Fig. 3

Resonance energies and phonon frequencies for the intralayer and interlayer el–ph processes. a, b Joint density of states that satisfy the intralayer and interlayer processes, respectively, for some twisting angles θ. c The dashed blue, green, pink and red curves represent the calculated values of \(E_{\mathrm{a}}^ -\), \(E_{\mathrm{e}}^ -\), \(E_{\mathrm{a}}^ +\) and \(E_{\mathrm{e}}^ +\) as a function of the twisting angle θ. The blue and black open circles correspond to the resonance energies of the extra peaks observed in the visible Raman spectra in samples with, respectively, small and intermediate angles. The light blue and red open squares correspond to the laser energies where the extra peaks observed in the Raman spectra in samples with θ in the ranges 3–5 and 6–10° have the maximum intensity. The black open squares correspond to the resonance energies of the extra peaks observed in the UV Raman spectra for samples with large twisting angles (22–28°). The black symbols × correspond to the energies of the peaks in the optical absorption spectra of TBG reported in ref. ¹⁷. d The red and black curves represent the frequencies of the phonons with wave vector q_M from the TO and LO branches, respectively, as a function of θ. The dark blue and black circles, and the light blue, red and black squares correspond to the frequencies of the extra peaks observed in the visible, IR and UV Raman spectra as described in c. The black triangles correspond to the results reported by Wang et al.^11. e, f are schematic representation of the low- and high-energy intralayer and interlayer processes, respectively. The error bars in c, d represent the standard deviation