Topological Phases in Graphene Nanoribbons: Junction States, Spin Centers, and Quantum Spin Chains

We show that semiconducting graphene nanoribbons (GNRs) of different width, edge, and end termination (synthesizable from molecular precursors with atomic precision) belong to different electronic topological classes. The topological phase of GNRs is protected by spatial symmetries and dictated by the terminating unit cell. We have derived explicit formulas for their topological invariants and shown that localized junction states developed between two GNRs of distinct topology may be tuned by lateral junction geometry. The topology of a GNR can be further modified by dopants, such as a periodic array of boron atoms. In a superlattice consisting of segments of doped and pristine GNRs, the junction states are stable spin centers, forming a Heisenberg antiferromagnetic spin $1/2$ chain with tunable exchange interaction. The discoveries here not only are of scientific interest for studies of quasi-one-dimensional systems, but also open a new path for design principles of future GNR-based devices through their topological characters.

Figure 1

Heterojunctions formed with $N=9$ and $N=7$ armchair graphene nanoribbons (9AGNR/7AGNR) between two (a) topologically equivalent segments and (b) topologically inequivalent segments. The red dashed line denotes the interface between the two nanoribbons. The carbon-carbon and carbon-hydrogen bonds are colored black and gray, respectively. The color scale shows the charge density of the localized midgap junction state. The charge density is integrated along the out-of-plane direction [in units of $1/{(\mathrm{a}.\mathrm{u}.)}^2$].

Figure 2

Boron-doped graphene nanoribbons. (a) Structure of an $N=7$ AGNR periodically doped with boron pairs (B2-7AGNR). The carbon-carbon, carbon-boron, and carbon-hydrogen bonds are colored black, red, and gray, respectively. The dashed black rectangle indicates a unit cell. (b) Band structure of the B2-7AGNR calculated with the ab initio $GW$ method. The top of the valence band is set at 0 eV. (c) Wave function of dopant orbitals with $s$ and $p$ symmetries, plotted at 1 Å above the nanoribbon plane. The red and blue color shows positive and negative amplitudes [in units of $1/{(\mathrm{a}.\mathrm{u}.)}^{3/2}$], respectively.

Figure 3

Doped-pristine AGNR superlattice. (a) Structure of a 7AGNR/B2-7AGNR superlattice. The carbon-carbon, carbon-boron, and carbon-hydrogen bonds are colored black, red, and gray, respectively. The color scale shows the charge density distribution of the lower junction-state band integrated over states in the superlattice Brillouin zone and integrated over the direction perpendicular to the ribbon plane [in units of $1/{(\mathrm{a}.\mathrm{u}.)}^2$]. (b) Schematics of the 1D antiferromagnetic Heisenberg spin $1/2$ chain corresponding to the system shown in (a). The arrows denote relative directions of electron spins. (c) Ab initio calculated exchange parameters in the Heisenberg model as a function of the separation distance, in log scale and linear scale (inset).