Topologically protected gap states and resonances in gated trilayer graphene

In Fig. 2 we present local density of states calculated for TLG with two minimal corrugations in the lower and upper layers, as shown in the middle and bottom panels of Fig. 1. The corrugations cause that the stacking of TLG changes from ABC (left side) to CBA (right side). Since TLG is periodic in the zigzag direction (along the corrugations) the LDOS is $k$-dependent, where $k$ is the wavevector corresponding to this periodicity. In the upper panels of the figure, gate voltages applied to external layers are $V=\pm 0.5$ eV, while in the lower panels $V=\pm 0.1$ eV. The left panels correspond to the case of spatially overlapping LBSDW and UBSDW, right panels concern separated LBSDW and UBSDW. In all cases there are three states connecting the valence and conduction bands across the energy gap. The figure shows LDOS calculated for $k$ close to the cone at $K$; for $K^{\prime}$ the figure is mirror-reflected, so the gap states are valley-protected.

Now, we analyze the spatial localization of the gapless states. We do it first for the case of $V=\pm 0.5$ eV. In Fig. 3 (a) and (b) we present the layer-resolved LDOS calculated for $k$-values for which $E=0$ line (the Fermi energy) crosses the left and right gapless states seen in Fig. 2 (a), i.e., for the case of overlapping LBSDW and UBSDW. The envelope of the vertical bars reflects the density of the corresponding wave function. For this gate voltage the states are almost entirely localized only in the lower or in the upper layer, just in place where the UBSDW and LBSDW occur. When the corrugations are separated (panels (c) and (d)) the localization of these states moves to the area between UBSDW and LBSDW. The LDOS for the middle gap state, presented in Fig. 4 (a), is almost equally distributed between the lower and the upper layers, with small contribution from the middle layer note2 . We can conclude that for this gate voltage the gapless states are due to stacking change between these two layers, which we call indirect stacking, since their nodes are not connected by $\gamma_{1}$.

Closer inspection of Fig. 2 (a) and (b) reveals also two resonance states below and above the Fermi level, i.e., in the valence and conduction band continua. The energy structure of these resonances reminds topological gapless states of bilayer graphene Zhang_2013 ; Pelc_2015 . Indeed, the corresponding layer-resolved LDOS, shown in Fig. 5, confirm this supposition. The left resonance state in the valence band is localized in the upper layer, while the right state is localized in the middle layer, just like the gapless states of the gated upper bilayer with UBSDW Jaskolski_2018 . In the case of resonances situated in the conduction band, the left one is localized in the middle layer and the right one in the bottom layer, again, like the gapless states of the lower bilayer with LBSDW and inverted gate polarization. The appearance of resonances can be explained as follows. The gate $V>\gamma_{1}$ moves the electron charge from the bottom to the middle layer, and from the middle layer to the top one. We thus get the bottom gated bilayer with the lower layer discharged - yielding two topological states in the unoccupied conduction band, and the upper gated bilayer with the top layer additionally loaded - yielding a couple of topological states in the valence band continuum. When $V<\gamma_{1}$ the electron charge is not moved between the layers and the resonances do not appear (Fig. 2 c,d).

The triplet of gapless states occurs for any gate voltage. However, for $V<\gamma_{1}$, their localization is different. The layer-resolved LDOS for the left and right gap states seen in Fig. 2 (c) are presented in Fig. 6 (a) and (b), respectively; the layer-resolved LDOS of the middle gap state is visualized in Fig. 4 (b). For $V=\pm 0.1$ eV the left and right gapless states are more diffused and more evenly distributed between all three layers. We have checked that when UBSDW and LBSDW are separated, the layer mixing and diffusion are even stronger. Now, the gap states do not have the character of bilayer topological states, but reflect the global ABC/CBA stacking change in TLG.

