## Topological Engineering of GNRs

Graphene nanoribbons (GNRs) are thin strips of graphene that can have different nanoscale widths and edge symmetries (for more introductory material on GNRs look here). One of the most exciting developments in the study of GNRs has been the discovery of topological phases in armchair GNRs. Topological behavior in GNRs was first predicted by Steven Louie’s group.^{1} One year later the predictions were confirmed by the Crommie group and collaborators.^{2} Topology in GNRs is classified by a Z_{2} index such that when Z_{2} = 0 GNRs are topologically trivial and when Z_{2} = 1 GNRs are are topologically nontrivial. What makes this important is that if a trivial GNR (Z_{2} = 0) is fused to a “topologically nontrivial” GNR (Z_{2} = 1) then a topologically-protected interface state holding a single unpaired electron forms at the boundary between the two GNR segments **(Fig. 1)**. This is analogous to the 2D metallic surface states that form at the surface of 3D topological insulators and the 1D metallic edge-states that form at the edges of 2D quantum spin Hall insulators.^{3,4} Since GNRs are 1D topological insulators the protected states at their interfaces can be thought of as 0D metals (i.e., localized states occupied by a single, unpaired electron).

This provides a completely new design criteria for engineering the quantum behavior of GNRs since it provides a roadmap for positioning unpaired electrons at any location we designate within a GNR, so long as it contains an interface between segments of differing topology.^{1} This can be used, for example, to create qubits **(Fig. 2a)**, quantum spin chains **(Fig. 2b)**, and new 1D band structures^{2} **(Fig. 2c)**. The Crommie group is currently pursuing all of these directions.

The origin of topology in GNRs is the quantization of the Berry phase.^{1} The Berry phase in a 1D system is calculated by integrating the Berry connection across the first Brillouin zone. If the 1D system has spatial symmetry (e.g., inversion or mirror) then the Berry phase is constrained to be either 0 or π. For a multiband system one simply adds the Berry phase for each band, so that the total Berry phase is *nπ* (where *n* is an integer). If *n* is even then Z_{2} = 0 (topologically trivial) while if *n* is odd then Z_{2} = 1 (topologically nontrivial). This leads to a simple set of topology rules for armchair nanoribbons as derived by Louie’s group^{1} **(Fig. 3)**

An example of a predicted topological interface can be seen in **Fig. 3** 4 which shows a topologically trivial N = 7 segment (i.e., 7 atoms wide) fused to a topologically nontrivial N = 9 segment (9 atoms wide). Because the two segments have different topological indices, a topologically-protected state is predicted to exist at the interface between these segments. The shaded region in **Fig. 4** shows the calculated spatial extent of the resulting interface state wavefunction (theory: S.G. Louie and coworkers).

Constructing such an interface experimentally, however, is quite tricky since it has to be synthesized with atomic precision. **Fig. 5** shows a strategy for accomplishing this via a linker molecule designed to link up with N = 7 precursors on one side and N = 9 precursors on the other side^{2} (this “7-9 linker” molecule was designed and fabricated by Greg Veber of F. Fischer’s group). Rather than making just a single topological interface, the 7-9 linker can also be

used to make a topological superlattice via self-polymerization **(Fig. 6a, b)**. This results in a structure **(Fig. 6c)** that is composed of a sequence of GNR segments having an alternating topological index. The rules of topology then demand that a topologically-protected interface state should exist at the interfaces between each GNR segment in this superlattice.

The Crommie group self-polymerized the 7-9 linker at the surface of a gold crystal and succeeded in growing a well-defined topological superlattice.^{2} The bond-resolved STM image shown in **Fig. 7b** shows that the 7-9 topological superlattice has the proper chemical structure.

**Fig. 8** shows an STM image of the 7-9 topological superlattice where the different segments can be clearly resolved. STM spectroscopy performed on the 7-9 superlattice has confirmed its topological behavior.

The STM *dI/dV* spectra of **Fig. 9a** show a series of peaks bracketing the Fermi energy (V = 0). The energy difference between peaks B and C show that the superlattice energy gap is just 0.7 eV, significantly less than the 2.3 eV energy gap of a uniform N = 7 GNR or the 1.4 eV of a uniform N = 9 GNR. *dI/dV* maps measured at peaks A-C reveal the wavefunction distribution of the electrons that reside in these topological states **(Fig. 9c)**. The wavefunction patterns are quite distinct and the intensity alternates between N = 7 and N = 9 segments. Theoretical simulations of both the GNR density of states **(Fig. 9b)** and the local density of states **(Fig. 9d)** performed by Louie’s group closely match the data obtained by the Crommie group **(Figs. 9a, c)**. The most important features are the two peaks marked B and C. These are the states that normally would not occur in either an N = 7 or an N = 9 GNR. They arise completely from topological considerations. The fact that these peaks and their associated wavefunctions match so well between theory and experiment confirms the topological nature of armchair graphene nanoribbons.^{2}

## References

1. Ting Cao, Fangzhou Zhao, and Steven G. Louie. Topological Phases in Graphene Nanoribbons: Junction States, Spin Centers, and Quantum Spin Chains. Phys. Rev. Lett. 119, 076401 (2017).

2. D. J. Rizzo, G. Veber, T. Cao, C. Bronner, T. Chen, F. Zhao, H. Rodriguez, S. G. Louie1, M. F. Crommie, F. R. Fischer. Topological Band Engineering of Graphene Nanoribbons. Nature 560, 204 (2018)

3. M. Z. Hasan and J. E. Moore. Three-Dimensional Topological Insulators. Annual Rev. Condens. Matter Phys., 2:1, 55-78 (2011).

4. J.E. Moore. The birth of topological insulators. Nature 464, 194–198 (2010).