Graphene flakes with defective edge terminations: Universal and topological aspects, and one-dimensional quantum behavior

Systematic tight-binding investigations of the electronic spectra (as a function of the magnetic field) are presented for trigonal graphene nanoflakes with reconstructed zigzag edges, where a succession of pentagons and heptagons, that is 5-7 defects, replaces the hexagons at the zigzag edge. For nanoflakes with such reczag defective edges, emphasis is placed on topological aspects and connections underlying the patterns dominating the spectra of these systems. The electronic spectra of trigonal graphene nanoflakes with reczag edge terminations exhibit certain unique features, in addition to those that are well known to appear for graphene dots with zigzag edge termination. These unique features include breaking of the particle-hole symmetry, and they are associated with nonlinear dispersion of the energy as a function of momentum, which may be interpreted as nonrelativistic behavior. The general topological features shared with the zigzag flakes include the appearance of energy gaps at zero and low magnetic fields due to finite size, the formation of relativistic Landau levels at high magnetic fields, and the presence between the Landau levels of edge states (the so-called Halperin states) associated with the integer quantum Hall effect. Topological regimes, unique to the reczag nanoflakes, appear within a stripe of negative energies ${\varepsilon }_b<\varepsilon <0$, and along a separate feature forming a constant-energy line outside this stripe. The ${\varepsilon }_b$ lower bound specifying the energy stripe is independent of size. Prominent among the patterns within the ${\varepsilon }_b<\varepsilon <0$ energy stripe is the formation of three-member braid bands, similar to those present in the spectra of narrow graphene nanorings; they are associated with Aharonov-Bohm–type oscillations, i.e., the reczag edges along the three sides of the triangle behave like a nanoring (with the corners acting as scatterers) enclosing the magnetic flux through the entire area of the graphene flake. Another prominent feature within the ${\varepsilon }_b<\varepsilon <0$ energy stripe is a subregion of Halperin-type edge states of enhanced density immediately below the zero-Landau level. Furthermore, there are features resulting from localization of the Dirac quasiparticles at the corners of the polygonal flake. A main finding concerns the limited applicability of the continuous Dirac-Weyl equation, since the latter does not reproduce the special reczag features. Due to this discrepancy between the tight-binding and continuum descriptions, one is led to the conclusion that the linearized Dirac-Weyl equation fails to capture essential nonlinear physics resulting from the introduction of a multiple topological defect in the honeycomb graphene lattice.

Figure 1

Distribution of the hopping matrix elements $t_k$ (see Table ) for the reczag edge.

Figure 2

Diagrams of corners used for the equilateral trigonal graphene flakes. (a) Corner for zigzag edges. (b) Type-I corner for reczag edges. (c) Type-II corner for reczag edges.

Figure 3

(a) TB single-particle spectrum for a zigzag trigonal graphene dot as a function of the magnetic field (the magnetic flux $\Phi$ over the whole dot). (b) Shape of the corresponding equilateral trigonal graphene dot with zigzag edges; it has 61 hexagons in the outer row along each side (the total number of carbon atoms is 3966). (c) TB single-particle spectrum for a type-I reczag trigonal graphene dot as a function of the magnetic field (the magnetic flux $\Phi$ over the whole dot). (d) Shape of the corresponding trigonal graphene dot with reczag edges (type-I corner); it has 60 hexagons in the outer unreconstructed row along each side (the total number of carbon atoms is 4731). Energy in units of the tight-binding hopping parameter $t=2.7$ eV. Lengths in units of the honeycomb graphene lattice constant $a=0.246$ nm. The magnetic flux is given in units of ${\Phi }_0=hc/e$.

Figure 4

(a) Enlarged section of the regime marked as D in Fig. 3c, showing the TB single-particle spectrum for a reczag trigonal graphene dot (with type-I corners), as a function of the magnetic field (the magnetic flux $\Phi$ over the whole dot). (b) The TB spectrum for the corresponding reczag trigonal GQD with type-II corners. Energy in units of the tight-binding hopping parameter $t$. The magnetic flux is given in units of ${\Phi }_0=hc/e$.

