Topological qubits based on square-root graphene nanoribbons induced by electric fields

Abstract

The present disclosure provides a graphene ribbon, comprising a planar monolayer of sp2-hybridized carbon atoms arranged in a hexagonal lattice characterized by a direction of periodicity (see formula (I), the hexagonal lattice structure comprising a unit cell. Each unit cell is symmetric with respect to a plane perpendicular to the plane of the monolayer (a mirror plane); each unit cell comprises 4n zigzags edges, wherein n is a natural number; and wherein the minimal number of hexagons intersected by a line lying in the plane of the monolayer parallel to the mirror plane is 3.

Description

UCI-00825 TOPOLOGICAL QUBITS BASED ON SQUARE-ROOT GRAPHENE NANORIBBONS INDUCED BY ELECTRIC FIELDS RELATED APPLICATION

[0001] This application claims the benefit of U.S. Provisional Application No.63 / 562,124, filed on March 6, 2024. The entire teachings of this application are incorporated herein by reference. STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] This invention was made with government support under N00014-21-1-2537 awarded by the U.S. Navy, Office of Naval Research. The government has certain rights in the invention. BACKGROUND

[0003] Embodiments of the present disclosure relate to graphene nanoribbons, and more specifically, to inducing topological qubits based on fractional fermions within graphene nanoribbons. BRIEF SUMMARY

[0004] In an example embodiment, the present invention is a graphene ribbon, comprising: a planar monolayer of sp2-hybridized carbon atoms arranged in a hexagonal lattice characterized by a direction of periodicity ^⃗^, the hexagonal lattice structure comprising a unit cell. Each unit cell is symmetric with respect to a plane perpendicular to the plane of the monolayer (a mirror plane); each unit cell comprises 4n zigzags edges, wherein n is a natural number. The minimal number of hexagons intersected by a line lying in the plane of the monolayer parallel to the mirror plane is 3.

[0005] In another example embodiment, the present invention is a method of initializing a qubit, comprising: providing a graphene ribbon of any embodiment described herein;UCI-00825 applying a first homogenous electric field to a first portion of the graphene ribbon, the first electric field having a component perpendicular to the direction of periodicity ^⃗^; applying a second electric field to a second portion of the graphene ribbon, the second electrical field having a component that is antiparallel to the first electric field, the second portion of the graphene ribbon disposed next to the first portion; applying a magnetic field to the graphene ribbon, the magnetic field having a component that is perpendicular to the first electric field and / or second electric field and the plane of the monolayer, thereby initializing a first qubit.

[0006] In another embodiment, the present invention is a method of causing an interaction between qubits, comprising: providing a graphene ribbon of any embodiment described herein; providing a means for generating an electric field, the means configured to generate: a first electric field, the first electric field being homogenous in a first region that comprises a first portion of the graphene ribbon, the first electric field having a component perpendicular to the direction of periodicity ^⃗^; a second electric field, the second electric field being homogenous in a second region that comprises a second portion of the graphene ribbon, the second electric field having a component that is antiparallel to the first electric field, a third electric field, the third electric field being homogenous in a third region that comprises a third portion of the graphene ribbon, the third electric field having a component that is parallel to the first electric field, wherein the second region is disposed next to the first region, and the third region is disposed next to the second region; and further wherein the means for generating the electric field is configured to translate the first region, the second region, and the third region, each along the plane of the monolayer.

[0007] The method may further comprise: applying a magnetic field to the first, the second, and the third regions, the magnetic field having a component that is perpendicular to the first electric field and / or second electric field and the plane of the monolayer, thereby initializing aUCI-00825 first and a second qubits; and translating the first region and / or the third region to be proximal to one another, thereby causing an interaction between the first qubit and the second qubit.

[0008] In another example embodiment, the present invention is a device, comprising: a graphene ribbon of any embodiment described herein; a means for generating an electric field, the means configured to generate: a first electric field, the first electric field being homogenous in a first region that comprises a first portion of the graphene ribbon, the first electric field having a component perpendicular to the direction of periodicity ^⃗^; a second electric field, the second electric field being homogenous in a second region that comprises a second portion of the graphene ribbon, the second electric field having a component that is antiparallel to the first electric field, a third electric field, the third electric field being homogenous in a third region that comprises a third portion of the graphene ribbon, the third electric field having a component that is parallel to the first electric field, wherein the second region is disposed next to the first region, and the third region is disposed next to the second region, wherein the means for generating the electric field is configured to translate the first region, the second region, and the third region, each along the plane of the monolayer; and a means for applying a magnetic field to the first, the second, and the third regions, the magnetic field having a component that is perpendicular to the first electric field and / or second electric field and the plane of the monolayer. BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0009] Fig.1A is a schematic diagram illustrating an exemplary structure of a unit cell of a pzc-GNR, according to embodiments of the present disclosure.

[0010] Fig.1B is a schematic diagram of the coupling strength between segments, according to embodiments of the present disclosure.

[0011] Fig.1C is a diagram of the electrostatic potential energy of atoms with a transverse electric field applied, according to embodiments of the present disclosure.

[0012] Fig.1D is an exemplary square-root model with four lattice sites, according to embodiments of the present disclosure.

[0013] Fig.2A is an exemplary reaction scheme of the synthesis of monomer and pzc-GNR according to embodiments of the present disclosure.UCI-00825

[0014] Fig.2B is an exemplary STM image for polymer-T at 280°C, according to embodiments of the present disclosure.

[0015] Fig.2C is an exemplary large-area STM scan of mostly planar pzc-GNR at 380°C, according to embodiments of the present disclosure.

[0016] Fig.2D is an exemplary STM image for completely planar pzc-GNR at 450°C, according to embodiments of the present disclosure.

[0017] Fig.2E is an exemplary close-up STM image of an isolated pzc-GNR at 450°C, according to embodiments of the present disclosure.

[0018] Fig.2F is an exemplary high-resolution nc-AFM image of pzc-GNR, according to embodiments of the present disclosure.

[0019] Fig.3A is an exemplary band structure of pzc-GNR without an electric field under a +0.1V / Å, and a -0.1V / Å electric field, according to embodiments of the present disclosure.

[0020] Fig.3B is an exemplary band structure of pzc-GNR without an electric field under a +0.1V / Å, and a -0.1V / Å electric field, according to embodiments of the present disclosure.

[0021] Fig.3C is a distribution of amplitude of wave functions, according to embodiments of the present disclosure.

[0022] Fig.3D is a band structure of the square root model with parameters chosen according to Fig.3B, according to embodiments of the present disclosure.

[0023] Fig.3E is an exemplary resultant parity of the wave functions at the middle of the Brillouin zone, according to embodiments of the present disclosure.

[0024] Fig.3F is an exemplary band structure of pzc-GNR without an electric field under a +0.1V / Å, and a -0.1V / Å electric field, according to embodiments of the present disclosure.

[0025] Fig.3G is a distribution of amplitude of wave functions, according to embodiments of the present disclosure.UCI-00825

[0026] Fig.3H is a band structure of the square root model with parameters chosen according to Fig.3G, according to embodiments of the present disclosure.

[0027] Fig.3I is an exemplary resultant parity of the wave functions at the middle of the Brillouin zone, according to embodiments of the present disclosure.

[0028] Fig.4A is an exemplary polarization calculation using the Wannier charge center methods, according to embodiments of the present disclosure.

[0029] Fig.4B is an exemplary system is filled up to the 155th band under positive field, according to embodiments of the present disclosure.

[0030] Fig.4C is a typical topological feature with quantization of 0 under negative electric field and 0.5 under positive electric field, according to embodiments of the present disclosure.

[0031] Fig.4D is a nontrivial system under a negative field, according to embodiments of the present disclosure.

[0032] Fig.4E is a diagram showing the switch from trivial to nontrivial under a negative field, according to embodiments of the present disclosure.

[0033] Fig.5A is a periodic supercell of pzc-GNR, according to embodiments of the present disclosure.

[0034] Fig.5B is a diagram illustrating the boundary state of the periodic supercell, according to embodiments of the present disclosure.

[0035] Fig.5C is a diagram of the density of the periodic supercell of pzc-GNR, according to embodiments of the present disclosure.

[0036] Fig.6A is an exemplary band structure of pzc-GNR without electric field calculated by DFT method, compared to the band structure of the square-root model, according to embodiments of the present disclosure.

[0037] Fig.6B is the same comparison as in Fig.6A, with an electric field of 0.1 V / Å, according to embodiments of the present disclosure.UCI-00825

[0038] Fig.7A is an exemplary band structure of pzc-GNR calculated in a tight-binding model assuming only nearest-neighbor hopping between carbon atoms, according to embodiments of the present disclosure.

[0039] Fig.7B is an exemplary band structure of pzc-GNR calculated in a tight-binding model assuming only nearest-neighbor hopping between carbon atoms, according to embodiments of the present disclosure.

[0040] Fig.7C is a graph illustrating a band structure in a positive field, according to embodiments of the present disclosure.

[0041] Fig.7D is an exemplary plot of amplitude of the wave function for the four bands near the Fermi level at the middle of the Brillouin zone under a positive and negative electric field, according to embodiments of the present disclosure.

[0042] Fig.7E is a comparison of phase and strength of amplitude at a gamma point, according to embodiments of the present disclosure.

[0043] Fig.7F is an amplitude plot, according to embodiments of the present disclosure.

[0044] Fig.7G is an exemplary amplitude of wave function calculated from the square-root model, according to embodiments of the present disclosure.