We begin with the Hamiltonian for trilayer graphene. Unlike the commonly applied reduced Hamiltonians for $N$-layer graphene Min_2008 ; Jung_PRB_2011 ; Chittari_PRL_2019 , we use the full TLG Hamiltonian in its most general form. This enables direct analysis of its symmetry when the stacking changes from ABC to CBA. The final formulas show how many terms in the eigenvalue expansion are sufficient for the calculation of of the valley Chern number. For momenta close to the Dirac point $K$, the Hamiltonian $H_{ABC}$ of trilayer graphene is a $3\times 3$ matrix of $2\times 2$ blocks:

where ${\rm{\bf\Pi}}=\nu\left(p_{x}{\rm{\bf\sigma_{x}}}+p_{y}{\rm{\bf\sigma_{y}}}\right)$, ${\rm{\bf T}}=\gamma_{1}\frac{1}{2}\left({\rm{\bf\sigma_{x}}}+i{\rm{\bf\sigma_{y}}}\right)$ and ${\rm{\bf I_{2}}}$ is the $2\times 2$ unit matrix, $\nu=\frac{\sqrt{3}a}{2\hbar}\gamma_{0}$ is the Fermi velocity, and $a$ is graphene lattice constant Min_2008 . Its (unnormalised) eigenvector $\Psi_{0}$ related to the lowest energy $\lambda_{0}$ is

where we use polar coordinates: $\nu(p_{x}+ip_{y})=P\exp(i\phi)$ and $V_{\pm}=V\pm\lambda_{0}$.

The valley Chern number $N_{\nu}=\gamma/2\pi$ at $K$ is defined by the Berry phase Jung_PRB_2011 ; Chittari_PRL_2019

with the integration performed over a circle $S_{1}(K,P)$. The calculation of $\mathrm{Im}\langle\Psi_{0}|\nabla\Psi_{0}\rangle$ yields $(\gamma_{1}^{2}(\lambda_{0}(P^{2}-V_{+}^{2})+\gamma_{1}^{2}V_{+})^{2}-(P^{2}-V_{-}^{2})^{2}(P^{2}-V_{+}^{2})^{2}-\gamma_{1}^{2}(P^{2}-V_{-}^{2})^{2}(V_{+}^{2}+2P^{2}))P^{2}\mathrm{d}\phi$ and $|\Psi_{0}|^{2}=(\lambda_{0}(P^{2}-V_{+}^{2})+\gamma_{1}^{2}V_{+})^{2}(\gamma_{1}^{2}V_{-}^{2}+\gamma_{1}^{2}P^{2}+(P^{2}-V_{-}^{2})^{2})+P^{2}(P^{2}-V_{-}^{2})^{2}((P^{2}-V_{+}^{2})^{2}+\gamma_{1}^{2}P^{2}+\gamma_{1}^{2}V_{+}^{2})$. To remove singularities, which appear when $P\to\infty$, we use the Rayleigh-Schrödinger perturbation theory (with the zero-order Hamiltonian ${\rm{\bf I_{3}\otimes\Pi}}$). We account up to the second correction (of the order $P^{-1}$) and get that $\lambda_{0}=-P-V+\frac{\gamma_{1}^{2}}{8P}+o(P^{-2})$. The leading term in $\mathrm{Im}\langle\Psi_{0}|\nabla\Psi_{0}\rangle$ gives $-96\pi\gamma_{1}^{2}V^{2}P^{6}$, while the leading term in $|\Psi_{0}|^{2}$ is equal $32\gamma_{1}^{2}V^{2}P^{6}$. Hence, the Berry phase is equal to $-3\pi$ and the contribution to the Chern number from $K$ is $-3/2$ Jung_PRB_2011 ; Chittari_PRL_2019 .

Switching from $K$ to $K^{\prime}$ causes transposition of the matrix ${\rm{\bf\Pi}}$ in the Eq. (1), which is equivalent to parity change and yields the opposite sign of the valley Chern number at $K^{\prime}$. The switching from ABC to CBA stacking causes transposition of the matrix ${\rm{\bf T}}$. Composition of these switchings is equivalent to a unitary symmetry ${\rm{\bf I_{3}}}\otimes{\rm{\bf\sigma_{x}}}$ on the Hamiltonian. This means that in case of CBA stacking, the contributions from neighbourhoods of $K$ and $K^{\prime}$ to the Chern number are $3/2$ and $-3/2$, respectively, i.e., they have reversed signs comparing to the ABC stacking. Therefore, the stacking change causes change of the local Chern number contribution by $\pm 3$ and thus the appearance of three gapless states localized at the stacking domain wall.

It is worth noting that $H_{CBA}$ can be unitarily transformed to $H_{ABC}(-V)$, i.e., with the sign of $V$ reversed. This means that three gapless states appear also when the gate polarization changes, as was shown in Ref. Jung_PRB_2011 .