Figure 5

Enlarged section of the regime marked as D1 in Fig. 4a showing the TB single-particle spectrum for a reczag trigonal graphene dot (with type-I corners), as a function of the magnetic field (the magnetic flux $\Phi$ over the whole dot). The horizontal arrows highlight the alternation 2-1-1-2 (1-2-2-1) in the state degeneracy between two successive braid bands at $\Phi /{\Phi }_0=n$ ($\Phi /{\Phi }_0=n+1/2$), $n=0,1,2,....$ (b) An example (for reasons of comparison) of a TB single-particle spectrum for a narrow trigonal graphene ring with zigzag edge terminations. Such nanorings were used in Ref. 19 to study the Aharonov-Bohm oscillatory patterns in graphene nanosystems.

Figure 6

TB electron densities (modulus) of reczag edge states participating (counting from the bottom) in the first (a), second (b), and fourth (c) braid bands of region D1 [see Fig. 5a], at $\Phi =15.9{\Phi }_0$. The A (red) and B (blue) sublattices are plotted separately. Green color denotes the density on the outer carbon dimers resulting from the edge reconstruction and connected by the hopping matrix element $t_4$ in Fig. 1. (Note that the color codings between Fig. 1 and the current Fig. 6 are unrelated.) The presence of azimuthal (along the sides of the triangle) nodes in the electronic densities is clearly visible. The number of nodes changes by unity from one braid band to the next, increasing with increasing energy. This behavior (including the fact that all three states within each braid band maintain the same number of nodes) is quite analogous to that of the edge states of a trigonal graphene nanoring at low magnetic fields (see Fig. 7 below). Energies in units of $t=2.7$ eV.

Figure 7

TB electron densities (modulus) of edge states of the zigzag ring participating (counting from the bottom) in (a) the first and (b) the third braid bands in Fig. 5b], at $\Phi /{\Phi }_0\sim 6$. The A (red) and B (blue) sublattices are plotted separately. The azimuthal nodes in the electronic densities are clearly visible. The number of nodes changes by one from one braid band to the next, increasing with increasing energy. Energies in units of $t=2.7$ eV. The ring has a total of 906 carbon atoms.

Figure 8

(a) Schematic diagram of the unit subcell associated with a single side of the trigonal graphene flake with reczag edge terminations. $V_2$ denotes the height of the potential barrier which mimicks the scatterer at the corners of the trigonal reczag flake. (b) Schematic diagram of the unit cell associated with the magnetic-field virtual lattice; it involves all three sides of the equilateral triangle, and thus it consists of three unit subcells in a series. (c), (d) The function $f\left(E\right)=\mathrm{Tr}\left[\mathbf{T}\right(E\left)\right]/2$ [see the right-hand side of Eq. (7)], which is associated with the unit cell [shown in (b)] of the magnetic-field virtual superlattice, as a function of the energy variable $E$. In calculating $f\left(E\right)$, the parameters for the unit subcell [shown in (a)] were taken as: $L_1=12.5$ nm, $L_2=1.5$ nm, $V_1=0$, and $V_2=0.1$ eV. The relevant values of $f\left(E\right)$ lie within $\pm 1$ (i.e., within the dotted lines). (c) $f\left(E\right)$ in the range $0\le E\le V_2$. (d) $f\left(E\right)$ in the range $V_2\le E\le 4V_2$. Energy in units of eV.

Figure 9

Single-particle spectrum (as a function of the total flux $\Phi$) from the semianalytic superlattice model considered in Sec. 3b2. The parameters for the unit subcell mimicking each reczag side of the trigonal graphene flake are $L_1=12.5$ nm, $L_2=1.5$ nm, $V_1=0$, and $V_2=0.1$ eV. Note that the total length $L_1+L_2=14$ nm is similar to the length of the trigonal flake in Fig. 3. The height of the potential barrier defining the scatterers at the corners $\left(V_2\right)$ is roughly one-fifth of the width ($0.2t$) of the D region [see Fig. 3c]. Energies in units of eV. The magnetic flux is given in units of ${\Phi }_0=hc/e$.

Figure 10

TB electron densities (modulus) of two characteristic states for the D2 regime of the reczag flake at $\Phi /{\Phi }_0=15.9$. (a) A state with near-zero energy $\varepsilon =-0.1272\times {10}^{-4}t$ exhibiting bulk zero-Landau-level behavior. (b) A state with lower energy $\varepsilon =-0.2207\times {10}^{-1}t$ exhibiting Halperin-edge behavior. Compared to Fig. 6, the absence of azimuthal nodes in the electronic densities here is noticeable. Energies in units of $t=2.7$ eV. The total number of carbon atoms is 4731. The A (red) and B (blue) sublattices are plotted separately. Green color denotes the density on the outer carbon dimers resulting from the edge reconstruction and connected by the hopping matrix element $t_4$ in Fig. 1.

