Localized Edge States Research Articles

Moiré edge states in twisted graphene nanoribbons

The edge physics of graphene based systems is well known to be highly sensitive to the atomic structure at the boundary, with localized zero mode edge states found only on the zigzag-type termination of the lattice. Here we demonstrate that the graphene twist bilayer supports an additional class of edge states, that (i) are found for all edge geometries and thus are robust against edge roughness, (ii) occur at energies coinciding with twist induced Van Hove singularities in the bulk and (iii) possess an electron density strongly modulated by the moir\'e lattice. Interestingly, these ``moir\'e edge states'' exist only for certain lattice commensurations and thus the edge physics of the twist bilayer is, in dramatic contrast to that of the bulk, not uniquely determined by the twist angle.

Inner edge magnetisms in carbon honeycombs

We show by comprehensive ab initio calculations that sp2 carbon honeycombs recently synthesised by Krainyukova et al. [Phys. Rev. Lett. 116, 055501 (2016)] exhibit antiferromagnetism, not only at the inner edge of the zigzag ribbon component but also at the chain of sp2 carbon that joins three zigzag ribbons. The π antiferromagnetism at the joint chain has spin orientations that alternatively change along the axis and stems from a super-exchange mechanism. Along with the spin-polarization, the joint chain conduction channel opens an energy gap. The spin-polarization of the zigzag edge due to the magnetic instability of the localized edge states is less stable in energy. Through hole doping, the zigzag edge antiferromagnetism is enhanced and stabilized as the magnetic ground state, along with the re-opening of the joint chain conduction channel. When the carbon honeycombs are reconstructed into sp3-sp2 hybrid honeycombs, the π states of the joint are diminished, but the zigzag edge magnetism is preserved. Our results propose carbon honeycombs as novel magnetic carbon with competing polarization configurations.

Edge states in mesoscopic Bi islands on superconducting Nb(110)

Islands of Bi showing (110)- and (111)-orientated facets were grown on a Nb(110) single crystal and were studied with a scanning tunneling microscope at low temperatures. On the (111) facets, several states localized at step edges are identified from maps of the local differential conductance. We find edge states on both types of zig-zag step edges that decay exponentially over a length scale of several nanometers into the terrace. These states agree with the necessary condition for topologically nontrivial edge states. Further, we find states localized at the atomic scale at the step edge, which we interpret as dangling bond states. To investigate the nature of these states, we studied the Bi(110) surface, where the sharply localized states form stripe patterns, while the more extended edge state does not show a strong lattice-periodic pattern. Thus, our findings indicate a causal link between the strongly localized edge states on Bi(111) and the stripe pattern on Bi(110) as they are both related to the zig-zag termination of the honeycomb structure of Bi. Finally, we observed a decay of the superconducting proximity gap of the Bi islands with island thickness.

Detecting Topological Invariants in Nonunitary Discrete-Time Quantum Walks.

We report the experimental detection of bulk topological invariants in nonunitary discrete-time quantum walks with single photons. The nonunitarity of the quantum dynamics is enforced by periodically performing partial measurements on the polarization of the walker photon, which effectively introduces loss to the dynamics. The topological invariant of the nonunitary quantum walk is manifested in the quantized average displacement of the walker, which is probed by monitoring the photon loss. We confirm the topological properties of the system by observing localized edge states at the boundary of regions with different topological invariants. We further demonstrate the robustness of both the topological properties and the measurement scheme of the topological invariants against disorder.

Plasmons in graphene nanoribbons

We calculate the dielectric function and plasmonic response of armchair (aGNRs) and zigzag (zGNRs) graphene nanoribbons using the self-consistent-field approach within the Markovian master equation formalism (SCF-MMEF). We accurately account for electron scattering with phonons, ionized impurities, and line-edge roughness and show that electron scattering with surface optical phonons is much more prominent in GNRs than in graphene. We calculate the loss function, plasmon dispersion, and the plasmon propagation length in supported GNRs. Midinfrared plasmons in supported (3N+2)-aGNRs can propagate as far as several microns at room temperature, with 4--5-nm-wide ribbons having the longest propagation length. In other types of aGNRs and in zGNRs, the plasmon propagation length seldom exceeds 100 nm. Plasmon propagation lengths are much longer on nonpolar (e.g., diamondlike carbon) than on polar substrates (e.g., ${\mathrm{SiO}}_{2}$ or hBN), where electrons scatter strongly with surface optical phonons. We also show that the aGNR plasmon density is nearly uniform across the ribbon, while in zGNRs, because of the highly localized edge states, plasmons of different spin polarization are accumulated near the opposite edges.