[0045] Fig.7H is an exemplary band structure of pzc-GNR calculated in a tight-binding model assuming only nearest-neighbor hopping between carbon atoms, according to embodiments of the present disclosure.

[0046] Fig.7I is a graph illustrating a band structure in a negative field, according to embodiments of the present disclosure.

[0047] Fig.7J is an exemplary plot of amplitude of the wave function for the four bands near the Fermi level at the middle of the Brillouin zone under a positive and negative electric field, according to embodiments of the present disclosure.UCI-00825

[0048] Fig.7K is a comparison of phase and strength of amplitude at a gamma point, according to embodiments of the present disclosure.

[0049] Fig.7L is an amplitude plot, according to embodiments of the present disclosure.

[0050] Fig.7M is an exemplary amplitude of wave function calculated from the square-root model, according to embodiments of the present disclosure.

[0051] Fig.8A is an exemplary chemical structure of 7-AGNR-S(1,3).

[0052] Fig.8B is an exemplary band structure of pzc-GNR calculated in a tight-binding model assuming only nearest-neighbor hopping between carbon atoms, according to embodiments of the present disclosure.

[0053] Fig.8C is a graph illustrating a band structure in a positive field, according to embodiments of the present disclosure.

[0054] Fig.8D is an exemplary plot of amplitude of the wave function for the four bands near the Fermi level at the middle of the Brillouin zone under a positive and negative electric field, according to embodiments of the present disclosure.

[0055] Fig.8E is a comparison of phase and strength of amplitude at a gamma point, according to embodiments of the present disclosure.

[0056] Fig.8F is an amplitude plot, according to embodiments of the present disclosure.

[0057] Fig.8G is an exemplary amplitude of wave function calculated from the square-root model, according to embodiments of the present disclosure.

[0058] Fig.8H is an exemplary band structure of pzc-GNR calculated in a tight-binding model assuming only nearest-neighbor hopping between carbon atoms, according to embodiments of the present disclosure.

[0059] Fig.8I is a graph illustrating a band structure in a negative field, according to embodiments of the present disclosure.UCI-00825

[0060] Fig.8J is an exemplary plot of amplitude of the wave function for the four bands near the Fermi level at the middle of the Brillouin zone under a positive and negative electric field, according to embodiments of the present disclosure.

[0061] Fig.8K is a comparison of phase and strength of amplitude at a gamma point, according to embodiments of the present disclosure.

[0062] Fig.8L is an amplitude plot, according to embodiments of the present disclosure.

[0063] Fig.8M is an exemplary amplitude of wave function calculated from the square-root model, according to embodiments of the present disclosure.

[0064] Fig.8N is an amplitude plot, according to embodiments of the present disclosure.

[0065] Fig.9A illustrates Eigenvalues near the Fermi level for a supercell with 10 repeating units, according to embodiments of the present disclosure.

[0066] Fig.9B is an isosurface plot of the wave function residing at an upper finite gap and the lower finite gap, according to embodiments of the present disclosure.

[0067] Fig.9C is an isosurface plot of the wave function residing at an upper finite gap and the lower finite gap, according to embodiments of the present disclosure.

[0068] Fig.10A illustrates Eigenvalues near the Fermi level for a large molecule, according to embodiments of the present disclosure.

[0069] Fig.10B is an isosurface plot of the wave function residing at an upper finite gap and the lower finite gap, according to embodiments of the present disclosure.

[0070] Fig.10C is an isosurface plot of the wave function residing at an upper finite gap and the lower finite gap, according to embodiments of the present disclosure.

[0071] Fig.11 is a schematic diagram illustrating a reaction scheme for synthesis of monomer-T, according to embodiments of the present disclosure.UCI-00825

[0072] Fig.12 illustrates high-resolution (MALDI FT-ICR) mass spectrum of monomer-T (matrix: TCNQ) with the insects show isotopic distributions for experimental and simulated spectra, according to embodiments of the present disclosure.

[0073] Fig.13 illustrates H-NMR (400MHz, CDCl3) of monomer-T, according to embodiments of the present disclosure.

[0074] Fig.14 illustrates C-NMR (100MHz, CDCl3) of monomer-T, according to embodiments of the present disclosure.

[0075] Fig.15 illustrates band structures of pzc-GNR, according to embodiments of the present disclosure.

[0076] Fig.16 illustrates band gap induced by the transverse electric field versus the strength of the field applied, according to embodiments of the present disclosure.

[0077] Fig.17 illustrates band structures and resultant parity of pzc-GNR, according to embodiments of the present disclosure.

[0078] Fig.18 illustrates chemical structure, band structures, and resultant parities of 7- AGNR-S(1,3), according to embodiments of the present disclosure.

[0079] Fig.19 illustrates various aspects of pzc-GNR under a step-function electric field, according to embodiments of the present disclosure.

[0080] Fig.20 illustrates various aspects of pzc-GNR under a step-function electric field, according to embodiments of the present disclosure.

[0081] Fig.21 illustrates various aspects of pzc-GNR under a step-function electric field, according to embodiments of the present disclosure.

[0082] Fig.22 illustrates various aspects of 5-sGNR, according to embodiments of the present disclosure.UCI-00825

[0083] Fig.23 illustrates various aspects of 5-sGNR, pzc-GNR with a mirror symmetric unit cell, and a non-mirror-symmetric unit cell, according to embodiments of the present disclosure.

[0084] Fig.24 is a schematic diagram of an embodiment of a device described herein.

[0085] Fig.25 illustrates isosurface plots of wavefunctions of boundary states generated in 5- sGNR and corresponding eigenvalues, according to embodiments of the present disclosure.

[0086] Fig.26 illustrates electronic band structures of pzc-GNR and band inversions by an electric field, according to embodiments of the present disclosure.

[0087] Fig.27 illustrates topological solitons, according to embodiments of the present disclosure. DETAILED DESCRIPTION

[0088] Graphene nanoribbons (GNRs) are unique quasi-one-dimensional carbon nano- materials and have garnered significant research interest in the field of topological insulators. Embodiments of the present disclosure are directed towards a new class of GNRs with unique topological properties induced by a transverse electric field. The GNRs in this class consist of four zigzag segments arranged in a mirror-symmetric way, in analogy to a four-lattice tight-binding square-root model. When a transverse electric field is applied, those segments obtain additional electrostatic potential, which opens up two finite gaps at the middle of the Brillouin zone. A dimethyl containing halogenated monomer precursor (monomer-T) is specifically designed to experimentally produce a partially zigzag chevron-type graphene nanoribbons (pzc-GNR) on-surface containing the desired mirror-symmetric four zigzag segments. This ribbon is synthesized using a bottom-up approach, where monomer-T undergoes on-surface polymerization and cyclodehydrogenation reactions on Au(111) surface in an ultra-high vacuum (UHV). The synthesis of GNRs from monomer-T and other structurally related monomers may be extended to other substrates, including various metals,UCI-00825 semiconductors, and dielectrics, and may even be realized in other environments, including solutions and gas phase. First-principle calculations on the pzc-GNR demonstrate different topological phases under electric field in the opposite direction, which is consistent with predictions from the square-root model. Topologically protected boundary states are created by joining different phases together, which can be achieved by tuning electric fields. Some embodiments may include the potential for realizing spatially controllable fractional charges within GNRs, thereby providing new opportunities for designing topological qubits based on fractional fermions.

[0089] GNRs can be regarded as quasi-one-dimensional materials cut from graphene or unrolled from carbon nanotubes. Over the years, advancements in bottom-up synthesis methods have led to a growing variety of GNRs with unique edge structures, such as zigzag, armchair, chevron, and chiral edges. Among these, GNRs exhibiting topological phases have obtained significant attention. They have served as a versatile platform for the exploration of fundamental topological models, including the Su-Schrieffer-Heeger (SSH) model. The bulk- boundary correspondence ensures that nontrivial topological GNRs can exhibit a robust localized state within the band gap when a boundary is made. Such a topologically protected boundary state can be utilized in spintronics and as qubits for quantum computing technology.

[0090] To engineer the topological phases of GNRs, there two general strategies. The first strategy is to control the chemical structures of GNRs by designing the reactions during the bottom-up synthesis. In this case, the boundary state is formed either at a structural heterojunction or at the two ends. The second strategy also has some requirements on the chemical structures but also uses an externally controllable factor to manipulate the electronic structure and the topological phase. As the factor is externally applied, it is expected that boundary states formed from such a strategy are more versatile and controllable, and can beUCI-00825 better integrated for potential applications. For GNRs the realization of the second strategy is limited to only a few examples.

[0091] Embodiments of the present disclosure broaden the scope where the second strategy can be effectively employed. A class of GNRs are introduced, of which topological phases are induced by a transverse electric field. The topological behaviors of GNRs in this class can be explained by the square-root model. Taking the square root of a Hamiltonian has been shown to generate novel nontrivial topological phases. In this work, the square-root model demonstrates that a process of taking the square-root of the Hamiltonian of two coupled Rice- Mele chains yields a new and topological model, at the expense of broken crystal symmetries. The technique possesses nonsymmorphic chiral symmetry and belongs to the BDI class of topological insulators. Through first-principles calculations, GNRs described by the square- root model are shown to have tunable topological phases and robust boundary states, as determined by electric fields.

[0092] As used herein, the term “zigzag edge” refers to a segment of the perimeter of a graphene ribbon having a direction of periodicity ^⃗^, such that a pair of atoms (atom, next nearest atom) that belong to this segment can be connected by a line intersecting the direction of periodicity ^⃗^at 30° angle.

