Direct Observation of Topological Phonons in Graphene

Phonons, as the most fundamental emergent bosons in condensed matter systems, play an essential role in the thermal, mechanical, and electronic properties of crystalline materials. Recently, the concept of topology has been introduced to phonon systems, and the nontrivial topological states also exist in phonons due to the constraint by the crystal symmetry of the space group. Although the classification of various topological phonons has been enriched theoretically, experimental studies were limited to several three-dimensional (3D) single crystals with inelastic x-ray or neutron scatterings. The experimental evidence of topological phonons in two-dimensional (2D) materials is absent. Here, using high-resolution electron energy loss spectroscopy following our theoretical predictions, we directly map out the phonon spectra of the atomically thin graphene in the entire 2D Brillouin zone, and observe two nodal-ring phonons and four Dirac phonons. The closed loops of nodal-ring phonons and the conical structure of Dirac phonons in 2D momentum space are clearly revealed by our measurements, in nice agreement with our theoretical calculations. The ability of 3D mapping (2D momentum space and energy space) of phonon spectra opens up a new avenue to the systematic identification of the topological phononic states. Our work lays a solid foundation for potential applications of topological phonons in superconductivity, dynamic instability, and phonon diode.

Figure 1

Symmetries and classification of TPs in a 2D hexagonal lattice. (a),(b) 2D crystal structure and BZ of the hexagonal lattice, respectively. (c)–(e) Illustrations of type-I, type-II, and type-III DPs, respectively. (f)–(h) Illustrations of type-I, type-II, and type-III NPs, respectively.

Figure 2

Graphene phonon spectra and experimental setup. (a) Calculated graphene phonon spectra along the high-symmetry paths. (b) A schematic of the experimental setup. (c) Experimental graphene phonon spectra corresponding to (a). (d) 3D HREELS mapping of the graphene phonon spectra. The topological NPs and DPs are indicated by the green and blue circles (a) or arrows (c), respectively.

Figure 3

Nodal-ring phonons of graphene. (a) Experimental observation of NP1 from the dispersion of LA and ZO branches in the 2D BZ from HREELS. The experimental data are marked by the black dots and the colored dispersions are reconstructed via linear interpolation from the experimental data. The data were taken by varying the azimuthal angle by 5° from the $\mathrm{\Gamma }\text{−}K$ direction each time until a total rotation of 60° (to the nearest $\mathrm{\Gamma }\text{−}K$ direction) is acquired. The entire spectrum is obtained by duplicating the experimental data according to the sixfold symmetry. (b) Projected shape of NP1 obtained from the measured energy gap between the LA and ZO branches in the first BZ. (c),(d) The calculated results corresponding to (a) and (b), respectively. (e)–(h) The corresponding results for NP2 from the experiments and calculations of the TA and ZO branches. The NPs are highlighted by bright cyan lines in (a), (c), (e), and (g).

Figure 4

Dirac phonons of graphene. (a) Demonstration of DP1 from the calculated 3D dispersion plot of the ZA and ZO branches. (b) The schematic of the measurement directions. #2 marks the direction along the $\mathrm{\Gamma }\text{−}K$, #1 and #3 mark the directions $\pm 15°$ off the $\mathrm{\Gamma }\text{−}K$ path. (c),(e),(g) The 2D phonon mapping corresponding to the #1, #2, and #3 directions, respectively. The white dotted lines mark the BZ boundary (BZB), and the boxes i, ii, and iii mark the corresponding cuts illustrated in (a). (d),(f),(h) The EDC stacks corresponding to the boxes i, ii, and iii in (c), (e), and (g), respectively.