Exact zero modes in frustrated Haldane chains

We show that the effective coupling between the spin-1/2 edge states of a spin-1 chain of finite length can be continuously tuned by frustration. For the $J_1-J_2$ model with nearest and next-nearest neighbor antiferromagnetic interactions, we show that the effective coupling in a chain of length $L$ changes sign $N\simeq 0.38 L$ times in the window $0.28\lesssim J_2/J_1 \lesssim 0.75$ where the short-range correlations are incommensurate. This implies that there are $N$ zero modes where the singlet and the triplet are strictly degenerate, i.e. $N$ values of $J_2/J_1$ where the spin-1/2 edge states are completely decoupled. We argue that this effect must be generic for all incommensurate phases with localized edge states, and we briefly discuss a few experimental implications.

Protocols for dynamically probing topological edge states and dimerization with fermionic atoms in optical potentials

Topological behavior has been observed in quantum systems including ultracold atoms. However, background harmonic traps for cold atoms hinder the direct detection of topological edge states arising at the boundary because the distortion fuses the edge states into the bulk. We propose experimentally feasible protocols to probe localized edge states and dimerization of ultracold fermions. By confining cold atoms in a ring lattice and changing the boundary condition from periodic to open using an off-resonant laser sheet to cut open the ring, topological edge states can be generated. A lattice in a topological configuration can trap a single particle released at the edge as the system evolves in time. Alternatively, depleting an initially filled lattice away from the boundary reveals the occupied edge states. Signatures of dimerization in the presence of contact interactions can be found in selected correlations as the system boundary suddenly changes from periodic to open and exhibit memory effects of the initial state distinguishing different configurations.

Theory of photon-driven correlated electrons in one dimension

In this paper we present a general theoretical framework to study interacting electrons under the influence of an external time-periodic driving, such as a homogeneous laser field. This is performed through a true many-body calculation and the use of Floquet theory. In particular, we consider a linear atomic chain using the Hubbard model to describe the short-ranged Coulomb interactions between electrons, plus Cluster Perturbation Theory to embed the many-body exact solution for a finite system into both an extended and an infinite lattice. Due to the presence of the external time-periodic perturbation, the electronic problem can be mapped into the study of photon-dressed quasiparticles thanks to Floquet theorem, keeping into account of all the virtual processes (absorption and emission of photons by electrons) with the laser field. This leads to an extension of the many-body static theories to out-of-equilibrium systems. This theoretical approach allowed us to show how the electronic properties of the system can be controlled and tuned varying the laser parameters. Above all, an inverse insulator-to-metal transition can be obtained for the one dimensional infinite lattice, and edge localized states appear as a finite size effect in an extended truncated chain.

Tip Functionalization of Finite Single-Walled Carbon Nanotubes and Its Impact on the Ground and Excited State Electronic Structure

We explore the effect of capping of finite (10,5) carbon nanotube (SWCNT) with various functional groups, including methylene, ether, ester, and carboxylic derivatives, having different electron withdrawing/donating properties. Using density functional theory (DFT) and time-dependent DFT (TDDFT), we found that the sp2-hybridization of the bonding atom of the capping groups and the number of such groups plays the predominant role in eliminating optically inactive edge-localized midgap states, while withdrawing/donating properties of the functional groups are less significant. Our calculations show that two sp2-groups, like methylene derivatives, combined with hydrogens are an optimal capping scheme for (10,5) SWCNT to provide the well-opened energy gap, since sp2-groups can be placed relatively far from each other to minimize the dipole moment at the edges, while preserving conjugation of the edge according to the tube chirality. Absorption spectra demonstrate negligible effects of electron donating/withdr...

