Zak phase and the existence of edge states in graphene

We develop a method to predict the existence of edge states in graphene ribbons for a large class of boundaries. This approach is based on the bulk-edge correspondence between the quantized value of the Zak phase $\mathcal{Z}\left(k_{\parallel }\right)$, which is a Berry phase across an appropriately chosen one-dimensional Brillouin zone, and the existence of a localized state of momentum $k_{\parallel }$ at the boundary of the ribbon. This bulk-edge correspondence is rigorously demonstrated for a one-dimensional toy model as well as for graphene ribbons with zigzag edges. The range of $k_{\parallel }$ for which edge states exist in a graphene ribbon is then calculated for arbitrary orientations of the edges. Finally, we show that the introduction of an anisotropy leads to a topological transition in terms of the Zak phase, which modifies the localization properties at the edges. Our approach gives a new geometrical understanding of edge states, and it confirms and generalizes the results of several previous works.

Figure 1

Examples of edges of different ribbons studied in this article. They are characterized by a translation vector $\mathbf{T}(m,n)$; see Sec. 3b. The values of the couples $(m,n)$ are specified in the figure. The edges of the ribbon $(1,-2)$ show dangling bounds since $mn<0$. Different ribbons can be obtained from the same vector $\mathbf{T}(m,n)$ in two different ways. For instance, the lattice vectors $\mathbf{C}$ connecting the left and right edges of the two ribbons defined by $\mathbf{T}(3,3)$ differ. Another possibility is to draw a different “unit-cell” pattern for the same vector $\mathbf{T}$, as for the two ribbons defined by $\mathbf{T}(2,5)$.

Figure 2

Chain of dimers $A$-$B$. The chain starts with an atom $A$ and ends with an atom $B$. $t$ and $t^{\prime }$ are the hopping parameters and $a_0$ is the lattice spacing. The unit cell $m$ is represented by a rectangle.

Figure 3

Two trajectories of the vector $\mathbf{g}\left(k\right)$ with different topologies when $k$ runs across the Brillouin zone. When $t^{\prime }/t<1$ ($t^{\prime }/t>1$) the curve $\mathbf{g}\left(k\right)$ does (not) wind around the origin.

Figure 4

Variation $\phi \left(k\right)$ for $t^{\prime }/t=1.1$ (bottom curve), $t^{\prime }/t=0.8$ (upper thick curve), and $t^{\prime }/t=0.95$ (thick dashed curve). The straight lines are lines of equation $f_{\kappa }\left(k\right)=(M+1)k{a}_0-\kappa \pi$. Here, we have chosen $M=10$, for which ${(t^{\prime }/t)}_c=0.9091$. The extreme values of $\kappa$ are indicated on the figure. In the latter case $t^{\prime }/t=0.95>{(t^{\prime }/t)}_c$, so there are $M$ bulk states although the Zak phase $\mathcal{Z}=\pi$.

Figure 5

Inverse localization length $\lambda$ as a function of the parameter $t^{\prime }/t$, for a chain of $M=10$ dimers. The full curve is the solution of Eq. (22). As expected, it diverges as the ratio $t/t^{\prime }$ reaches the critical value given by Eq. (15). The dashed line corresponds to the approximation (25) valid far from the transition and corresponding to $\epsilon =0$.

Figure 6

Energy levels as a function of the parameter $t^{\prime }/t$, for a chain of $M=10$ dimers. There is an edge state with an energy close to 0 when $t^{\prime }/t<1-1/(1+M)$. The dashed lines correspond to the gap $\pm (t^{\prime }-t)$ in the limit $M\to \infty$. The thick (light-purple) curve is the energy of the lowest energy state, which becomes an edge state(light-green curve) when $t^{\prime }/t<{(t^{\prime }/t)}_c$.

Figure 7

Chains of dimers coupled in the $y$ direction by a hopping parameter $t^{\prime \prime }$.

Figure 8

Energy levels of a ribbon built from chains of $M=20$ dimers. The chains are coupled to one other in the $y$ direction with a hopping parameter $t^{\prime \prime }=0.1$, and the ribbon is invariant by translation along this direction. We took $t=1$, (left) $t^{\prime }=1,5$, and (right) $t^{\prime }=0,5$. In the first case the Zak phase is $\mathcal{Z}=0$ and there is no edge state. In the other case, the Zak phase is $\mathcal{Z}=\pi$ and dispersive edge states have emerged in the gap.

Figure 9

(Left) Unit cell (dimer) $A$-$B$ of the graphene sheet with the basis vectors of the Bravais lattice ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ and the hopping parameters $t_1$, $t_2$, and $t_3$. (Right) Example of an edge obtained by translating the dimer along ${\mathbf{a}}_1$ and then twice along ${\mathbf{a}}_2$. This edge is characterized by the periodicity vector $\mathbf{T}={\mathbf{a}}_1+2{\mathbf{a}}_2$.

Figure 10

Schematic representation of a zigzag ribbon. The zigzag edge is obtained by translating the dimer $A$-$B$ (represented by a rectangle) with the periodicity $\mathbf{T}={\mathbf{a}}_1$. The two edges ${\mathit{\beta }}_1$ and ${\mathit{\beta }}_2$ are represented by thick lines. The first vacant sites outside of the ribbon where we impose the wave function to vanish are represented by circles. $\mathbf{C}$ is the vector of the Bravais lattice that connects two sites on both edges, and ${\mathbf{C}}^{\prime }$ connects the first vacant site on one side of the ribbon to the first vacant site on the other side.

