Comment on “Theory of phonon-assisted adsorption in graphene: Many-body infrared dynamics”

Two approximations used by Sengupta [Phys. Rev. B 100, 075429 (2019)] in numerically computing the adsorption rate of cold hydrogen atoms on suspended graphene are critically examined. The independent boson model approximation (IBMA) was used to compute the atom self-energy, and the single-pole approximation (SPA) was used to obtain the adsorption rate from the self-energy. It is shown explicitly that there are additional contributions to the self-energy appearing at the same order of the atom-phonon coupling as the IBMA terms that alter the value of the real part of the self-energy at low energies by several orders of magnitude in the regime of interest. This shift in the self-energy, consequently, renders the use of SPA invalid.

Figure 1

Real (solid) and imaginary (dotted) parts of the atom self-energy in the IBMA ${\overline{\mathrm{\Sigma }}}^{\left(\mathrm{IBM}\right)}$ versus $E/{\omega }_D$ for $\epsilon =2.0\mathrm{K}$. The (dimensionless) self-energy is defined $\overline{\mathrm{\Sigma }}\equiv \mathrm{\Sigma }/g_{kb}^2{\rho }_0$ where ${\rho }_0$ is the partial (axisymmetric) vibrational density of states for the membrane.

Figure 2

Real part of the (scaled) atom self-energy ${\overline{\mathrm{\Sigma }}}_r$ versus $E/{\omega }_D$. IBMA (right) compares favorably to the $O\left(g_{kb}^2\right)$ (left) for $\epsilon =2\mathrm{K}$ (long dashed line). The real part of the $O\left(g_{kb}^2\right)$ self-energy shifts downward as $\epsilon$ is reduced: $\epsilon =1.5$ (dot-dashed line), 1 (dashed line), and 0.8 K (short dashed line). Order $O\left(g_{kb}^2\right)$ self-energy (left) changes substantially over the range of $\epsilon$. IBMA self-energy (right) changes little with $\epsilon$ over the same range.

Figure 3

Imaginary part of the atom self-energy ${\overline{\mathrm{\Sigma }}}_i$ versus $E/{\omega }_D$. Order $O\left(g_{kb}^2\right)$ self-energy ${\overline{\mathrm{\Sigma }}}_i$ (left) becomes positive near $E=0$ with decreasing $\epsilon$, whereas IBMA self-energy ${\overline{\mathrm{\Sigma }}}_i^{\mathrm{IBM}}$ changes little for $\epsilon =0.25$ (solid line), 0.3 (dashed line), 0.5 (dotted line), and 0.6 K (dot-dashed line).

Figure 4

Real part of the (scaled) atom self-energy ${\overline{\mathrm{\Sigma }}}_r$ to order $O\left(g_{kb}^2\right)$ versus $E/{\omega }_D$ for $\epsilon =0.25$ (solid line), 0.3 (dashed line), 0.5 (dotted line), and 0.6 K (dot-dashed line).

Figure 5

Graphical solution for the quasiparticle energy $E_p$. Real part (solid line) of the atom self-energy in IBMA $\mathrm{Re}{\overline{\mathrm{\Sigma }}}^{\left(\mathrm{IBM}\right)}$ for $\epsilon =2.0\mathrm{K}$ versus $E/{\omega }_D$ crosses the line (dot-dashed line) where ${\overline{E}}_p-{\overline{E}}_k=\mathrm{Re}\overline{\mathrm{\Sigma }}\left({\overline{E}}_p\right)$. In the single-pole approximation, the adsorption rate depends on $\mathrm{Im}{\mathrm{\Sigma }}^{\left(\mathrm{IBM}\right)}\left(E_p\right)$ and is determined by Eq. (8).

Figure 6

Graphical solution for the quasiparticle energy $E_p$ fails with large negative shifts of the real part of the self energy. Real part (solid line) of the atom self-energy $\mathrm{Re}\overline{\mathrm{\Sigma }}$ for $\epsilon \le 0.4\mathrm{K}$ versus $E/{\omega }_D$ does not intersect with the line (dot-dashed line) where ${\overline{E}}_p-{\overline{E}}_k=\mathrm{Re}\overline{\mathrm{\Sigma }}\left({\overline{E}}_p\right)$. The single-pole approximation fails for low atom energy $[E_k<|{\mathrm{\Sigma }}_r\left(0\right)\left|\right]$ when using the order $O\left(g_{kb}^2\right)$ self-energy but does not using the IBMA self-energy (horizontal dashed) line.