Collapse of the vacuum in hexagonal graphene quantum dots: A comparative study between tight-binding and mean-field Hubbard models

In this paper, we perform a systematic study on the electronic, magnetic, and transport properties of the hexagonal graphene quantum dots (GQDs) with armchair edges in the presence of a charged impurity using two different configurations: (1) a central Coulomb potential and (2) a positively charged carbon vacancy. The tight-binding and the half-filled extended Hubbard models are numerically solved and compared with each other in order to reveal the effect of electron interactions and system sizes. Numerical results point out that off-site Coulomb repulsion leads to an increase in the critical coupling constant to ${\beta }_c=0.6$ for a central Coulomb potential. This critical value of $\beta$ is found to be independent of the GQD size, reflecting its universality even in the presence of electron-electron interactions. In addition, a sudden downshift in the transmission peaks shows a clear signature of the transition from subcritical $\beta <{\beta }_c$ to the supercritical $\beta >{\beta }_c$ regime. On the other hand, for a positively charged vacancy, collapse of the lowest bound state occurs at ${\beta }_c=0.7$ for the interacting case. Interestingly, the local magnetic moment, induced by a bare carbon vacancy, is totally quenched when the vacancy is subcritically charged, whereas the valley splittings in electron and hole channels continue to exist in both regimes.

Figure 1

Energy values of the lowest bound states as a function of the coupling constant $\beta$. (a) The critical coupling constant ${\beta }_c$ is 0.6 within the MFH model for all samples that differ in size. The inset contains a sketch of the problem for the hexagonal GQD that consists of 114 atoms. Here, sublattices $A$ and $B$ are red and blue filled circles, and a positively charged impurity is at the center. Green triangles show how the leads are connected to samples throughout our paper to determine the transmission coefficients. (b) shows a zoomed view of the energy eigenvalues. (c) contains a comparison between the TB and the MFH models for a GQD consisting of 5514 carbon atoms.

Figure 2

The transmission coefficients in (a)–(c) for the number of 546, 1626, and 10 806 atoms, respectively. The behavior of transmission coefficients obviously corresponds to two different regimes. The inset in (c): The critical coupling constant ${\beta }_c$ is at the point of intersection of two lines on a linear scale.

Figure 3

(a) Total densities of states are shown for a pristine hexagonal GQD consisting of 5514 atoms, (b) total TB DOS belongs to the same GQD that contains a bare vacancy placed near the center, and (c) contains the spin and valley splittings for both spin components.

Figure 4

(a) clearly shows the spin and valley splittings as a function the size of the hexagonal GQDs. Additional symmetry in Eq. (8) can be followed by the overlapped lines. (b) The spin splitting disappears as a function of $\beta$, whereas the valley splittings do not completely vanish.

Figure 5

The energy spectrum of the TB model as a function of $\beta$ is shown in (a). The positions of the leads and the bare carbon vacancy are sketched in (b). Scaled electronic densities per lattice of the vacancy state, i.e., LDOS, for $\beta =0$, 0.1, 0.2, and 0.3 can be seen in (c), from top to bottom. The quasilocalization of the lowest bound state is demonstrated in (d) for $\beta =0$, 0.3, 0.6, 0.8, and 1.0 from top to bottom.

Figure 6

The energy spectra of the spin up and spin down are shown in (a). ${\beta }_c$ equals 0.7 for a Coulomb charged vacancy. Scaled electronic densities for the vacancy states can be seen in (b) and (c) for $\beta =0$, 0.2, and 0.4 from top to bottom. In (d) and (e), the behavior of the critical states for the $\beta$ values of 0, 0.7, and 1.2 can be seen starting from the top.

Figure 7

Transmission coefficients of the critical states of the TB in (a), spin-up in (c), and spin-down in (e) spectra, whereas the vacancy states are given in (b), (d), and (f).

Figure 8

The quenching of staggered magnetization ${\mu }_s^z$ is given as a function of $\beta$. Up to $\beta =0.4$, a large portion of ${\mu }_s^z$ disappears.