Figure 1

The layer breathing mode for bilayer graphene. (a) Schematic representation of the atomic displacement for the LBM in 2LG. (b) Normalized 2ZO${}^{\prime }$-mode spectra of 2LG at different laser excitation energies $E_{\mathrm{exc}}$. The high- and low-energy components of the 2ZO${}^{\prime }$ band are denoted as 2ZO${}^{\prime +}$ and 2ZO${}^{\prime -}$, respectively. (c) The main electronic scattering processes in the two-phonon double-resonance Raman mechanism of the 2ZO${}^{\prime }$ mode in 2LG. (d) The ZO${}^{\prime }$-mode phonon dispersion (symbols) extracted from (b). The phonon frequency is obtained as half of the value of the Raman shift for the peaks in (b) through fits by a double Gaussian function. The error bars reflect the widths of the Gaussian fit functions. The 2ZO${}^{\prime +}$ and 2ZO${}^{\prime -}$ peaks are ascribed to the P11 and P22 processes in (c), respectively. The corresponding error bars represent uncertainties in estimating the phonon momenta in these scattering processes. The experimental data are compared with the theoretically predicted dispersion (Ref. 16) for the LBM in 2LG (lines).Reuse & Permissions

Figure 2

The layer breathing modes for 4LG. (a) Schematic representation of the atomic displacements for the three ZO${}^{\prime }$ modes in 4LG. The length of the arrows represents the magnitude of the vibration. (b) Raman spectra of the three 2ZO${}^{\prime }$ modes in 4LG for $E_{\mathrm{exc}}$ $=$ 1.58 (upper trace), 1.96 (middle trace), and 2.33 eV (lower trace). (c) ZO${}^{\prime }$-mode phonon dispersion, extracted from (b). In determining the phonon momenta, we assume that the high- (low)-frequency components of the overtone modes arise from electronic processes with the maximum (minimum) phonon wave vector defined by the band structure of 4LG for the double-resonance Raman scattering. The experimental data are compared with the theoretically predicted dispersion (Ref. 16) of the low-frequency phonon modes in 4LG.Reuse & Permissions

Figure 3

The layer breathing modes for 11LG. (a) Schematic representation of the atomic displacements of the ten different ZO${}^{\prime }$ modes in 11LG. The left panel shows the layer structure of 11LG sample. The right panel shows the layer displacement in each ZO${}^{\prime }$ mode, with arrows drawn in proportion to magnitude of the displacement. (b) The 2ZO${}^{\prime }$-mode Raman spectrum of 11LG for $E_{\mathrm{exc}}$ $=$ 2.33 eV. (c) The ZO${}^{\prime }$-mode phonon frequency, obtained as half of the frequency of the corresponding overtone Raman peaks in (b). The experimental data are compared with the predictions of the model described in the text [Eq. (1)] for the ZO${}^{\prime }$-mode frequencies of 11LG.Reuse & Permissions

Figure 4

The layer breathing modes for FLG. (a) The 2ZO${}^{\prime }$-mode Raman spectra for FLG with $N$ $=$ 2 to 20 and for the bulk graphite. The spectra are shifted for clarity. (The sharp features below 150 cm${}^{-1}$ in the graphite spectrum arise from the Raman scattering of air. The spikes are more visible in graphite than in FLG because of the reflection of the laser from the graphite surface.) (b) The ZO${}^{\prime }$-mode phonon frequencies, obtained from the overtone Raman peaks in (a), as a function of layer thickness. The experimental results (symbols) are compared with theory (lines) as described in the text [Eq. (1)]. The dashed line denotes the ZO${}^{\prime }$-mode frequency measured in bulk graphite. (c) The LBM phonon dispersion along the $A$-Γ line in graphite, constructed from the data in (b) within a zone-folding scheme, in comparison with the theoretically predicted dispersion (Ref. 16) of the low-frequency phonons in graphite.Reuse & Permissions