Screening of long-range Coulomb interaction in graphene nanoribbons: Armchair versus zigzag edges

We study the electronic screening of the long-range Coulomb interaction in graphene nanoribbons (GNRs) with armchair and zigzag edges as a function of the ribbon width by employing ab initio calculations in conjunction with the random-phase approximation. We find that in GNRs with armchair edges quantum confinement effects lead to oscillatory behavior of the on-site screened Coulomb interaction with the ribbon width. Furthermore, the reduced dimensionality and the existence of a band gap result in a nonconventional screening of the Coulomb interaction; that is, it is screened at short distances and antiscreened at intermediate distances, and finally, it approaches the bare (unscreened) interaction at large distances. In the case of GNRs with zigzag edges the presence of edge states strongly affects the screening, which leads to a strong reduction of the effective on-site Coulomb interaction (Hubbard $U)$ parameters at the edge. We find that the interactions turn out to be local; the nonlocal part is strongly screened due to edge states, making GNRs with zigzag edges correlated materials. On the basis of the calculated effective Coulomb interaction parameter $U$, we discuss the appearance of ferromagnetism at zigzag edges of GNRs within the Stoner model.

Figure 1

Two types of edges for graphene nanoribbons: (a) $N_a=6$ (6-AGNR), (b) $N_a=7$ (7-AGNR), (c) $N_a=8$ (8-AGNR), and (d) $N_a=9$ (9-AGNR) are armchair ribbons with different widths. $N_a$ is the number of dimer lines across the ribbon width. (e), (f), and (g) correspond to zigzag ribbons of $N_z=6$ (6-ZGNR), $N_z=7$ (7-ZGNR), and $N_z=8$ (8-ZGNR) respectively. $N_z$ is the number of zigzag chains across the ribbon width.

Figure 2

(a) Total and orbital-projected band structures for edge and inner C atoms of 7-AGNR. The Fermi level is set to zero energy. (b) The same as (a) for edge and inner C atoms of 6-ZGNR.

Figure 3

(a) and (c) DFT-PBE and Wannier-interpolated band structure of 7-AGNR and 6-ZGNR, respectively. (b) and (d) The $p_z$-like MLWF for edge C atoms of these two systems.

Figure 4

Calculated partially screened on-site interaction $U$ and fully screened Coulomb interaction $W$ parameters for carbon $p_z$ electrons for four GNRs with armchair edges (a) 6-AGNR, (b) 7-AGNR, (c) 8-AGNR, and (d) 9-AGNR. For comparison, $U$ and $W$ of pristine graphene are also included (dashed lines).

Figure 5

(a) DFT-GGA band gaps as a function of armchair GNR width. (b) Average partially screened on-site Coulomb interaction $U$ for $p_z$ electrons as a function of ribbon widths of GNRs with armchair edges. (c) The same as (b) for the fully screened Coulomb interaction $W$. Projected DOS for (d) C1 and (e) C8 in 7-AGNR.

Figure 6

(a) Partially screened Coulomb interaction $U$ for C $p_z$ electrons as a function of distance $r$ for an armchair GNR (7-AGNR). Here, $U(\parallel )$ and $U(\perp )$ correspond to interactions along the ribbon and across the ribbon, respectively. For comparison the bare interaction is presented. (b) The same as (a) for the fully screened Coulomb interaction $W$. (c) and (d) The same as (a) and (b) for zigzag GNR (6-ZGNR).

Figure 7

The difference $\left(V-W\right)$ between bare interaction $V$ and fully screened interaction $W$ as a function of distance $r$ between two C atoms along the ribbon for the edge and central atoms of (a) 6-AGNR, (b) 7-AGNR, (c) 8-AGNR, and (d) 9-AGNR.

Figure 8

The same as Fig. 5 for (a) 6-ZGNR, (b) 7-ZGNR, and (c) 8-ZGNR. The projected non-spin-polarized DOS for (d) C1 and (e) C6 in 6-ZGNR.

Figure 9

(a) Stoner criterion corresponding to two ZGNR systems, 6-ZGNR and 8-ZGNR. (b) Calculated magnetic moments (in units of ${\mu }_B)$ of different carbon atoms for these two systems (see Fig. 1).