[0093] In a 1stexample embodiment, the present invention is a graphene ribbon. In a 1staspect of the 1stexample embodiment, the graphene ribbon comprises a planar monolayer of sp2-hybridized carbon atoms arranged in a hexagonal lattice characterized by a direction of periodicity ^⃗^, the hexagonal lattice structure comprising a unit cell. Each unit cell is symmetric with respect to a plane perpendicular to the plane of the monolayer (a mirror plane); each unit cell comprises 4n zigzags edges, wherein n is a natural number. The minimal number of hexagons intersected by a line lying in the plane of the monolayer parallel to the mirror plane is 3.UCI-00825

[0094] In a 2ndaspect of the 1stexample embodiment, n = 1. The remainder of the features and example features of the 2ndaspect are as described above with respect to the 1staspect.

[0095] In a 3rdaspect, the unit cell is represented by the following structural formula:.

[0096] The remainder of the features and example features of the 3rdaspect are as described above with respect to the 1stor 2ndaspects.

[0097] In a 2ndexample embodiment, the present invention is a method of initializing a qubit. In a 1staspect, the method comprises: providing a graphene ribbon according to any of the aspects of the 1stexample embodiment; applying a first homogenous electric field to a first portion of the graphene ribbon, the first electric field having a component perpendicular to the direction of periodicity ^⃗^; applying a second electric field to a second portion of the graphene ribbon, the second electrical field having a component that is antiparallel to the first electric field, the second portion of the graphene ribbon disposed next to the first portion; applying a magnetic field to the graphene ribbon, the magnetic field having a component that is perpendicular to the first electric field and / or second electric field and the plane of the monolayer, thereby initializing a first qubit.

[0098] In a 2ndaspect, the method further comprises applying a third homogenous electric field to a third portion of the graphene ribbon, the third electric field being having a component that is parallel to the first electric field, the third portion disposed next to the second portion of the graphene ribbon, thereby initializing a second qubit. The remainder ofUCI-00825 the features and example features of the 2ndaspect are as described above with respect to the 1staspect.

[0099] In a 3rdaspect, the first electric field is perpendicular to the direction of periodicity ^⃗^; and

[0100] the second electric field is antiparallel to the first electric field. The remainder of the features and example features of the 3rdaspect are as described above with respect to the 1stor 2ndaspects.

[0101] In a 4thaspect, the first electric field is perpendicular to the direction of periodicity ^⃗^; the second electric field is antiparallel to the first electric field; and the third electric field is parallel to the first electric field. The remainder of the features and example features of the 4thaspect are as described above with respect to any of the 1stthrough 3rdaspects.

[0102] In a 5thaspect, the magnetic field perpendicular to the first electric field and the plane of the monolayer. The remainder of the features and example features of the 5thaspect are as described above with respect to any of the 1stthrough 4thaspects.

[0103] In a 3rdexample embodiment, the present invention is a method of causing an interaction between qubits. In a 1staspect, the method comprises: providing a graphene ribbon of any aspect of the 1stexample embodiment; providing a means for generating an electric field, the means configured to generate: a first electric field, the first electric field being homogenous in a first region that comprises a first portion of the graphene ribbon, the first electric field having a component perpendicular to the direction of periodicity ^⃗^; a second electric field, the second electric field being homogenous in a second region that comprises a second portion of the graphene ribbon, the second electric field having a component that is antiparallel to the first electric field, a third electric field, the third electric field being homogenous in a third region that comprises a third portion of the graphene ribbon, the third electric field having a component that is parallel to the first electric field,UCI-00825 wherein the second region is disposed next to the first region, and the third region is disposed next to the second region; and further wherein the means for generating the electric field is configured to translate the first region, the second region, and the third region, each along the plane of the monolayer; the method further comprising: applying a magnetic field to the first, the second, and the third regions, the magnetic field having a component that is perpendicular to the first electric field and / or second electric field and the plane of the monolayer, thereby initializing a first and a second qubits; and translating the first region and / or the third region to be proximal to one another, thereby causing an interaction between the first qubit and the second qubit.

[0104] In a 2ndaspect, the first electric field is perpendicular to the direction of periodicity ^⃗^; the second electric field is antiparallel to the first electric field; and the third electric field is parallel to the first electric field. The remainder of the features and example features of the 2ndaspect are as described above with respect to the 1staspect.

[0105] In a 3rdaspect, the magnetic field perpendicular to the first electric field and the plane of the monolayer. The remainder of the features and example features of the 3rdaspect are as described above with respect to the 1stor 2ndaspects.

[0106] In a 4thexample embodiment, the present invention is a method of reading a qubit. In a first aspect, the method comprises providing a qubit initialized according to any of the aspects of the 2ndexample embodiment, the qubit characterized by a spin; and measuring the spin of the qubit.

[0107] In a 5thexample embodiment, the present invention is a device. In a 1staspect, the device comprises: a graphene ribbon according to any of the aspects of the 1stexample embodiment; a means for generating an electric field, the means configured to generate: a first electric field, the first electric field being homogenous in a first region that comprises a first portion of the graphene ribbon, the first electric field having a component perpendicularUCI-00825 to the direction of periodicity ^⃗^; a second electric field, the second electric field being homogenous in a second region that comprises a second portion of the graphene ribbon, the second electric field having a component that is antiparallel to the first electric field, a third electric field, the third electric field being homogenous in a third region that comprises a third portion of the graphene ribbon, the third electric field having a component that is parallel to the first electric field, wherein the second region is disposed next to the first region, and the third region is disposed next to the second region, wherein the means for generating the electric field is configured to translate the first region, the second region, and the third region, each along the plane of the monolayer. The device further comprises a means for applying a magnetic field to the first, the second, and the third regions, the magnetic field having a component that is perpendicular to the first electric field and / or second electric field and the plane of the monolayer.

[0108] In a 2ndaspect, the first electric field is perpendicular to the direction of periodicity ^⃗^;

[0109] the second electric field is antiparallel to the first electric field; and the third electric field is parallel to the first electric field. The remainder of the features and example features of the 2ndaspect are as described above with respect to the 1staspect.

[0110] In a 3rdaspect, the magnetic field is perpendicular to the first electric field and the plane of the monolayer. The remainder of the features and example features of the 3rdaspect are as described above with respect to the 1stor 2ndaspects.

[0111] An embodiment of a device according to the 5thexample embodiment is shown in Fig.24. The device 100 comprises: a graphene ribbon 102 according to any of the aspects of the 1stexample embodiment; a means 104 (shown as 104a, 104b, 104c) for generating an electric field, the means 104 configured to generate: a first electric field 106a, the first electric field being homogenous in a first region 108a that comprises a first portion of the graphene ribbon 110a, the first electric field being perpendicular to the direction ofUCI-00825 periodicity ^⃗^; a second electric field 106b, the second electric field being homogenous in a second region 108b that comprises a second portion of the graphene ribbon 110b, the second electric field being antiparallel to the first electric field, a third electric field 106c, the third electric field being homogenous in a third region 108c that comprises a third portion of the graphene ribbon 110c. Although shown as parallel to the second electric field, the third electric field 106c can be parallel to the first electric field 106a and antiparallel to the second electric field 106b. The second region 108b is disposed next to the first region 108a, and the third region 108c is disposed next to the second region 108b. The means 104 for generating the electric field is configured to translate the first region, the second region, and the third region, each along the plane of the monolayer 120. The device further comprises a means 112 for applying a magnetic field 114 to the first, the second, and the third regions 108a,b,c, the magnetic field 114 being perpendicular to the electric fields 106a,b,c and the plane of the monolayer 120. EXEMPLIFICATION

[0112] To demonstrate the design is experimentally accessible; experimental evidence on the synthesis and characterization of a representative GNR with partially zigzag chevron type edges (pzc-GNR) is provided in Fig.1A.

[0113] Fig.1A illustrates the structure of a unit cell 100 of the pzc-GNR. It has a mirror plane 102 and contains four zigzag segments, mirrored across axis a of the mirror plane 102.

[0114] The band structures of GNRs are known to depend strongly on the width and edge structures. Zigzag edges, as shown in Fig.1A, lead to localized states with energies near the Fermi level. Lining up zigzag-like segments in a strategic way can therefore form bands with dispersion and modulate the density of states (DOS) of graphene near the Fermi energy. For square-root GNRs, we place four zigzag segments in a mirror symmetric way. The coupling between those segments alternate between two distinct values (t1 and t2). An example pzc-UCI-00825 GNR is illustrated in Figs.1A and 1B. When a transverse electric field is applied, the electrostatic potential of electrons is changed, including those near the zigzag edges. It is assumed two zigzag segments receieve an increase of potential by +m (the two get the same due to the mirror symmetry) and another two by –m, given that the center atom obtains zero. This allows the mapping of the bands of GNRs near the Fermi energy to the one-dimensional square-root model.

[0115] Fig.1B illustrates the coupling strength between the zigzag segments, which alternate and are illustrated by straight lines 104 and dashed lines 106.

[0116] Fig.1C is a graph illustrating a transverse electric field applied to the GNR and changes in the electrostatic potential energy of atoms, especially for the four zigzag segments. The electronic bands near the Fermi energy can be mapped to a square-root model as shown in Fig.1D, which is characterized by four lattice sites containing mirror symmetric on-site energy terms and are coupled by alternating hopping strength t1 and t2.

