Polarons in highly doped atomically thin graphitic materials

Polaron spectral functions are computed for highly doped graphene-on-substrate and other atomically thin graphitic systems using the diagrammatic Monte Carlo technique. The specific aim is to investigate the effects of interaction on spectral functions when the symmetry between sublattices of a honeycomb lattice has been broken by the substrate or ionicity, inducing a band gap. Introduction of electron-phonon coupling leads to several polaronic features, such as band-flattening and changes in particle lifetimes. At the K point, differences between energies on each sublattice increase with electron-phonon coupling, indicating an augmented transport gap, while the spectral gap decreases slightly. Effects of phonon dispersion and long-range interactions are investigated, and found to lead to only quantitative changes in spectra.

Figure 1

Graphene-substrate system annotated with interactions and sublattices. Electron-phonon interactions between the graphene layer and substrate are poorly screened, and large interactions are possible.

Figure 2

Image plots of the graphene spectral function across the Brillouin zone. Various $\Delta$ and $\lambda$ are shown. The noninteracting dispersion is overlaid. At $\Delta =0$, $A$ and $B$ sublattices are symmetrical, leading to identical spectral functions, so only a single panel is shown for each $\lambda$. Comparison with the noninteracting dispersion shows that there is some flattening of the band close to the K point. The spectral function is sharp close to the K point. In the vicinity of the $\Gamma$ point, the spectral function is also sharp for states within an energy $\hslash \Omega$ of the bottom of the band. Quasiparticle lifetime (related to the inverse of the width) is greatly reduced close to the tops of the bands. Increase in $\Delta$ breaks the $AB$ symmetry. Some band flattening is seen. There is also a weak excitation associated with $B$ sites at higher energies, which touches the lower band. Panel (a) $\Delta =0$, $\lambda =0.7$, (b) $\Delta =0$, $\lambda =2.8$ where $A$ and $B$ site electrons are identical. (c) and (d) $\Delta =0.5t$, $\lambda =0.7$, and (e) and (f) $\Delta =t$, $\lambda =0.7$, showing $A$ and $B$ site electrons, respectively. There is only a single electron, so the chemical potential is at the bottom of the band, $E_0$. The origin is arbitrary, but for consistency has been taken as the point where the unperturbed bands cross.

Figure 3

Same as Fig. 2 with the leading nearest neighbor corrections from Fröhlich interactions included. There are only minor changes to the results.

Figure 4

Variation of the graphene spectral function across the Brillouin zone for various $\Delta$ and $\lambda =0.7$. The same data as Fig. 2 is shown with both $A$ and $B$ sublattices superimposed on the same plot. From top to bottom, $\Delta =0$, $\Delta =t/2$, and $\Delta =t$. The gap opening is clearly visible. Broadening of the spectral functions can be seen, especially at larger energies.

Figure 5

Image plots of the graphene spectral function across Brillouin zone. $\Delta =t$ and various $\lambda$. Panels (a) and (b) $\lambda =0.7$, (c) and (d) $\lambda =1.4$, (e) and (f) $\lambda =2.1$, and (g) and (h) $\lambda =2.8$. $A$ sites can be seen on the left and $B$ on the right. A flat polaron band can be seen forming at an energy around $\hslash \Omega$ above the bottom of the band for very large $\lambda$. The quasiparticle lifetime decreases dramatically with increased $\lambda$.

Figure 6

Graphene spectral function across Brillouin zone. $\Delta =t$ and various $\lambda$. Data is as Fig. 5 but with results for both sublattices plotted together. It can be seen that the gap is robust against increase in $\lambda$, although a number of additional excitations appear.

Figure 7

Same as Fig. 5 with the leading corrections from Fröhlich interactions included. There are only moderate changes to the results, consistent with a small reduction in effective $\lambda$.

Figure 8

Graphene spectral function at the K point. $\Delta =t$ and $\lambda$ is varied. The spectral gap (seen at approximately $E=0$) is robust. The spectral functions broaden with increased $\lambda$, and the polaron band rapidly drops in energy leading to an enhanced transport gap.

Figure 9

Effects of dispersion, $\Delta \Omega$, and phonon frequency, $\Omega$, on the spectral function. $\lambda =2.8$ throughout. Broadening the phonon dispersion has no major effects on the spectral function. On the other hand, decreasing the phonon frequency leads to a broadening of all features corresponding to a sharp decrease in quasiparticle lifetime, and indicating a significant increase in scattering.