Radiative corrections to quantum sticking on graphene

We study the sticking rate of atomic hydrogen to suspended graphene using four different methods that include contributions from processes with multiphonon emission. We compare the numerical results of the sticking rate obtained by: (i) the loop expansion of the atom self-energy; (ii) the noncrossing approximation (NCA); (iii) the independent boson model approximation (IBMA); and (iv) a leading-order soft-phonon resummation method (SPR). The loop expansion reveals an infrared problem, analogous to the infamous infrared problem in QED. The two-loop contribution to the sticking rate gives a result that tends to diverge for large membranes. The latter three methods remedy this infrared problem and give results that are finite in the limit of an infinite membrane. We find that for micromembranes (sizes ranging 100 nm to $10\mu \mathrm{m})$, the latter three methods give results that are in good agreement with each other and yield sticking rates that are mildly suppressed relative to the lowest-order golden rule rate. Lastly, we find that the SPR sticking rate decreases slowly to zero with increasing membrane size, while both the NCA and IBMA rates tend to a nonzero constant in this limit. Thus, approximations to the sticking rate can be sensitive to the effects of soft-phonon emission for large membranes.

Figure 1

Feynman diagram for the one-loop atom self-energy ${\mathrm{\Sigma }}_{kk}^{\left(1\right)}$ (left). Symbolic elements used in the construction of diagrams are pictured (on right).

Figure 2

Feynman diagrams for the two-loop terms for ${\mathrm{\Sigma }}_{kk}$: (a) nested (or rainbow), (b) overlap, and (c) loop-after-loop.

Figure 3

The real and imaginary parts of scaled IBM Green's function $g_b{G}_{bb}^{\mathrm{IBM}}(E_s/g_b)$ vs $E_s/g_b$ for IR cutoff $\epsilon =0$. $G_{bb}^{\mathrm{IBM}}(E_s/g_b)$ is a smooth, complex-valued function in the limit $\epsilon \to 0$.

Figure 4

Diagrams in the NCA. Evaluation of these diagrams results in a set of coupled nonlinear integral equations that may be solved self-consistently. This is equivalent to a sum of all diagrams with no phonon lines crossing.

Figure 5

Dependence of the (normalized) adsorption rate $\mathrm{\Gamma }/{\mathrm{\Gamma }}_0$ with IR cutoff $\epsilon$ (top). The two-loop approximation to $\mathrm{\Gamma }$ diverges as $\epsilon \to 0$. Under NCA and IBMA, $\mathrm{\Gamma }/{\mathrm{\Gamma }}_0$ converges to a constant in this limit (bottom). In SPR [9], $\mathrm{\Gamma }/{\mathrm{\Gamma }}_0$ behaves asymptotically as ${ln}^{-2}({\omega }_D/\epsilon )$ as $\epsilon \to 0$.

Figure 6

Left: Real part of the atom self-energy $\mathrm{Re}{\mathrm{\Sigma }}_{kk}\left(E\right)$ vs energy $E$. Results are plotted for IR cutoff $\epsilon =0,g_{kb}^2{\rho }_0=0.7\mu \mathrm{eV}$ using three different approximation methods: one-loop self-energy ${\mathrm{\Sigma }}_{kk}^{\left(1\right)}$, NCA, and IBMA. Right: Imaginary part of the atom self-energy $\mathrm{Im}{\mathrm{\Sigma }}_{kk}\left(E\right)$ vs energy $E$. The sticking rate is obtained from self-energy using Eq. (8).

Figure 7

Top: Normalized adsorption rate $\mathrm{\Gamma }/{\mathrm{\Gamma }}_0$ from the SPR method vs energy transfer $E_s\equiv E+E_b$ for various membrane sizes ranging 100 nm to $10\mu \mathrm{m}$. Bottom: Normalized adsorption rate $\mathrm{\Gamma }/{\mathrm{\Gamma }}_0$ from SPR, IBMA, and NCA vs $E_s$ for a $1\mu \mathrm{m}$ micromembrane. For shallow bound states, the SPR result shows appreciable suppression in sticking relative to the other methods. However, for deep bound states, the three methods (IBMA, NCA, and SPR) give sticking rates that are in good agreement with each other.