[0117] The unit cell in the square-root model contains four spinless lattice sites with on-site potentials and are coupled by nearest-neighbor hopping (Fig.1D). The tight-binding Hamiltonian in the momentum space is given by:where m and -m are the on-site potentials, ^^^and ^^ଶare hopping parameters, as indicated by dash and solid lines, s is the distance between the first and second lattice sites, and b the length of the unit cell. Because the model has a mirror symmetry, the distance between thesecond and third sites is thus ( ^ଶ − ^^). It should be noted that there are many other possiblestructures of GNRs where the square-root model can be realized and they share the same setUCI-00825 of topological properties, as long as there are four zigzag segments arranged in a mirror symmetric way with alternating hopping strength. The zigzag segment is conceptual and may contain chemically a mixture of one or more zigzag edges with other edges such as armchair and chevron.

[0118] The synthesis of pzc-GNR is shown in Fig.2A-F. Fig.2A illustrates a reaction scheme for the synthesis of monomer and pzc-GNR during on-surface synthesis of pzc-GNR. A dimethyl was specifically designed to contain halogenated monomer precursor, 2,7- dibromo-10,13-dimethyl-9,14-diphenylbenzo[f]tetraphene, (monomer T), to produce a partially zigzag chevron-type graphene nanoribbon (pzc-GNR) containing four zigzag segments arranged in a mirror symmetric manner, similar to a four-lattice tight-binding square-root model. The monomer-T was synthesized from 5,10-dibromo-1,3-diphenyl-2H- cyclopenta[l]phenanthren-2-one (1) using Diels-Alder reaction with 2-amino-3,6- dimethylbenzoic acid (Fig.2A, top). A technique for synthesizing pzc-GNRs, begins by depositing monomer-T onto an Au(111) surface by sublimation in an ultra-high vacuum (UHV). The molecules were then coupled using Ullmann coupling by annealing at 280°C for 10 minutes, forming long-chain homo-polymers (polymer-T in Fig.2A, middle). These polymers were transformed into pzc-GNRs through a high-temperature on-surface cyclodehydrogenation reaction. Most of the polymers transformed during the second step annealing process at 380°C for 5 minutes, and a complete transformation occurred during the final, third step annealing at 450°C for 2 minutes, where the polymers underwent cyclization to form the desired graphene nanoribbons (Fig.2A bottom).

[0119] Fig.2B displays a representative large-area topographic image recorded by scanning tunneling microscopy (STM) on a polymer-T sample after it was annealed at 280°C. In this STM image, the edges of the polymer chains demonstrate bright dots, which correspond to the nonplanar uncyclized fragments of the polymer. A careful observation shows that theseUCI-00825 bright features are not regular and the chains also contain extended darker regions, suggesting that at this temperature the polymer-T is already partially cyclized. The large-area STM feedback scan taken on a sample after annealing at 380°C shows that most of the bright features at the edges of the polymer chains disappeared (Fig.2C), although some rare bright dots can still be observed. A low-coverage STM image recorded on a sample after annealing at 450°C (Fig.2D) demonstrates a complete cyclodehydrogenation reaction of the polymer chains, where all bright dots completely disappeared. This process results in the formation of the desired planar pzc-GNRs. Fig.2E shows a close-up topographic STM image of a representative isolated pzc-GNR on Au(111) after annealing at 450°C. The chemical structure overlay on this STM image exhibits the formation of the desired ribbon. A high-resolution non-contact atomic force microscopy (nc-AFM) image shown in Fig.2F demonstrates a perfect agreement of the structure of pzc-GNR with the chemical structure shown in the reaction scheme in Fig.2A.

[0120] A band structure of the pzc-GNR without any electric field is shown in Fig.3A. It has a central gap of 0.83 eV, according to density-functional theory (DFT) calculations performed with the Quantum Espresso package. The pzc-GNR has 154 valence bands and a primary focus on the middle four bands (red) near the Fermi level (numbered 153 to 156). The applying of a transverse electric field of 0.1V / ˚A changes the central gap width only very slightly (0.82 eV). As can be easily seen, the field opens up two gaps at k = π / a. These two new gaps (0.20 eV) are denoted as finite gaps, to differentiate them from the central gap. In Fig.3D and Fig.3H, the parameters for the corresponding square-root model are obtained by fitting the gap width to DFT results. The resultant band structure has four bands in total and closely matches Fig.3B and Fig.3F.

[0121] The Zak phase (^^^) is often used to characterize one-dimensional systems. It is in general an origin-dependent value. For a quasi-one-dimensional mirror-symmetric systemUCI-00825 with the origin set to the mirror plane, the Zak phase only takes 0 or π (mod 2π). Under fthis circumstance, the Zak phase is directly related to the number of edge states and can be used as a topological invariant. It is closely related to the electronic polarization along the periodic direction by:where n is the band index, and the sum is over occupied bands. A total nonzero (π) Zak phase indicates a nonzero polarization, and thus an nontrivial phase. The Zak phase for each of the four bands can be calculated according to:where ^^^,^is the nth periodic part of the Bloch function ^^^,^and BZ is the Brillouin zone in one dimension. With mirror symmetry, the calculation can be simplified to:where M is the mirror symmetry operator and X is the middle of the BZ with k = π / a.

[0122] Eq.4 relates the parity of the wave function to the Zak phase of the band. By comparing Figs.3C to 3G, it is noticed that the lower pair of bands switches parity under opposite directions of electric field, and so does the upper pair. The parity at the gamma point, however, persists. This indicates a band inversion at the two finite gaps caused by flipping the direction of the electric field. A horizontal comparison between Figs.3C and Fig.3E, or between Figs.3G and Fig.3I shows that the band inversion is exactly what can be expected from the square-root model. The Zak phase for the middle four bands in GNRUCI-00825 can then be obtained according to Eq.4. If the system is in ground state, i.e., filled to the Fermi level, then the topological phase is not affected by the transverse electric field. The summation of Zak phase for the lower two bands remains constant at 0 (mod 2π), regardless of variations in the electric field. This is consistent with previous work showing that the topological behavior of such a GNR is determined by the relative strength of t1 and t2, in analogy to the SSH model. In contrast, the Zak phase of the lowest band and the summation of the Zak phase for the lower three bands are determined by the direction of the electric field. When the system is gated such that a typical highest occupied band (HOMO) becomes unoccupied or the lowest unoccupied band (LUMO) becomes occupied, the topological phase then can be successfully tuned by the externally applied electric field.

[0123] Figs.3A, 3B, and 3F are band structures of pzc-GNR without electric field, under a +0.1V / ˚A, and a -0.1V / ˚Aelectric field, respectively. The structure has 154 valence bands and the middle four bands (numbered in a) are compared to the square-root model. The applying of a transverse electric field opens up two finite gaps at k = π / a between 153 and 154, and 155 and 155 bands. A band inversion at the two finite gaps is induced by switching the direction of the electric field, which is supported by the change of parity of the wave functions for the middle four bands at k = π / a. This can be seen by comparing Fig.3C and Fig.3G, which are the distribution of amplitude of wave functions. As the GNR is mirror symmetric, the Zak phase of the four bands can therefore be calculated. It suggests a topological phase change by electric field if the system is filled up to 153th or 155th bands. Boundary states are expected to exist within the gap labeled in red. Figs.3D and 3H are band structures of the square-root model with parameters chosen according to Figs.3B and 3F. The resultant parity of the wave functions at the middle of the Brillouin zone matches that of pzc-GNR, as shown by comparing Figs.3E to 3C, and Figs.3I to 3G. Fermi energies are labeled by red dashed lines. The green plus and orange minus signs indicate the parity of the wave function.UCI-00825

[0124] The Zak phase calculated above is for the middle four bands only. To see if it is able to predict the behavior of pzc-GNR, an approach of Wannier charge centers is taken to get the polarization from DFT calculations. The polarization is related to the location of Wannier centers according to:where ^^^is the Wannier charge center. As discussed, some embodiments that may be of interest have n iterated from 1 to 153, and from 1 to 155. Utilizing the WANNIER90 package, the Wannier centers of the lower 153 and 155 bands are obtained. It may be of interest to look at those projected from the ^^௭orbitals of carbon atoms. In Fig.4A, when GNR has 153 occupied bands and is under a positive field, it is found that the location of Wannier charge centers from 35 ^^௭orbitals is in analogy of that in the squre-root model (The unit cell has 72 carbon atoms in total). They are distributed in a mirror symmetric way except for the one being cut by the left cell boundary, which gives a total polarization of 0.5. Similarly, for GNR with 155 occupied bands, 37 of the wannier centers are plot in Fig.4D. There are two being crossed by the boundary line, which together yields a polarization of 0. Next, all Wannier centers are summed up, giving the polarization of the whole system according to Eq.5. With 153 occupied bands, a quantized polarization of 0.5 is found for positive field, which goes to 0 when the direction of the field is flipped (as shown in Fig. 4C). A similar quantized behavior but with opposite trend is observed when the occupation is 155 (Fig.4D). Meanwhile, a gap closing and reopening is observed between the 153 and 154 bands and between the 155 and 156 bands as the electric field sweeps from a positive value to a negative value (Fig.4E). The results demonstrate that the topological behavior of 7Z-GNR is predicted by the tight-binding square-root model.UCI-00825

[0125] In Fig.4A, assuming the pzc-GNR is filled up to the 153th band, the polarization is calculated using the Wannier charge center methods. The location of the Wannier charge centers from the 35 ^^௭orbitals of carbon atoms under a positive field is consistent with the simple squre-root model with one center at the boundary of the unit cell.

[0126] In Fig.4C, a graph illustrates typical topological feature with quantization of 0 under negative electric field and 0.5 under positive electric field. Instead, if the system is filled up to the 155th band under positive field, the 37 Wannier centers add up to zero, which suggests a trivial phase, as shown in Fig.4B. In Fig.4D, similar to that shown in Fig.4C but in an opposite trend, the system can be switched to nontrivial under a negative field. As the field sweeps from a negative value to a positive one, the band gap (between 153th and 154th bands) decreases first and then increases, shown in Fig.4E. The results are consistent with Zak phase obtained from the square-root model.

