Antiferromagnetism in hexagonal graphene structures: Rings versus dots

The mean-field Hubbard model is used to investigate the formation of the antiferromagnetic phase in hexagonal graphene rings with inner zigzag edges. The outer edge of the ring was taken to be either zigzag or armchair, and we found that both types of structures can have a larger antiferromagnetic interaction as compared with hexagonal dots. This difference could be partially ascribed to the larger number of zigzag edges per unit area in rings than in dots. Furthermore, edge states localized on the inner ring edge are found to hybridize differently than the edge states of dots, which results in important differences in the magnetism of graphene rings and dots. The largest staggered magnetization is found when the outer edge has a zigzag shape. However, narrow rings with armchair outer edge are found to have larger staggered magnetization than zigzag hexagons. The edge defects are shown to have the least effect on magnetization when the outer ring edge is armchair shaped.

Figure 1

(a) The dot (red color) with four atoms at the zigzag edge removed from the larger dot (black color) which has seven dimers at the dot edge. (b) The formed ring is labeled by $7_{AC}:4_{ZG}$. (c) The distribution of magnetic moments in the $9_{AC}:N_{ZG}$ ring shown in a sextant of the ring for $N$ taking the values 3, 4, 5, 9, 10, and 11. The majority spin is labeled by both orientation and color of a triangle centered at an atomic site. The local magnetic moment value is proportional to the color intensity.

Figure 2

Density of states in the $9_{AC}:N_{ZG}$ rings for $N$ taking the values 3, 4, 5, 9, 10, and 11 in the noninteracting system (black lines) and the interacting system (purple lines).

Figure 3

Distribution of magnetic moments in the ${13}_{ZG}:N_{ZG}$ rings for $N$ ranging from 6 to 11.

Figure 4

(a) Staggered magnetization ${\mu }_s^z$, (b) maximum moment $m_{\max }$, and (c) energy shift $\Delta E$ as they vary with the length of the side of the inner ring edge.

Figure 5

(a) Contour plot of WLDOS at several stages of the carving process forming the $7_{AC}:7_{ZG}$ ring and the $7_{ZG}$ hexagonal dot; the number in the upper-left corner indicates the number of bonds cut. (b) Summed WLDOS in the ring, the dot, and in the whole structure as well as the zero-energy density of states versus the number of bonds cut. (c) Stacked plot of the density of states; red depicts the densities of the stages displayed in panel (a).

Figure 6

(a) Perfect edges (black region) are randomly perturbed (red lines) to produce a random set of defects. (b) Final outlook of the deformed ring. (c) Magnetic moments distributions in ${25}_{ZG}:{20}_{ZG}$ ring for several values of $f_{\text{def}}$. The scale is the same as in Fig. 3. (d) Staggered magnetization of an ensemble of randomly defected structures as a function of defect ratio for ${25}_{ZG}:{20}_{ZG}$ ring (black dots), ${16}_{AC}:{20}_{ZG}$ ring (red dots) ring, and ${20}_{ZG}$ hexagonal dot (blue dots). The polynomial fitting curves are added to guide the eye.