Spectral butterfly, mixed Dirac-Schrödinger fermion behavior, and topological states in armchair uniaxial strained graphene

An exact mapping of the tight-binding Hamiltonian for a graphene nanoribbon under any armchair uniaxial strain into an effective one-dimensional system is presented. As an application, for a periodic modulation we have found a gap opening at the Fermi level and a complex fractal spectrum, akin to the Hofstadter butterfly resulting from the Harper model. The latter can be explained by the commensurability or incommensurability nature of the resulting effective potential. When compared with the zig-zag uniaxial periodic strain, the spectrum shows much bigger gaps, although in general the states have a more extended nature. For a special critical value of the strain amplitude and wavelength, a gap is open. At this critical point, the electrons behave as relativistic Dirac fermions in one direction, while, in the other direction, a nonrelativistic Schrödinger behavior is observed. Also, some topological states were observed which have the particularity of not being completely edge states since they present some amplitude in the bulk. However, these are edge states of the effective system due to a reduced dimensionality through decoupling. These states also present the fractal Chern beating observed recently in quasiperiodic systems.

Figure 1

Mapping of armchair strained graphene into coupled chains. The strain in the $y$ direction distorts the graphene hexagons, while the boundary of the unitary cell in the $x$ direction is shown by red dotted lines. Inside of the cell, four inequivalent atoms appear (shown with different colors inside the rectangles) denoted by $A_1^{\left(m\right)},A_2^{\left(m\right)},B_1^{\left(m\right)}$, and $B_2^{\left(m\right)}$. The effective Hamiltonian of the armchair path in the $y$ direction can be mapped into the coupled chains that appear to the right, where the label $j$ corresponds to each step of the ladder along the $y$ direction as indicated.

Figure 2

Spectrum as a function of $\sigma$ for $\lambda =1$ and $\varphi =\pi \sigma$ obtained by solving the Schrödinger equation for a system of 160 atoms, using 250 grid points for sampling $k_x$ and with fixed boundary conditions. The different colors represent the normalized localization participation ratio $\alpha \left(E\right)$. Inset: $\sigma =1/2$ near $E=0$.

Figure 3

Spectrum as a function of $\sigma$ for $\lambda =1$ and $\varphi =\pi \sigma$ obtained by solving the Schrödinger equation for a system of (a) 20 atoms and (b) 40 atoms, using 250 grid points for sampling $k_x$ and with fixed boundary conditions. The different colors represent the normalized localization participation ratio $\alpha \left(E\right)$.

Figure 4

Band structure (left column) and density of states (right column) using $\varphi =\pi \sigma$ and $\lambda =1$ for (a) an unstrained graphene lattice, (b) strained graphene with $\sigma =\sqrt{3}/4$, (c) strained graphene with $\sigma =\sqrt{3}\tau /2$, and (d) strained graphene with $\sigma =1/2$. Fixed boundary conditions were used in this plot.

Figure 5

Energy spectrum of graphene as a function of $\lambda$ for (a) $\sigma =\sqrt{3}/4$, (b) $\sigma =\sqrt{3}\tau /2$, and (c) $\sigma =1/2$. For $\lambda >1/2$ a gap at the Fermi level is opened. Fixed boundary conditions were used in this plot.

Figure 6

Different perspectives of the energy surface as a function of $k_x$ and $k_y$ for $\lambda =1/2$ and $\sigma =1/2$, using a linearized version of $t_j$. Notice how the electron has a mixed Schrödinger parabolic behavior with a Dirac linear fermion behavior at the Fermi level corresponding to $E=0$.

Figure 7

Evolution of the energy surface as a function of $k_x$ and $k_y$ for $\sigma =1/2$ near the critical point. (a) $\lambda =0.9{\lambda }_c$. (b) $\lambda ={\lambda }_c$. (c) $\lambda =1.1{\lambda }_c$. In case (a), two Dirac cones are seen, which are merged in (b), and in (c) the cones disappear. The arrows indicate the position of the Fermi level.

Figure 8

Upper panels show the energy spectrum with 160 sites with $\lambda ={\lambda }_C$ and $\sigma =1/2$. (a) Energy spectrum using cyclic boundary conditions. (b) Energy spectrum using fixed boundary conditions. The colors represent the normalized localization participation ratio $\alpha \left(E\right)$. Two $E=\pm 1$ energy modes are localized on either one of the edges and on the middle of the chain in $0\le \varphi <\pi /2$. For $\varphi =\pi /2$ the localized energy modes become extended. The lower panel displays the eigenstates for $E=-1$ energy modes using $\varphi =0$ and $\pi /2$. Notice how the wave function is modulated with an envelope of bigger wavelength, a phenomena called Chern beating [19].