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

Abstract: Knowledge of the topology of the electronic ground state of materials has led to deep insights to novel phenomena such as the integer quantum Hall effect and fermion-number fractionalization, as well as other properties of matter. Joining two insulators of different topological classes produces fascinating boundary states in the band gap. Another exciting recent development is the bottom-up synthesis (from molecular precursors) of graphene nanoribbons (GNRs) with atomic precision control of their edge and width. Here we connect these two fields, and show for the first time that semiconducting GNRs of different width, edge, and end termination belong to different topological classes. The topology of GNRs is protected by spatial symmetries and dictated by the terminating unit cell. We have derived explicit formula for their topological invariants, and show 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 consisted 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 are not only 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.

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

This paper explores the topological classification of graphene nanoribbons (GNRs), linking the electronic ground states of these materials to their spatial symmetries and distinctive edge configurations. By utilizing concepts from the topological band theory, the authors assert that the topology of GNRs can significantly influence their electronic properties—a feature harnessed to design novel GNR-based devices with enhanced functionalities.

Graphene nanoribbons, quasi-one-dimensional strips of graphene, exhibit unique electronic properties determined by their edge configurations and width. A pivotal highlight of this study is the establishment of distinct topological classes for GNRs based on these characteristics. Through the derived formulae for topological invariants—specifically, the Z₂ invariant, a mod 2 count of the intercell component of the Zak phase—the paper delineates conditions under which GNRs become topological insulators.

The importance of symmetry-protected topological (SPT) phases in GNRs cannot be understated. Armchair graphene nanoribbons (AGNRs) are comprehensively analyzed regarding their topological characteristics, and it is shown that the topology varies with the ribbon width and termination type. This variation is mathematically captured using the Z₂ invariant calculated from wavefunction parities at pivotal k-points in the Brillouin zone. These invariants guide the identification and classification of SPT phases, which manifest as localized states at junctions between topologically distinct segments.

A notable implication of this research is its elucidation of how the topological properties of GNRs are altered by periodic doping. The introduction of dopants, such as boron, induces a change in the Zak phase, thereby transforming the material's topological class. The construction of superlattices comprising alternating segments of doped and pristine GNRs showcases these topological transitions. The resulting mid-gap junction states in such configurations exhibit considerable exchange interactions, enabling the emergence of one-dimensional antiferromagnetic Heisenberg spin 1/2 chains. This finding holds significant implications for uncovering novel quantum spin phenomena and potential quantum computation applications.

Through rigorous density functional theory (DFT) calculations and the implementation of the ab initio GW method, this study provides a robust validation of theoretical predictions and computational models. The calculated exchange interactions emphasize the practical feasibility of harnessing these topologically induced properties for developing devices that exploit the complex interplay of spin and electronic states in GNR systems.

Looking forward, this research paves the way for further exploration into the incorporation of other doping elements and the synthesis of GNR structures with varied topological characteristics. The demonstrated stability and tunability of spin centers and junction states present valuable opportunities for advancing quantum material design and creating a platform for investigating topological quantum computing architectures.