Lattice density-functional theory on graphene

A density-functional approach on the hexagonal graphene lattice is developed using an exact numerical solution to the Hubbard model as the reference system. Both nearest-neighbor and up to third nearest-neighbor hoppings are considered and exchange-correlation potentials within the local density approximation are parameterized for both variants. The method is used to calculate the ground-state energy and density of graphene flakes and infinite graphene sheet. The results are found to agree with exact diagonalization for small systems, also if local impurities are present. In addition, correct ground-state spin is found in the case of large triangular and bowtie flakes out of the scope of exact diagonalization methods.

Figure 1

(a) The 12-atom supercell of the infinite graphene sheet, chosen for the determination of the exchange-correlation energy and potential. The dashed lines show the connections over the periodic boundaries. (b) A ${\text{C}}_{16}$ flake chosen as the first test system of our LDFT method.

Figure 2

(Color online) The XC energy and potential fitted to the infinite graphene sheet with nearest-neighbor hopping for $S_z=0$. Top row: 1NN hopping, bottom row: 3NN hopping (a/e) and (c/g): numerical XC energy and potential from ED (b/f) and (d/h): fitted analytic functions of form Eq. (9).

Figure 3

(Color online) A comparison between the ED, HF, and LDFT ground-state energies of the ${\text{C}}_{16}$ flake (Fig. 1). (a) At quarter filling. (b) With $S_z=\frac{1}{2}$ $\left(N_{\uparrow }=6,N_{\downarrow }=5\right)$. LDFT calculation performed both using a potential fitted for $S_z=0$ and $S_z=\frac{1}{2}$ reference systems. (c) At quarter-filling with an added on-site impurity potential $ϵ=-\left|t\right|$ on the two middle sites of the structure. (d) At half-filling. The insets show the relative errors (in percentage) with respect to the ED energy.

Figure 4

(Color online) The ground-state energy of an infinite graphene sheet (12-atom supercell) at quarter filling, perturbed with a single spin-dependent on-site energy, using ED (full line), HF (dashed line), and third nearest-neighbor LDFT (dashed-dotted line). In (a) the strength of the on-site energy ${ϵ}_0$ is varied at fixed $U$ and in (b) the on-site energy is fixed and $U$ varied.

Figure 5

(Color online) Comparison of PD and LDFT within the nearest-neighbor hopping approach for the ${\text{C}}_{22}$ flake having a spin-polarized ground state at half-filling. In (a) the comparison of the ground-state energy for different values of $S_z$. Solid line: LDFT and dotted line: PD. The inset shows the deviation in percentage from the PD $S_z=0$ energy. In (b) and (c), the ground-state density at $U=t$ from PD and LDFT, respectively. In (d) and (e), the density at $U=2.8t$. Red (light gray) and blue (dark) signify the up- and down-spin densities, respectively. Half-filling corresponds to the radius of the blue (dark) circles in the lower left corner in (b), (c) and (d) and to the black dotted circle in lower left corner in (e), drawn to facilitate comparison.

Figure 6

(Color online) The energy gap between the lowest energy state and the other $S_z$ states for the 286-atom triangular graphene flake at $U=t$ with different total numbers of electrons, $N_{\text{el}}=N_{\uparrow }+N_{\downarrow }$. As the gap jumps when crossing the diagonal, the gap is shown only up to the $S_z$ value with minimal energy. The $x$ axis shows the difference in $S_z$ from the minimal spin, $S_{z,min}=0$ for even $N_{\text{el}}$ and $S_{z,min}=1/2$ for uneven $N_{\text{el}}$.

Figure 7

(Color online) The ground-state density $\left(S_z=0\right)$ of a 78-atom bowtie flake at half-filling, previously studied by Wang et al. in Ref. 18. Up to third nearest-neighbor hopping is included and the value of $U$ is $t$. Red (light gray) and blue (dark) correspond to spin-up and spin-down densities, respectively.