Copropagating Edge States Produced by the Interaction between Electrons and Chiral Phonons in Two-Dimensional Materials

Unlike the chirality of electrons, the intrinsic chirality of phonons has only surfaced in recent years. Here, we report on the effects of the interaction between electrons and chiral phonons in two-dimensional materials by using a nonperturbative solution. We show that chiral phonons introduce inelastic Umklapp processes resulting in copropagating edge states that coexist with a continuum. Transport simulations further reveal the robustness of the edge states. Our results hint on the possibility of having a metal embedded with hybrid electron-phonon states of matter.

Figure 1

Scheme depicting the chiral phonon mode considered during this work. The left panel shows the phonon band structure of a honeycomb lattice, in arbitrary units, with chiral phonons at high symmetry points. Breaking inversion symmetry lifts the degeneracy between the two middle bands at the valleys, and generates circular motion of the sublattices with different amplitudes, allowing for phonon modes with net chirality [20]. We focus on the mode represented in the right panel, where sublattice A (B), depicted by blue (red) circles, exhibits counterclockwise (clockwise) circular motion, as indicated by the corresponding arrows. Because of the nonzero phonon momentum, the movement acquires a phase $e^{\pm i2\pi /3}$ between neighboring unit cells, represented by the arrows.

Figure 2

(a) Considering the phonon Fock space, the system can be viewed as a semi-infinite series of pure electronic Hamiltonian replicas centered at energies $n_{\mathrm{ph}}\hslash \omega$, with $n_{\mathrm{ph}}\in {\mathbb{N}}_0$ the corresponding phonon population. $e$-ph interactions generate nonvertical transitions in the BZ, between momentum $\mathit{k}$ in replica $n_{\mathrm{ph}}=0$ to momentum $\mathit{k}-\mathit{G}$ in replica $n_{\mathrm{ph}}=1$, with $\mathit{G}=K^{+}$ the phonon momentum. A pseudogap opens in the nonvertical intersection between the Dirac cones at valleys $K^{-}$ and $K^{+}$ in replicas $n_{\mathrm{ph}}=0$ and $n_{\mathrm{ph}}=1$, respectively. (b) Band structure of the bulk system where the valley selective gap is seen at the replica crossing. The color scale indicates the weight of the bands on $n_{\mathrm{ph}}=0$ (black) and $n_{\mathrm{ph}}=1$ (red). (c) A zone-folding scheme may be developed, where areas B and C of the BZ of the hexagonal lattice are folded into area A [as indicated in panels (a) and (b)], forming the rBZ where the system presents vertical transitions. The band structure in the rBZ is represented in panel (c), where the $k$ path plotted corresponds to that of the cyan shaded area in panel (b).

Figure 3

(a) Density of states (DoS), shown in logarithmic scale $\ln (1+\mathrm{DoS})$, near the edges of a semi-infinite zigzag ribbon, in arbitrary units. A valley selective gap opens in the replica crossing, bridged by two edge states with parallel velocities. These states coexist with a continuum of extended states of the ungapped valleys. (b) Band structure of a zigzag ribbon (of width $150a_0$). The color scale indicates the weight of the bands on $n_{\mathrm{ph}}=0$ (black) and $n_{\mathrm{ph}}=1$ (red), while the curves’ transparency is related to the localization of the states (ranging from light curves for extended states to opaque curves for states with an inverse participation ratio larger than 0.1) The edge states described in (a), at energy $E_F=\hslash \omega /2$, are indicated with a blue circle. The horizontal axis in (a) and (b) corresponds to the rBZ projected to the periodicity direction. (c),(d) Projection of the edge states in (b) to zone C of the BZ in $n_{\mathrm{ph}}=0$, and zone B in $n_{\mathrm{ph}}=1$, showing that they are localized at opposite borders of the ribbon. Other parameters used are ${\gamma }_1=0.025{\gamma }_0$, $\hslash \omega =0.3{\gamma }_0$, and $\mathrm{\Delta }=0.01{\gamma }_0$.

Figure 4

Transmittance, in arbitrary units, between noninvasive probes located on different transverse lines across the zigzag ribbon (of width $150a_0$) separated by a distance $300\sqrt{3}{a}_0$ (the rest of the parameters are taken as in Fig. 3). The input site is taken at the edge of the corresponding line, while the output site runs over the entire output line, with transverse position $x$. For easier visualization, the situation for each panel is represented as an inset, where the input sites are marked in red and signaled by downward red arrows. The transmittance includes both the elastic and inelastic contributions and the energy of the incoming electrons is set to $\hslash \omega /2$. Panels on the top row [(a)–(d)] correspond to the pristine system while those on the lower row [(e)–(h)] include disorder (random vacancies at 0.1%), where each set of plots is normalized separately to its maximum value.