Figure 11

TB electron densities (modulus) of two characteristic states for the E1 and E2 regimes of the reczag flake at $\Phi /{\Phi }_0=15.9$. Note the concentration of the electron densities at the corners of the triangle. (a) A state in the E2 regime with energy $\varepsilon =-0.6826\times {10}^{-1}t$. (b) A state in the E1 regime with energy $\varepsilon =-0.297t$. Energies in units of $t=2.7$ eV. The A (red) and B (blue) sublattices are plotted separately. Green color denotes the density on the outer carbon dimers resulting from the edge reconstruction and connected by the hopping matrix element $t_4$ in Fig. 1.

Figure 12

(a) TB single-particle spectrum for a very small (with type-I corners) reczag trigonal graphene dot, as a function of the magnetic field (the magnetic flux $\Phi$ over the whole dot). (b) Shape of the equilateral trigonal graphene dot which corresponds to (a); it has 10 hexagons in the outer unreconstructed row along each side (the total number of carbon atoms is 195). (c) TB single-particle spectrum for a larger (with type-I corners) reczag trigonal graphene dot, as a function of the magnetic field (the magnetic flux $\Phi$ over the whole dot). (d) Shape of the trigonal graphene dot which corresponds to (c); it has 38 hexagons in the outer unreconstructed row along each side (the total number of carbon atoms is 1819). The thick black line in (c) denotes the Fermi level in the canonical ensemble corresponding to $N=60$ holes (spin included). Energy in units of the tight-binding hopping parameter $t=2.7$ eV. Lengths in units of the honeycomb graphene lattice constant $a=0.246$ nm. The magnetic flux is given in units of ${\Phi }_0=hc/e$.

Figure 13

(a) Reczag flake: an enlarged part of the TB single-particle spectrum shown in Fig. 12c, as a function of the magnetic field (the magnetic flux $\Phi$ over the whole dot). The shape of the corresponding reczag flake is displayed in Fig. 12d. (b) Reczag flake: Landau magnetization (at zero temperature) for $N=60$ holes (spin included) exhibiting Aharonov-Bohm oscillations superimposed on larger ones generated by the rapid variation of the background Halperin-type edge states which cross the braid bands. The thick black line in (a) denotes the corresponding Fermi level in the canonical ensemble. (c) Zigzag flake: a similar part of the TB single-particle spectrum for the corresponding zigzag trigonal flake (with 38 hexagons in the outer row along each side), as a function of the magnetic flux $\Phi$. (d) Zigzag flake: Landau magnetization for $N^{*}=4$ effective holes (spin included); the absence of Aharonov-Bohm oscillations is apparent. The thick black line in (c) denotes the corresponding Fermi level in the canonical ensemble. For a meaningful comparison, this Fermi level was chosen to fall within the ${\varepsilon }_b=-0.205t<\varepsilon <0$ energy band. Note that for the reczag flake this energy band contains the special D region; for the zigzag flake this energy band is reduced to being part of region C1. The total number of holes is $N=N_0+N^{*}$, where $N_0$ is the number of strictly zero-energy states present in the zigzag trigonal flake (also, see text). Energy in units of the tight-binding hopping parameter $t=2.7$ eV. The magnetic flux is given in units of ${\Phi }_0=hc/e$.

Figure 14

Single-particle spectra as a function of the magnetic flux $\Phi$, according to the continuous Dirac-Weyl description for a circular GQD with a reczag edge termination. (a) The $K$ valley. (b) The $K^{\prime }$ valley. The radius of the dot is $R=8$ nm. Energies in units of $t=2.7$ eV.

Figure 15

Single-particle energies as a function of the angular momentum $m$, according to the continuous Dirac-Weyl description for a circular GQD with a reczag edge termination. The total flux is fixed at $\Phi =15{\Phi }_0$. Both the $K$ and $K^{\prime }$ valleys are considered. Note the four dispersive branches of edge states [(a), (b), (c), and (d)] associated with the zeroth Landau level. We note that only two dispersive branches of edge states [(a) and (c)], associated with the zeroth Landau level, appear in a circular GQD with a zigzag edge termination. The radius of the dot is $R=8$ nm. Energies in units of $t=2.7$ eV.