[0127] A nontrivial Zak phase is a bulk property and it ensures the existence of robust boundary states, according to the bulk-boundary correspondence. To directly demonstrate such a topologically protected boundary state, an edge or an interface must be created. This can be achieved either by terminating the structure (joining with vacuum, which is a trivial insulator), or joining with a trivial phase. The first method requires the structure to have end structure commensurate with the unit cell, which is experimentally not realistic for pzc-GNR. Instead, a spatially varying electric field can be utilized to create an interface between an nontrivial and a trivial phase. In Fig.5A, a periodic 10-unit-cell-long pzc-GNR supercell is simulated under a step-function static electric field shown in Fig.5B using the cp2k package. Fig.5A also shows the isosurface of wave function of a boundary state with energy within the lower finite gap. Fig.5B illustrates the boundary state resulted from the applying of a step-function electric field. The state has electron density localized at the interface where the change of electric field occurs. In Fig.5C, when compared to the density of states (DOS) ofUCI-00825 pzc-GNR under homogeneous transverse electric field (black and gray curves), DOS under a step-functioned electric field shows four in-gap states in total, two in each finite gap. All are found to be localized, just like the one shown in Fig.5A. According to the bulk properties of pzc-GNR filled up to 153th, there should be a boundary state coming from the nontrivial phase (i.e., GNR under positive field). The boundary state should be localized and have an energy level within the lower finite gap. Similarly, GNR filled up to the 155th band is nontrivial under a negative electric field (and trivial under a positive field), which should result in boundary states within the upper finite gap. These are indeed supported by the first- principle calculations. As shown by the plot of density of states in Fig.5C, there are several states residing with the two finite gaps. In contrast, if the system is under a homogeneous transverse electric field, there is no sign of such in-gap states. In addition, these in-gap states are found to be localized at the interfaces where the change of electric field occurs. The isosurface and electron density distribution for one of the states are plotted in Fig.5A and 5B. A finite pzc-GNR molecule with the two end structures is also simulated and experimentally verified. The in-gap state localized at the interface still exist, as expected.

[0128] Computational Details:

[0129] First-principles calculations were carried out using Quantum Espresso package. Structures were relaxed until all forces fall below 0.025 eV / Å. Standard solid-state pseudopotentials (SSSP[1.3.0][PBE][efficiency]), a kinetic energy cutoff of 100 Ry for wave functions, a kinetic energy cutoff of 800 Ry for charge density, 16 k-points for scf, 24 k- points for nscf were used.20 Å of vacuum was added in the non-periodic directions. Electric field was added in the form of a saw-tooth potential by setting tefield to true. Amplitude of wave function 2604 and 2612 of Fig.26 were obtained by tight-binding calculations implemented using the PythTB package assuming nearest-neighbour hopping of ^^௭orbitals of carbon atoms. The hopping parameter was set to -2.7 eV. Calculations in Fig.27 wereUCI-00825 implemented using the CP2K package. Structures were relaxed until all forces are below 0.015 eV / Å. Perdew-Burke-Ernzerhof (PBE) functional, double-zeta valence basis set, and pseudopotentials of Goedecker- Teter- Hutter (GTH) were used. External potential was added according to the function described in Sec.5. To get DOS from eigenvalues, a smearing of 0.5 meV was adopted.

[0130] Experimental Details:

[0131] All the starting materials and reagents used were purchased from commercial sources and used without further purification unless otherwise noted. The dry solvent 1,2-dichloro- ethane (DCE) was freshly distilled, employing a standard method before use. Chemical shifts are stated in parts per million (ppm, ^^), downfield from tetramethyl silane (TMS, ^^ = 0.00 ppm) and are referenced to residual solvent (CDClଷ, ^^ = 7.26 ppm (^H) and 77.00 ppm (^ଷC)). Mass spectrum was collected using Bruker Solari X.

[0132] 2,7-dibromo-10,13-dimethyl-9,14-diphenylbenzo[f]tetraphene (monomer-T) may be synthesized following a reported procedure. Fig.11 is a schematic diagram illustrating a reaction scheme for synthesis of monomer-T, according to embodiments of the present disclosure. As shown in Fig.11, synthesis of compound 1: 5,10-dibromo-1,3-diphenyl-2H- cyclopenta[l]phenanthren-2-one (1) was synthesized according to a reported procedure. As shown in Fig.11, in the first step, 1 (1080.5 mg, 2.0 mmol), and 2-amino-3,6- dimethylbenzoic acid (346.9 mg, 2.1 mmol) were added to a two-neck round bottom flask with a magnetic stirring bar, followed by 10 mL of 1,2-dichloro-ethane (DCE). The mixture was heated to 90°C with constant stirring. A solution of isopentyl nitrite (0.6 mL, 4.6 mmol) in 4 mL DCE was slowly added into the reaction mixture via syringe, and stirring was continued for approximately 30 minutes leading to a color change. After cooling to room temperature, the solvent evaporated, and the residue was precipitated out using hexanes. The precipitate was filtered and washed with hexanes and dried under vacuum, yielding a light-UCI-00825 yellow powder. In the second step, the crude powder was transferred to a reaction tube, followed by the addition of 1.0 mL of diphenyl ether and heated to reflux for 30 minutes. The reaction mixture turned dark brown in color and was cooled to room temperature. Purification by flash column chromatography using 10% of dichloromethane (DCM) in hexanes as eluent (R^= 0.61) to give monomer-T as a pale-yellow powder (678 mg, 55% yield).

[0133] ^H-NMR (400 MHz, CDClଷ, ^^): 7.99 (d, 2H, and Ar-H), 7.52-7.42 (m, 10H, and Ar- H), 7.39 (d, 1H, and Ar-H), 7.36 (d, 1H, and Ar-H), 7.14 (s, 2H, and Ar-H), 7.11 (d, 2H, and Ar-H), and 1.82 (s, 6H, and methyl-H) ppm.

[0134] ^ଷC-NMR (100 MHz, CDClଷ, and ^^): 142.29, 135.21, 134.50, 133.54, 133.21, 132.96, 132.81, 130.15, 129.73, 129.47, 129.10, 128.23, 125.89, 124.70, 120.08, and 23.65 (methyl-C), ppm.

[0135] HR-MS (MALDI FT-ICR, TCNQ Matrix), m / z: 614.02208 (calculated m / z = 614.02448). MALDI FT-ICR and NMR spectra are provided in Figs. S1 - 3.

[0136] To synthesize pzc-GNRs, we deposited monomer-T onto an Au(111) surface by sublimation in an ultra-high vacuum (UHV). The molecules were then coupled using Ullmann coupling by annealing at 280°C for 10 minutes, forming long-chain homo-polymers. These polymers were transformed into pzc-GNRs through a high-temperature on-surface cyclodehydrogenation reaction. Most of the polymers transformed during the second step annealing process at 380°C for 5 minutes, and a complete transformation occurred during the final, third step of annealing at 450°C for 2 minutes, where the polymers underwent cyclization to form the desired GNRs.

[0137] Square-root Tight-binding Model

[0138] Here, the square-root tight-binding model is considered, as shown in Fig.1D. Solving the Hamiltonian given in Eq.6 (below) gives the four energy bands:UCI-00825The model has a chiral symmetry Γ^^Γற = −^^ characterized by the operator:the matrix of the operator is written in the canonical convention, and obtains k dependency in the periodic convention. The chiral symmetry is nonsymmophic in nature. The time-reversalsymmetry is satisfied by ^^^^(^^)^^ = ^^⋆(^^), withTherefore, the system can be categorized to the BDI class. The system also has a spatial mirror symmetry MH(k)M = H(−k):

[0139] The Zak phase is a value dependent on the choice of unit cell and the location of origin. In this work the unit cell is the one drawn in Fig.1D and the origin is set to the centerof the cell. Under this circumstance, the mirror symmetry requires Zak phase for each band:^^^, ^^ଶ, ^^ଷ, ^^ସ ∈ 0, ^^ ^^^^^^ 2^^ (10)And the chiral symmetry requires:^^ସ − ^^^ = ^^ଷ − ^^ଶ = ^^ ^^^^^^ 2^^ (11)UCI-00825

[0140] All possible combinations of Zak phase from the requirements of symmetry are listed in Table 1. In the four-lattice square-root model, the four cases in Table 1 can be achievedby controlling the sign of m and ^^ଶ + ^^ଶ ଶଶ − ^^^ , as elaborated in Table 2. The values of Zakphase are summarized in Table 1, where it is assumed ^^^and ^^ଶare negative and a positive up-pointing electric field leads to a positive potential for the first lattice site in Fig.1D. Table 1: Possible combinations of Zak phase from symmetry combinationsTable 2: Zak phase in the square-root model

[0141] From Table 2, it can be seen that when keeping t^and tଶconstant, switching the signof m always alternates the Zak phase. This means that (γ^ + γଶ + γଷ) also alternates,whereas (^^^ + ^^ଶ) remains constant. Therefore, it is the finite gaps, instead of the central gap,UCI-00825 that can host topologically protected boundary state. Meanwhile, the SSH model can be regarded as a special case of the square-root model where m equals zero. The observation thatthe topological phase at half-filling depends on |^^ଶ| − |^^^| is consistent with the presentdisclosure.