Effect of edge vacancies on performance of planar graphene tunnel field-effect transistor

The influence of edge vacancies on the working ability of the planar graphene tunnel field-effect transistor (TFET) is studied at various concentrations and distributions (normal, uniform, periodic) of defects. All calculations are performed by using the Green's function method and the tight-binding approximation. It is shown that the transistor performance depends critically on two important factors associated with the defects: the destruction of the edge-localized electronic states and the emergence of subpeaks near the Fermi level. The supportable operation conditions of the TFET are found to be ensured at 30 percent or less of edge vacancies regardless of the type of their distribution.

Spin splitting and electric field modulated electron-hole pockets in antimonene nanoribbons

The one-dimensional nanoribbons have great potential applications in novel multifunctional electronic devices. Herewith, we investigate the electronic structure of antimonene nanoribbons using the first-principles calculations. Four ribbon models are considered. All of them are semiconductors with appropriate band gaps. Because of the loss of inversion symmetry, spin-orbit splitting appears in the asymmetric washboard nanoribbons. Significantly, spin splittings of 62 and 44 meV are found at the valence band maximum and conduction band minimum for zigzag asymmetric washboard nanoribbon (aW-zSbNR). Further applying an in-plane electric field introduces a potential difference between the two zigzag edges. The energies for the localized edge states are modified. Eventually spin-orbit-coupling induced band inversion and electron-hole pockets with appropriately the same size could be triggered in a W-zSbNR, which is interesting for exploiting the magnetoresistance effect.

Spatially resolving and energy splitting of edge state in zigzag edged triangle graphene quantum dots on Cu(111) surface

The electronic structure of graphene nanoribbons (GNRs) and graphene quantum dots (GQDs) has been predicted to depend sensitively on the crystallographic orientation of their edges. However, direct observation of edge state for triangle graphene quantum dots (TGQDs) has not been verified experimentally. Here we explore, using the scanning tunneling spectroscopy (STS), the zigzag edged electronic property of varisized TGQDs. Predominantly zigzag-edged TGQDs exhibit edge-localized states with the energy splittings of about 0.2–0.3V when its lateral dimension is less than 7nm. The measured energy splittings agree with theoretical calculations, and show that these edge states originate from a hybridization effect of the substrate, and not from a magnetic splitting of the edge state.

Modified participation ratio approach: application to edge-localized states in carbon nanoclusters

Modified participation ratio approach: application to edge-localized states in carbon nanoclusters

Long-range entangled zero-mode state in a non-Hermitian lattice

In contrast to a Hermitian system, in which zero modes are usually degenerate and localized edge state in the thermodynamic limit, the zero mode of a finite non-Hermitian system can be a single nontrivial long-range entangled state at the exceptional point (EP). In this work, we demonstrate this feature with a concrete example based on exact solutions. Numerical simulations show that the entangled state can be generated through the dynamic process with a high fidelity.

Structural, electronic and magnetic properties of carbon doped boron nitride nanowire: Ab initio study

Using spin-polarized density functional theory calculations, we demonstrated that carbon doped boron nitride nanowire (C-doped BNNW) has diverse electronic and magnetic properties depending on position of carbon atoms and their percentages. Our results show that only when one carbon atom is situated on the edge of the nanowire, C-doped BNNW is transformed into half-metal. The calculated electronic structure of the C-doped BNNW suggests that doping carbon can induce localized edge states around the Fermi level, and the interaction among localized edge states leads to semiconductor to half-metal transition. Overall, the bond reconstruction causes of appearance of different electronic behavior such as semiconducting, half-metallicity, nonmagnetic metallic, and ferromagnetic metallic characters. The formation energy of the system shows that when a C atom is doped on surface boron site, system is more stable than the other positions of carbon impurity. Our calculations show that C-doped BNNW may offer unique opportunities for developing nanoscale spintronic materials.

Effective spin dephasing mechanism in confined two-dimensional topological insulators

A Kramers pair of helical edge states in quantum spin Hall effect (QSHE) is robust against normal dephasing but not robust to spin dephasing. In our work, we provide an effective spin dephasing mechanism in the puddles of two-dimensional (2D) QSHE, which is simulated as quantum dots modeled by 2D massive Dirac Hamiltonian. We demonstrate that the spin dephasing effect can originate from the combination of the Rashba spin-orbit coupling and electron-phonon interaction, which gives rise to inelastic backscattering in edge states within the topological insulator quantum dots, although the time-reversal symmetry is preserved throughout. Finally, we discuss the tunneling between extended helical edge states and local edge states in the QSH quantum dots, which leads to backscattering in the extended edge states. These results can explain the more robust edge transport in InAs/GaSb QSH systems.