Figure 11

Density plot of the phase $\tilde{\phi }\left(\mathbf{k}\right)$. The discontinuities of the phase are shown by horizontal white lines connecting pairs of Dirac points. These singularities separate the values $\tilde{\phi }=+\pi$ (light region) to the value $\tilde{\phi }=-\pi$ (dark region). Consequently, the only paths ${\Gamma }_{\perp }$ that contribute to a nonvanishing Zak phase are those that cross these singularities. The black lines, thick lines, and dashed lines represent the iso-$\tilde{\phi }$ lines, respectively, for $\tilde{\phi }=0$, $\tilde{\phi }=-\pi /2$, and $+\pi /2$.

Figure 12

On top of the density plot of the phase $\tilde{\phi }\left(\mathbf{k}\right)$, we have represented several values of the vector ${\mathit{\Gamma }}_{\perp }(m,n)=n{\mathbf{a}}_1^{*}-m{\mathbf{a}}_2^{*}$. The left extremity of a discontinuity of $\tilde{\phi }\left(\mathbf{k}\right)$ is taken as the origin. We have explicitly plotted the vector ${\mathit{\Gamma }}_{\perp }(1,0)$ as an example.

Figure 13

Surfaces $|{\mathit{\Gamma }}_{\parallel }\wedge {\mathit{\Gamma }}_{\perp }|$ associated with different ribbon vectors $\mathbf{T}(m,n)$. ${\mathit{\Gamma }}_{\perp }$ is obtained as ${\mathit{\Gamma }}_{\perp }(m,n)=n{\mathbf{a}}_1^{*}-m{\mathbf{a}}_2^{*}=(n,-m)$; see text for the construction of ${\mathit{\Gamma }}_{\parallel }$. (a) ${\mathit{\Gamma }}_{\perp }(m=1,n=2)=(2,-1)$, (b) ${\mathit{\Gamma }}_{\perp }(-2,3)=(3,2)$, and (c) ${\mathit{\Gamma }}_{\perp }(0,1)=(1,0)$. In these three cases, $m$ and $n$ are coprime, and the surfaces $|{\mathit{\Gamma }}_{\parallel }\wedge {\mathit{\Gamma }}_{\perp }|$ represented by a shaded rectangles are Brillouin zones. (d) For $\mathbf{T}(0,2)$ the surface obtained by ${\mathit{\Gamma }}_{\parallel }$ and ${\mathit{\Gamma }}_{\perp }$ is not a Brillouin zone because $m$ and $n$ are not coprime. The corresponding Brillouin zone (shaded rectangle) is given by ${\mathit{\Gamma }}_{\perp }(0,1)={\mathit{\Gamma }}_{\perp }(0,2)/2$ and ${\mathit{\Gamma }}_{\parallel }(1,0)=2{\mathit{\Gamma }}_{\parallel }(2,0)$.

Figure 14

Projection of the discontinuity of $\tilde{\phi }\left(\mathbf{k}\right)$ onto the $k_{\parallel }$ axis. This projection gives the range $\Delta k_{\parallel }$ for the existence of edge states.

Figure 15

Schematic band structures of graphene ribbons that exhibit edge states. Edge states are represented as zero-energy flat bands as expected in the large width limit within the tight-binding model (assuming the chiral symmetry is preserved). The number of pairs of edge states is indicated on the three figures (a), (b) and (c). In Fig. (a), the parameter $\mathcal{R}<1$ and in Fig. (b) $\mathcal{R}>1$. Our analysis based on the Zak phase predicts that the configuration (c) is not allowed.

Figure 16

Density plot of the phase $\tilde{\phi }\left(\mathbf{k}\right)$ for $t_2=1.5t_1=1.5t_3$. The discontinuities of the phase are shown by curved white lines that pair the Dirac points. The black lines, thick lines, and dashed lines represent the iso-$\tilde{\phi }$ lines, respectively, for $\tilde{\phi }=0$, $\tilde{\phi }=-\pi /2$, and $+\pi /2$.

Figure 17

Projection of the discontinuity line of $\tilde{\phi }\left(\mathbf{k}\right)$ onto the $k_{\parallel }$ axis. The asymmetry of the hopping induces a modification of the discontinuity locations as compared with Fig. 14.

Figure 18

(Left) Density plot of the phase $\tilde{\phi }\left(\mathbf{k}\right)$ for $t_1=1$, $t_2=1.5$, and $t_3=1$ represented in the Brillouin zone corresponding to armchair edges. The two horizontal lines delimit the region where $\mathcal{Z}\left(k_{\parallel }\right)=\pi$. (Right) Band structure of an armchair ribbon with the same hopping parameters. In this case, $k_{\parallel }=k_y$. The two vertical lines delimit the same range as in the left panel, which is such that edge states at zero energy, clearly separated from the bulk bands, have emerged.

Figure 19

Density plot of the phases $\tilde{\phi }\left(\mathbf{k}\right)$ at the merging of the Dirac points, and the band structure for a zigzag ribbon with $\theta =\pi /6$ [that is, $\mathbf{T}(m=0,n=1)\to {\mathit{\Gamma }}_{\perp }(m=0,n=1)=(1,0)$]. (Top) $t_1=1$, $t_2=1$, and $t_3=2$; (center) $t_1=2$, $t_2=1$, and $t_3=1$; (bottom) $t_1=1$, $t_2=2$, and $t_3=1$. The scale is given in units of $1/a_0$.