[0142] Realizing the Square-root Model in pzc-GNR

[0143] The Zak phase may be calculated in two ways. One is utilizing Eq.9 by comparing the parity of the wave function. Another one is the Wannier charge center method based on DFT results. The two results are the same and listed in Table 3 here: Table 3: Zak phase for the middle four bands in GNR

[0144] It is noted that the pattern matches that of case C and D in Table 1. As C and D canbe achieved by letting |t^| < |tଶ| (see Table 2), the results of square-root model arecompared to that of pzc-GNR, assuming 0 > t^> tଶ. The parameters t^, tଶ, min the square- root model are chosen such that its gap width at the edge and middle of the Brillouin zone is the same as the band structure calculated by DFT method. Meanwhile, a positive m GNRs isUCI-00825 assumed under a positive field along the direction shown in Figs.1A-D. The results are shown in Figs.6A-B, which demonstrate good agreement between the two band structures. In addition, the hopping parameters for the GNR with and without electric field are very close, further suggesting it is reasonable to treat the four zigzag segments in a tight-binding way.

[0145] Fig.6A illustrates the band structure of pzc-GNR without electric field calculated by DFT method, compared to the band structure of the square-root model. The parameters for the square-root model are obtained by matching the gap width indicated in blue color. Fig. 8B shows the same comparison between band structures, with an electric field of 0.1 V / Å.

[0146]

[0001] Next the parity of wave function of the square-root model and pzc-GNR are compared. To do so, the band structure of pzc-GNR is calculated by the tight-binding model assuming only nearest-neighbor hopping between carbon atoms, using the PythTB package. In Figs.7A-M, the hopping parameter is set to -2.7 eV. To simulate the effects of electric field, an on-site potential proportional to the distance of the atom to the center along the yaxis is assumed, i.e., ^^ · (^^ − ^^^), where E is the strength of the field set to +0.05V / Å forFig.7B and –0.05V / Å for Fig.7E, y is the vertical coordinate of the carbon and ^^^is the vertical coordinate of the center carbon atom. The results show gap opening at the middle of the Brillouin zone, especially for the four bands near the Fermi level. The amplitude of the wave functions at k = π / a is plotted in Fig.7D. At k = π / b, it can be seen that the parity for all four bands are the same for the GNR and the square-root at both negative and positive field (see Figs.7D and 7G, 7J and 7M). The difference between the two is that the parity at k = 0, which results in different Zak phase. This is due to the fact that at gamma point, the transverse electric field does not change much of the band structure, which is mainly determined by the specific chemical structures of the GNR, such as how many columns of benzene rings exist between nearby zigzag segments. Regardless, at this point it should be noted that the square-root model ensures a topological phase change by electric field for aUCI-00825 class of GNRs, given that the normal LUMO is occupied or the HOMO is unoccupied. Below another GNR is shown to further demonstrate this point.

[0147] Constructing Effective Square-Root Model In Pzc-GNR:

[0148] The Zak phase may be calculated in two ways. One is utilizing Eq.9 by comparing the parity of the wave function, which gives the Zak phase for the middle four bands (Table 3). Another one is the Wannier charge center method based on DFT results, which can yield a result encompassing contributions from all low-lying bands. We note here that the two results are consistent. To get meaningful parameters (m, ^^^, ^^ଶ) to describe GNR, we first notice that the pattern in Table 3 matches that of case C and D in Table 1. As C and D can be achievedby letting |^^^| < |^^ଶ| (see Table 2), we next compare the results of square-root model to thatof pzc-GNR, assuming 0 > ^^^ > ^^ଶ. The parameters ^^^, ^^ଶ, ^^ in Eq. 6 are chosen such that itsgap width at the edge and middle of the Brillouin zone is the same as the band structure of GNR calculated by DFT methods. Meanwhile, we assume a positive m for GNRs under a positive field along the direction shown in Fig.1. The results are shown in Fig.15, which demonstrates good agreement between the two band structures. The left half of Fig.15 illustrates band structure of pzc-GNR without electric field calculated by DFT method, compared to the band structure of the tight-binding square-root model. The parameters for the square-root model are obtained by matching the gap width indicated in blue color. The right half of Fig.15 illustrates the same comparison between band structures, with an electric field of 0.1 V / Å. In addition, the hopping parameters for the GNR with and without electric field are very close, further suggesting it is reasonable to use a square-root Hamiltonian to predict properties of GNRs.

[0149] Although we used DFT results from GNRs under a transverse electric field of 0.1 V / Å, a smaller, more achievable strength can also be applied without affecting the analysis presented disclosed herein. As demonstrated in Fig.16, an electric field of 0.02 V / Å inducesUCI-00825 a gap of 40.8 meV, which is sufficiently large for detecting in-gap states by common techniques such as STM.

[0150] In Fig.2, the parity of wave functions is obtained by solving the tight-binding model carried out with the PythTB package assuming only nearest-neighbor hopping between carbon atoms. The hopping parameter is set to -2.7 eV. To simulate the effects of electric field, we assume an on-site potential proportional to the distance of the atom to the centeralong the y axis, i.e., ^^ ⋅ (^^ − ^^^), where E is the strength of the field set to 0 for 1702 of Fig.17, +0.05 V / Å for Fig.18c, and –0.05 V / Å for 1720 of Fig.17, y is the vertical coordinate of the carbon and ^^^is the vertical coordinate of the center carbon atom.1708 and 1722 of Fig. 17 are replicates of Fig.2604 and 2612 of Fig.26. For comparison, in 1716 of Fig.17 we plot the band structure from the DFT calculations again. The band structures of pzc-GNR in 1702 and 1716 are quite similar. The tight-binding results show gap opening at the middle of the Brillouin zone, especially for the four bands near the Fermi level.

[0151] At ^^ = ^^ / ^^, the parity for all four bands in GNRs are the same as in the square-rootmodel at both negative and positive fields (see 1708, 1714, 1722, and 1728 of Fig.17). Nowwhat’s different between the two is the parity at ^^ = 0 (see 1704, 1710, 1718, and 1724 ofFig.17), which results in different Zak phase. This is due to the fact that at gamma point, the transverse electric field does not affect much of the band structure. The parity of wavefunctions at ^^ = 0 is mainly determined by the specific chemical structures of the GNRs,such as whether they contain other types of edge configurations. The square-root model onlypredicts the parity at ^^ = ^^ / ^^, but not that at ^^ = 0. Regardless, the model guarantees atopological phase change by switching the direction of electric fields for a class of GNRs, given that the normal LUMO is occupied or the HOMO is unoccupied. Below we show another GNR to further demonstrate this point.UCI-00825

[0152] 7-AGNR-S (1,3)

[0153] 7-AGNR-S(1,3) is shown in Fig.8A. A similar mapping to the square-root model can be made. The Zak phase for the middle four bands are labeled. The pattern agrees with case A and B in Table 1, which suggest a larger |t^| than |tଶ|. Similar to pzc-GNR, with k = π / b, the two models give the same parity (see Figs.8D and 8G, 8J and 8M. In addition, the parity at gamma point is also the same, which results in exactly the same Zak phase for the four bands in GNR and in square-root (see Fig.8A). It shares the same topological feature as pzc- GNR that it has a phase change upon changing the direction of electric field with 203 or 205 occupied bands.

[0154] Another Example Of Square-Root GNR: 7-AGNR-S(1,3)

[0155] 7-AGNR-S(1,3) shown in 1802 of Fig.18 is named according to the present disclosure. The band structures in 1806, 1816, and 1820 of Fig.18 were obtained in the same way as in Fig.17. The Zak phase for the middle four bands are labeled. We see the pattern agrees with case A and B in Table 1, which suggest a larger |^^^| than |^^ଶ| (agrees with the present disclosure). For the corresponding square-root model, the parameters are chosen suchthat the gap width in 1812 of Fig. 18 matches that in 1806, given that 0 > ^^ଶ > ^^^, and ^^ >0 for a positive electric field. Similar to pzc-GNR, with ^^ = ^^ / ^^, the two models give thesame parity (see 1808 and 1814, 1822 and 1828). In addition, the parity at gamma point is also the same, which results in exactly the same Zak phase for the four bands in GNR and in square-root tight-binding model (see Fig.1804 and 1810, 1818 and 1824). It shares the same topological feature as pzc-GNR that it has a phase change upon changing the direction of electric field with 203 or 205 occupied bands (The structure has 204 valence bands). Boundary States

[0156] In Fig. 4A a step-function electric field described by 0.3 · 2 / ^^ [^^^^^^^^^^^^(2^^) −^^^^^^^^^^^^(2^^ − ^^^ ) + ^^^^^^^^^^^^(2^^ − 2^^^ )] (^^ / Å) is applied, where x is the horizontal πUCI-00825 coordinate and ^^^is the length of the supercell equaling to 172.7Å. The maximum field strength is thus 0.3 V / Å. In this periodic system, two interfaces are created by such a electric field (orange dashed line in Fig.9B). Therefore, there should be four boundary states in total. Two have an energy within the lower finite gap and another two within the upper finite gap. In Fig.8A, the eigenvalues of states is plotted around the Fermi level. There are two localized states appearing in the lower finite band gap, each located at one interface created by the step-function electric field (Fig.8C). Similarly, other two localized states have a energy within the upper finite gap (Fig.8B).