On-surface synthesis of graphene nanoribbons with zigzag edge topology.

Graphene-based nanostructures exhibit electronic properties that are not present in extended graphene. For example, quantum confinement in carbon nanotubes and armchair graphene nanoribbons leads to the opening of substantial electronic bandgaps that are directly linked to their structural boundary conditions. Nanostructures with zigzag edges are expected to host spin-polarized electronic edge states and can thus serve as key elements for graphene-based spintronics. The edge states of zigzag graphene nanoribbons (ZGNRs) are predicted to couple ferromagnetically along the edge and antiferromagnetically between the edges, but direct observation of spin-polarized edge states for zigzag edge topologies--including ZGNRs--has not yet been achieved owing to the limited precision of current top-down approaches. Here we describe the bottom-up synthesis of ZGNRs through surface-assisted polymerization and cyclodehydrogenation of specifically designed precursor monomers to yield atomically precise zigzag edges. Using scanning tunnelling spectroscopy we show the existence of edge-localized states with large energy splittings. We expect that the availability of ZGNRs will enable the characterization of their predicted spin-related properties, such as spin confinement and filtering, and will ultimately add the spin degree of freedom to graphene-based circuitry.

Edge-state transport in Floquet topological insulators

Floquet topological insulators are systems in which the topology emerges out of equilibrium when a time periodic perturbation is applied. In these systems one can define quasi-energy states which replace the quilibrium stationary states. The system exhibits its non-trivial topology by developing edge localized quasi-energy states which lie in a gap of the quasi energy spectrum. These states represent a non-equilibrium analogue of the topologically protected edge-states in equilibrium topological insulators. In equilibrium these edge-states lead to very specific transport properties, in particular the two-terminal conductivity of these systems is $2e^2/h$. Here we explore the transport properties of the edge-states in a Floquet topological insulator. In stark contrast to the equilibrium result, we find that the two terminal conductivity of these edge states is significantly different from $2e^2/h$. This fact notwithstanding, we find that for certain external potential strengths the conductivity is smaller than $2e^2/h$ and robust to the effects of disorder and smooth changes to the Hamiltonian's parameters. This robustness is reminiscent of the robustness found in equilibrium topological insulators. We provide an intuitive understanding of the reduction of the conductivity in terms of scattering by photons. This leads us to consider a previously proposed Floquet sum rule which recovers the equilibrium value of $2e^2/h$ for the conductivity when edge states are present. We show that this sum rule holds in our system using both numerical and analytic techniques.

Topological edge states in two-gap unitary systems: a transfer matrix approach

We construct and investigate a family of two-band unitary systems living on a cylinder geometry and presenting localized edge states. Using the transfer matrix formalism, we solve and investigate in detail such states in the thermodynamic limit. Analytic considerations then suggest the construction of a family of Riemann surfaces associated to the band structure of the system. In this picture, the corresponding edge states naturally wind around non-contractile loops, defining a topological invariant associated to each gap of the system.

Edge state magnetism in zigzag-interfaced graphene via spin susceptibility measurements.

Development of graphene spintronic devices relies on transforming it into a material with a spin order. Attempts to make graphene magnetic by introducing zigzag edge states have failed due to energetically unstable structure of torn zigzag edges. Here, we report on the formation of nanoridges, i.e., stable crystallographically oriented fluorine monoatomic chains, and provide experimental evidence for strongly coupled magnetic states at the graphene-fluorographene interfaces. From the first principle calculations, the spins at the localized edge states are ferromagnetically ordered within each of the zigzag interface whereas the spin interaction across a nanoridge is antiferromagnetic. Magnetic susceptibility data agree with this physical picture and exhibit behaviour typical of quantum spin-ladder system with ferromagnetic legs and antiferromagnetic rungs. The exchange coupling constant along the rungs is measured to be 450 K. The coupling is strong enough to consider graphene with fluorine nanoridges as a candidate for a room temperature spintronics material.