. M.C. acknowledges support from ICSC—Centro Nazionale di Ricerca in HPC, Big Data and Quantum Computing, funded by the European Union under NextGenerationEU. U.A. is thankful to the Material Physics Center for a predoctoral fellowship. J.D. thanks the Department of Education of the Basque Government for a predoctoral fellowship (grant no. PRE-2020-1-0220). L.M. acknowledges funding support from the European Union under the Marie Curie Individual Fellowship THERMOH. Computer facilities were provided by the Donostia International Physics Center (DIPC).
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U.A. and J.D. contributed equally. U.A. performed the atomistic calculations, whereas U.A., J.D. and T.C. performed the calculations on the membrane and developed the theoretical adaptation of SCHA theory to the membrane. L.M., T.C., R.B. and I.E. developed the mathematical proof of the relation between Green’s function and free energy Hessian. I.E. and F.M. supervised the full project. The manuscript was written by U.A., J.D. and I.E., with input from all authors.
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Extended data
Extended Data Fig. 1 Negligible dynamic effects in the ZA mode frequencies.
Harmonic, and SCHA auxiliary and physical phonons (static and dynamic) calculated at 0 K (a) and 300 K (b) with the atomistic potential for the ZA mode.
Extended Data Fig. 2 Height-height correlation function as a function of the biaxial strain δa.
Impressively, the behavior for small q deviates from the q−4 law even for very small strains, for example δa = 10−5.
Extended Data Fig. 3 Bubble approximation in the static limit of the SCHA theory.
(a) Physical phonons in the static approach with the atomistic potential at 500 K including and neglecting \(\mathop{{{{\boldsymbol{\Phi }}}}}\limits^{(4)}\) in Eq. (11). (b) panel only includes the ZA modes and it is in logarithmic scale. The calculation is done in a 6 × 6 supercell.
Extended Data Fig. 4 Empirical potential benchmark I.
Harmonic phonon spectrum of graphene calculated with the machine learning empirical potential and ab initio. The calculations are done in a 6 × 6 supercell.
Extended Data Fig. 5 Empirical potential benchmark II.
Harmonic, and SCHA auxiliary and physical frequencies (static) using the DFT and machine learning (ML) forces. (a) panel shows the out-of-plane optical frequency at the Γ point and (b) panel the in-plane one.
Extended Data Fig. 6 Negative thermal expansion of graphene in the membrane model.
δa as a function of temperature in the membrane model.
Extended Data Fig. 7 Justification of the bubble and Lorentzian approximations in the calculation of spectral properties of the LA mode within the membrane model.
(a) Linewidth (full width at half maximum, FWHM) contribution of the term containing the fourth-order tensor of the LA mode calculated in the membrane model at 100 K using the harmonic and SCHA auxiliar phonons. The value of the smearing is in the legend. (b) Spectral function of the LA mode with momentum 0.01 Å−1 with and without considering the frequency dependence of the self energy.
Extended Data Fig. 8 Anharmonicity does not remove the quasiparticle picture of ZA modes.
Linewidth (full width half maximum) of ZA phonon mode divided by its frequency at 300K calculated within the membrane model. The ratio never gets bigger than 1.
Extended Data Fig. 9 Height-height correlation function showing a pure quantum behaviour.
Fourier transform of the height-height correlation function at 0 K in the membrane model evaluated at different levels of approximation: harmonic (black dots), anharmonic rotationally invariant (RI) result (green filled dots) and anharmonic no rotationally invariant (No RI) result (green empty dots). The dashed lines correspond to the linear fitting in each case.
Extended Data Fig. 10 Height-height correlation function showing a pure classical behaviour.
Fourier transform of the height-height correlation function at 300 K in the membrane model evaluated at different levels of approximation: harmonic (black dots), anharmonic RI result (green filled dots) and anharmonic No RI result (green empty dots). The dashed lines correspond to the linear fitting in each case.
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Source Data Fig. 2
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Source Data Fig. 3
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Source Data Fig. 4
Numerical data. Comments for each panel are indicated with #.
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Aseginolaza, U., Diego, J., Cea, T. et al. Bending rigidity, sound propagation and ripples in flat graphene. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02441-z
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DOI: https://doi.org/10.1038/s41567-024-02441-z