[0157] Fig.8A illustrates the chemical structure of 7-AGNR-S(1,3). It has a mirror plane and four zigzag segments per unit cell. Figs.8C, 8F, and 8I are band structures of 7-AGNR- S(1,3) calculated in a tight-binding model assuming only nearest-neighbor hopping between carbon atoms. Numbers in Fig.8F indicate sequence of bands. Figs.8D and 8J are plots of amplitude of the wave function for the four bands near the Fermi level at the middle of the Brillouin zone under a positive and negative electric field, respectively. Color stands for relative phase and saturation of color indicates strength of amplitude. Figs.8G and 8M are amplitude of wave function calculated from the square-root model. They match well with Figs.8D and 8J. Similarly, Figs.8E and 8K agree with Figs.8B and 8H at gamma point. The minus and plus symbols in the plot indicate relative phase. The topological invariant can be obtained by comparing the parity of the wave function at Γ and X.

[0158]

[0002] To simulate a finite molecule of pzc-GNR, we create a long molecule containing 10 unit cells with the two ends terminated by a structure detected in experiments.A similar step-function according to 0.3 · 2 / ^^ ^^^^^^^^^^^^(^^ − ^^ி) (^^ / Å) is then applied,where ^^ிis the length of the cell equaling to 250Å and the GNR is placed in the middle of the cell. In Fig.9A, there are two in-gap boundary states in total, as there is only oneUCI-00825 interface created. The isosuface of the wave function of the two states is plot in Fig.9B and Fig.9C.

[0159] Fig.10A illustrates Eigenvalues near the Fermi level for a large molecule consisting of 10 unit cells of pzc-GNR under a step-function electric field. The left and right end structures are drawn according to experimental observations. Figs.10B and 10C are isosurface plots of the wave function residing at the upper finite gap, and the lower finite gap. They are localized at the interface where the switch of the electric field occurs.

[0160] Solitons In A Finite Molecule:

[0161] In Fig.27 the step-function electric field described by the following is applied: ^^ = 0.2where ^^ is the horizontal coordinate and ^^^is the length of the supercell equaling to 345.4 Å. The maximum field strength is thus 0.2 V / Å. ^^ determines how abrupt the electric field changes sign and is denoted as the width of domain wall. The electrostatic potential added toelectrons is then given by ^^ = ^^ ⋅ (^^ − ^^^), where ^^^ is the vertical coordinate of the centralatom. In this way we have equal number of electrons with positive and negative change of potential.

[0162] To simulate a finite molecule of pzc-GNR, we create a long molecule containing 10 unit cells with the two ends terminated by structures detected in experiments (1910 and 1912of Fig. 19). We then apply a similar step-function according to 0.2 ⋅ ଶగ^^^^^^^^^^^^(^^ −^^ி)](V / Å), where ^^ி is the length of the cell equaling to 250 Å. The GNR is placed such thatthe change of the field occurs at the boundary of the mirror symmetric unit cell (1906 of Fig. 19). There are now two in-gap boundary states in total (1902 of Fig.19). The isosufaces of the wave function of the two states are plotted in 1904 and 1908.UCI-00825

[0163] Solitons And Choice Of Unit Cell:

[0164] There are two types of unit cell that can be drawn in pzc-GNR, one is mirror symmetric (Fig.1) and the other is not (Fig.22). The properties of boundary states depend on which unit cell the boundary is commensurate with. With mirror symmetric unit cell, boundary states are topologically protected and are degenerate. Fig.20 illustrates the energy levels and spatial profile of boundary states calculated with DFT methods. Each of the finite gap owns two-fold degenerate states with localized spatial distribution. The degeneracy remains regardless of the width of the domain wall (which is set to 0.5 Bohr and 4 Bohr in 2008 and 2016 of Fig.20, respectively).2006 of Fig.20 is a replicate of 2702 of Fig.4.

[0165] With the second choice of unit cell without mirror symmetry, localized boundary states can still be generated but they are no longer degenerate (2102 of Fig.21). Comparing 2102 to 2108, it is obvious that the energy level for this type of soliton is influenced by the width of domain wall. These properties can be expected by considering a Rice-Mele model described bywhere ^^^is the on-site potential and ^^ is the hopping parameter. Note the term Rice-Mele model can also refer to model with alternating hopping strength. In this work the name is used to refer to the Hamiltonian above.

[0166] With a non-mirror-symmetric unit cell, solitons from pzc-GNR behave similarly to solitons in a Rice-Mele model. Their structural similarity is shown Fig.22. Soliton boundary states from Rice-Mele model have been well discussed before. Their existence is related to the nonsymmorphic chiral symmetry. The energy level is reported to be dependent on the width of domain wall. Here we use tight-binding model assuming nearest-neighbor hoppingUCI-00825 between carbon atoms only to simulate boundary states from GNRs when a step-function field is applied. Details of the tight-binding calculations have been mentioned in Sec.3. The effect of the electric field is simulated by modifying the on-site potential of carbon atoms according to the coordinates of carbon atoms. The maximum strength of the electric field is adjusted so that an effective gap (such as the one shown in 1706 of Fig.17) of 0.2 eV is created for all GNRs. We also consider another GNR (5-sGNR) that directly represents the Rice-Mele model when a transverse electric field is applied (2202 of Fig.22). It is named according to the original publication. The DFT calculated band structure of 5-sGNR is shown in 2210 of Fig.22, which is consistent with previous report. The applied field with a strength of 0.3 V / Å can open up a gap of 0.09 eV at X near the Fermi level. The boundary states are thus expected to exist near the Fermi level.

[0167] In Fig.23, it can be seen that boundary states in 5-sGNR and pzc-GNR with an non- mirror symmetric unit cell have similar behaviors. They indeed have a much stronger dependence on the width of domain wall, compared to pzc-GNR with mirror symmetric unit cell. A smoother domain wall pushes the energy of boundary states closer to the Fermi level (see inset 2302 and 2306 in Fig.23). The width of domain wall also affects the spread of the boundary state, which can be evaluated by calculating the second moment of the wavefunction. Fig.2304 and 2312 suggest a smoother domain yields a more localized boundary state for 5-sGNR, but a more spread state for pzc-GNR (with mirror symmetric terminating unit cell).

[0168] We further supplemented the above tight-binding results with DFT calculated boundary states in 5-sGNR. The electric field is applied as shown in Fig.25 with the same width of domain wall as in Fig.20 and Fig.21. The maximum strength is 0.3 V / Å. The 0.5% isosurface of the wavefunction of the two boundary states are plotted in 2502 and 2504, 2506 and 2508 of Fig.25 with an abrupt and a smooth domain wall, respectively. A smootherUCI-00825 domain wall results in an energy level slightly closer to the Fermi level, which is consistent with results obtained from the tight-binding method, although the energy dependency is not as obvious.

[0169] The above results imply that boundary states in 5-sGNR and non-mirror-symmetric pzc-GNR are of the same type, whereas they are different from mirror-symmetric square-root GNR. Although both types are localized and reside within gaps, they have a few distinctions. On one hand, the Rice-Mele type of boundary state can reside at the central gap, i.e., near the Fermi level at half filling. Samples don’t have to be gated to observe boundary states. In comparison, the square-root type only exist at the finite gaps below or above the Fermi level at half filling. On the other hand, the square-root type of boundary states have robust quantized charge, resulting from quantized Zak phase due to mirror symmetry (Eq.10), whereas Rice-Mele type does not. Depending on the specific requirements of potential applications, a square-root type, or a Rice-Mele type of boundary state can be chosen alternatively.

[0170] Referring now to Fig.15, band structure of pzc-GNR without electric field calculated by DFT method, compared to the band structure of the tight-binding square-root model is depicted at (a). The parameters for the square-root model are obtained by matching the gap width indicated in blue color. The same comparison between band structures, with an electric field of 0.1 V / Å, is depicted at (b). Referring now to Fig.16, band gap induced by the transverse electric field versus the strength of the field applied is depicted.

[0171] Referring now to Fig.17, 1702, 1706, and 1720 are band structures of pzc-GNR calculated in a tight-binding model assuming only nearest-neighbor hopping between ^^௭orbitals of carbon atoms. For comparison, 1716 is the band structure calculated with DFT methods.1708 and 1722 are plots of amplitude of the wave function for the four bands near the Fermi level at the middle of the Brillouin zone under a positive and negative electric field,UCI-00825 respectively. Color stands for relative phase and saturation of color indicates strength of amplitude.1714 and 1728 are amplitude of wave function calculated from the square-root model. They match well with 1708 and 1722.1704, 1710, 1718, and 1724 are the amplitude of wave function at the gamma point for the middle four bands. The square-root model only ensures the parity of wave function of GNR at X, but not at Γ.

[0172] Referring now to Fig.18, 1802 is the chemical structure of 7-AGNR-S(1,3). It has a mirror plane and four zigzag segments per unit cell.1806, 1816, and 1820 are band structures of 7-AGNR-S(1,3) calculated in a tight-binding model assuming only nearest-neighbor hopping between carbon atoms.1808 and 1822 are plots of amplitude of the wave function for the four bands near the Fermi level at the middle of the Brillouin zone under a positive and negative electric field, respectively. Color stands for relative phase and saturation of color indicates strength of amplitude.1814 and 1828 are amplitude of wave function calculated from the square-root model. They match well with 1808 and 1822.1804, 1810, 1818, and 1824 are the amplitude of wave function at the gamma point for the middle four bands. For this particular GNR, the parity of the middle four bands at both X and ^^ match that in the square-root model. As a result, the Zak phases are exactly the same (e.g., see text in 1806 and 1812).

