Robust edge states induced by electron-phonon interaction in graphene nanoribbons

The search of new means of generating and controlling topological states of matter is at the front of many joint efforts, including band-gap engineering by doping and light-induced topological states. Most of our understading, however, is based on a single particle picture. Topological states in systems including interaction effects, such as electron-electron and electron-phonon, remain less explored. By exploiting a nonperturbative and nonadiabatic picture, here we show how the interaction between electrons and a coherent phonon mode can lead to a band gap hosting edge states of topological origin. Further numerical simulations witness the robustness of these states against different types of disorder. Our results contribute to the search of topological states, in this case in a minimal Fock space.

Figure 1

Armchair GNR with phonon mode $E_{2g}$. (a) ac-GNR of $N_y=4$ carbon atoms wide $\left(L_y=8.61\text{Å}\right)$. The shaded rectangle encloses a “transversal layer,” i.e., a single line of $N_y$ carbon atoms. The longitudinal optical phonon is depicted by red and blue arrows. (b) Eigenmode decomposition of the lattice. In the noninteracting case, each eigenmode consists of a dimer chain with intracell hoppings ${\gamma }_{q,\mathrm{a}}^{\left(0\right)}$ and intercell hoppings ${\gamma }_{q,\mathrm{b}}^{\left(0\right)}$. (c) Interacting case: Each dimer chain (eigenmode $q)$, now splits into infinite replicas with different number of phonon excitations.

Figure 2

Band-gap estimation according to Eq. (8) as a function of the eigenmode number in the limit $N_y\to \infty$, for several values of the phonon frequency, and ${\gamma }_x=0.1{\gamma }_0$. The $n=0$ and $n=1$ phonon bands are shown as shaded regions in black and red, respectively. The dashed blue lines delimit the gapped regions.

Figure 3

LDoS plots for an ac-GNR of $N_y=11$ carbon atoms wide $\left(L_y=25.82\text{Å}\right)$ and coupled to a longitudinal optical mode. (a) LDoS in log scale weighted over the zero-phonon replica (solid red, shaded area), evaluated at the border of a semi-infinte ac-GNR. (b) Zero-phonon LDoS in bulk situation, i.e., evaluated at the center of an infinite ac-GNR. In (a) and (b), the black lines show the noninteracting case $\left({\gamma }_x=0\right)$. Panels (c) and (d) show the same densities (in grayscale) as in panels (a) and (b), respectively, but resolved in eigenmodes. The vibration parameters are $\hslash \omega =5{\gamma }_0$ and ${\gamma }_x=0.1{\gamma }_0$.

Figure 4

(a) LDoS evaluated at $\varepsilon =\hslash \omega /2$ as a function of the transversal layer $j$ for a semi-infinite ac-GNR with $N_y=11$ carbon atoms wide and normalized to its maximum value at $j=3$. The used phonon energy is $\hslash \omega =5{\gamma }_0$ and the e-ph interaction is ${\gamma }_x=0.1{\gamma }_0$. Inset: Same LDoS as in (a), for eigenmodes $q=1$, 3, and 5 and normalized to their respective maximum values. (b) Eigenenergy spectrum for a finite ac-GNR of dimensions $N_x=200\left(L_x=211.58\text{Å}\right)$ and $N_y=11\left(L_y=25.82\text{Å}\right)$. The other parameters coincide with those of panel (a). We use a color scale to indicate the eigenstates weight with respect to the zero-phonon replica: Full weight $\left(p_{q,k}=1\right)$ is in black, while zero weight $\left(p_{q,k}=0\right)$ is in red. (c) Probability density $|{\psi }_j{|}^2$ (normalized to its maximum value and weighted over the $n=0$ replica) as a function of the transversal layer $j$ for one of the two eigenenergies at $\varepsilon =\hslash \omega /2$ and eigenmode $q=4$ in panel (b).

Figure 5

Zero-phonon LDoS evaluated around the edge of different ribbon geometries, including vacancy disorder of $0.5\%$. The evaluation region consists of ten transversal layers of carbon atoms starting from the left edge of the ribbon. Black lines show the case without vacancies for comparison reasons, blue lines (shaded area) show a single disorder realization, and red lines show an ensamble average over 200 disorder realizations. The used geometries are (see schemes): (a) armchair-zigzag $\left(N_y=11\to L_y=25.82\text{Å}\right)$, (b) armchair-Klein $\left(N_y=11\to L_y=25.82\text{Å}\right)$, (c) armchair-armchair $\left(N_y=12\to L_y=20.91\text{Å}\right)$, and (d) zigzag-armchair $\left(N_y=12\to L_y=22.01\text{Å}\right)$. The insets in each panel are zoom areas around $\varepsilon =0$ (bottom) and $\varepsilon =\hslash \omega /2$ (top). We used two phonon replicas and the other parameters are $\hslash \omega =5{\gamma }_0,{\gamma }_x=0.1{\gamma }_0$.

Figure 6

Impurity disorder effects in the zero-phonon LDoS for three ribbon geometries: (a) armchair-armchair $\left(N_y=12\to L_y=20.91\text{Å}\right)$, (b) armchair-zigzag $\left(N_y=11\to L_y=25.82\text{Å}\right)$, and (c) zigzag-armchair $\left(N_y=12\to L_y=22.01\text{Å}\right)$. In all panels, we evaluated the LDoS along 40 transversal layers of carbon atoms from the left border and other 40 lines from the right border. Solid lines show the average LDoS over 250 disorder realizations, where we used $W=0.01{\gamma }_0$ (red), $0.02{\gamma }_0$ (blue), and $0.03{\gamma }_0$ (green). Black lines correspond to the case without disorder, while orange dotted lines (shaded) show a single realization for $W=0.03{\gamma }_0$. Main panels sweep the e-ph induced band-gap region, centered at $\varepsilon =\hslash \omega /2$, while the insets show the Dirac point region, centered at $\varepsilon =0$. The other parameters coincide with those of Fig. 5.