[0173] Referring now to Fig.19, 1902 illustrates Eigenvalues near the Fermi level for a finite molecule consisting 10 repeating unit cells of pzc-GNR under a step-function electric field. The left and right terminating structures are drawn according to experimentally observed STM images (see 1810 and 1812).1806 illustrates spatial profile of the externally applied electric field.1804 and 1808 are 0.2% isosurfaces plots of the wave function residing at the upper finite gap, and the lower finite gap, respectively. They are localized at the interface where the switch of the electric field occurs. Scale bars: 2 nm.UCI-00825

[0174] Referring now to Fig.20, 2002 illustrates Eigenvalues near the Fermi level for a supercell with 20 repeating unit cells of pzc-GNR under a step-function electric field with the spatial profile shown in 2008.2004 and 2006 are 0.5% isosurface plots of the wave function of boundary states residing at the upper finite gap, and the lower finite gap (red dots). They are localized at the interface where the switch of the electric field occurs (middle and the side of the supercell). Boundary states are two-fold degenerate in each finite gap. An electric field with a larger width of domain wall as illustrated in 2016 is applied to the same supercell. Eigenvalues and corresponding localized boundary states are shown in 2010, 2012, and 2014. Boundary states remain two-fold degenerate.

[0175] Referring now to Fig.21, 2102 illustrates Eigenvalues near the Fermi level for a supercell with 20 repeating unit cells of pzc-GNR under a step-function electric field with the spatial profile shown in 2108. The unit cell is shifted from the one in Fig. S9 and is no longer mirror symmetric.2104 and 2106 are 0.5% isosurface plots of the wave function of boundary states residing at the upper finite gap, and the lower finite gap (blue dots). They are localized at the interface and are nondegenerate. An electric field with a larger width of domain wall as illustrated in 2116 is applied to the same supercell. Eigenvalues and corresponding localized boundary states are shown in 2110, 2112, and 2114. Energy level of the boundary states is influenced by the width of the domain wall.

[0176] Referring now to Fig.22, 2202 illustrates chemical structure of 5-sGNR as reported in.2204 illustrates pzc-GNR drawn in a non-mirror symmetric unit cell.2206 illustrates Rice-Mele model with a nearest neighbor hopping of t and on-site potentials of –m and +m. As illustrated at 2208, with the unit cell in 2204, solitons from pzc-GNR behave in a similar way to that from the Rice-Mele model.2210 DFT calculated band structure of 2202 with and without a transverse electric field. The applied field opens up a gap in the middle at X. TheUCI-00825 middle two bands can be explained by the Rice-Mele model 2206. Energy is with respect to the Fermi level.

[0177] Referring now to Fig.23.2302, 2306, and 2310 are plots of energy levels of boundary states versus the width of domain wall for 5-sGNR, pzc-GNR with a mirror symmetric unit cell, and a non-mirror-symmetric unit cell, respectively. Insets show eigenvalues of modes near the center of the gap. Black and red correspond to the smallest and the largest width of domain wall considered.2304, 2308, and 2312 are corresponding plots of second moment of the boundary states versus the width of domain wall. Results are obtained by solving tight- binding Hamiltonian considering nearest-neighbour hopping between ^^௭orbitals of carbon atoms.

[0178] Referring now to Fig.25, 2502 illustrates 0.5% isosurfaces of wavefunctions of boundary states generated in 5-sGNR when a step-function electric field 2504 is applied. The corresponding eigenvalues are shown in 2510.2506 is the same isosurface plot when a different electric field with a more smooth domain wall is applied 2508.2512 illustrates corresponding eigenvalues. Although not obvious, states number 3 and 4 are closer to zero comparing to 1 and 2.

[0179] Referring now to Fig.26, electronic band structures of pzc-GNR and band inversions by an electric field are shown.2602 and 2610 are band structures of pzc-GNR under a +0.1V / ˚A, and a -0.1V / ˚A electric field, respectively. The middle four bands (black solid line) are compared to the system without an electric field (blue dashed line). The GNR has 154 valence bands and the applying of a transverse electric field opens up two finite gaps at k = π / a between 153 and 154, and 155 and 155 bands. A band inversion at the two finite gaps is induced by switching the direction of the electric field, which is supported by the change of parity of the wave functions for the middle four bands at k = π / a. This can be seen by comparing 2604 to 2612, which are the distributions of amplitude of wave functions. As theUCI-00825 GNR is mirror symmetric, the Zak phase of the four bands can therefore be calculated according to Eq.4. It suggests a topological phase change by electric field if the system is filled up to 153th or 155th bands. Boundary states are expected to exist within the gap labeled in red.2606 and 2614 are band structures of the square-root model with parameters chosen according to the gap width in 2602 and 2610. The resultant parity of the wave functions at the middle of the Brillouin zone matches that of pzc-GNR, as shown by comparing 2608 to 2604, and 2616 to 2612. Energy is with respect to the Fermi level. The green plus and orange minus signs indicate the parity of the wave function.

[0180] Referring now to Fig.27, Topological solitons are shown.2702 shows periodic supercell of pzc-GNR comprising 20 mirror-symmetric unit cells is subject to a step-function electric field, which results in a localized boundary state (soliton state) with the 0.5% isosurfaces of the wave function plotted.2704 shows the same electric field is applied on a supercell consisting of 20 non-mirror-symmetric unit cells, which also yields a localized boundary state.2706 shows a positive and constant electric field of 0.2V / ˚A is applied to the left domain of the supercell, which is transitioned to a negative one for the right domain. 2708 shows, compared to DOS of pzc-GNR under homogeneous transverse electric field without domain walls (black and gray), DOS under a step-functioned electric field (red) shows four in-gap soliton states in total, two in each finite gap and are degenerate (peaks filled in red).2710 shows, with another choice of domain wall configuration, the corresponding unit cell commensurate with it becomes non-mirror-symmetric. As a result, the soliton states within the finite gap can still be localized but are no longer degenerate (peaks filled in blue). Vertical dashed line indicates the center of the domain wall where the electric field changes sign. The star and triangle symbols in 2708 and 2710 refer to soliton states of which the wave functions are plotted in 2702 and 2704.UCI-00825

[0181] The descriptions of the various embodiments of the present disclosure have been presented for purposes of illustration, but are not intended to be exhaustive or limited to the embodiments disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The terminology used herein was chosen to best explain the principles of the embodiments, the practical application or technical improvement over technologies found in the marketplace, or to enable others of ordinary skill in the art to understand the embodiments disclosed herein.

Claims

UCI-00825 CLAIMS What is claimed is:

1. A graphene ribbon, comprising: a planar monolayer of sp2-hybridized carbon atoms arranged in a hexagonal lattice characterized by a direction of periodicity ^⃗^, the hexagonal lattice structure comprising a unit cell; wherein: each unit cell is symmetric with respect to a plane perpendicular to the plane of the monolayer (a mirror plane); each unit cell comprises 4n zigzags edges, wherein n is a natural number; and wherein the minimal number of hexagons intersected by a line lying in the plane of the monolayer parallel to the mirror plane is 3.

2. The graphene ribbon of Claim 1, wherein n = 1.

3. The graphene ribbon of any one of Claims 1 or 2, wherein the unit cell is represented by the following structural formula:.

4. A method of initializing a qubit, comprising: providing a graphene ribbon of any one of Claims 1 through 3;UCI-00825 applying a first homogenous electric field to a first portion of the graphene ribbon, the first electric field having a component perpendicular to the direction of periodicity ^⃗^; applying a second electric field to a second portion of the graphene ribbon, the second electrical field having a component that is antiparallel to the first electric field, the second portion of the graphene ribbon disposed next to the first portion; applying a magnetic field to the graphene ribbon, the magnetic field having a component that is perpendicular to the first electric field and / or second electric field and the plane of the monolayer, thereby initializing a first qubit.

5. The method of Claim 4, further comprising: applying a third homogenous electric field to a third portion of the graphene ribbon, the third electric field being having a component that is parallel to the first electric field, the third portion disposed next to the second portion of the graphene ribbon; thereby initializing a second qubit.

6. The method of any one of Claims 4-5, wherein: the first electric field is perpendicular to the direction of periodicity ^⃗^; and the second electric field is antiparallel to the first electric field.

7. The method of Claim 5, wherein: the first electric field is perpendicular to the direction of periodicity ^⃗^; the second electric field is antiparallel to the first electric field; and the third electric field is parallel to the first electric field.

8. The method of any one of Claims 4-7, wherein the magnetic field perpendicular to the first electric field and the plane of the monolayer.UCI-00825 9. A method of causing an interaction between qubits, comprising: providing a graphene ribbon of any one of Claims 1 through 3; providing a means for generating an electric field, the means configured to generate: a first electric field, the first electric field being homogenous in a first region that comprises a first portion of the graphene ribbon, the first electric field having a component perpendicular to the direction of periodicity ^⃗^; a second electric field, the second electric field being homogenous in a second region that comprises a second portion of the graphene ribbon, the second electric field having a component that is antiparallel to the first electric field, a third electric field, the third electric field being homogenous in a third region that comprises a third portion of the graphene ribbon, the third electric field having a component that is parallel to the first electric field, wherein the second region is disposed next to the first region, and the third region is disposed next to the second region; and further wherein the means for generating the electric field is configured to translate the first region, the second region, and the third region, each along the plane of the monolayer; the method further comprising: applying a magnetic field to the first, the second, and the third regions, the magnetic field having a component that is perpendicular to the first electric field and / or second electric field and the plane of the monolayer, thereby initializing a first and a second qubits; and translating the first region and / or the third region to be proximal to one another, thereby causing an interaction between the first qubit and the second qubit.

10. The method of Claim 9, wherein: the first electric field is perpendicular to the direction of periodicity ^⃗^; the second electric field is antiparallel to the first electric field; and the third electric field is parallel to the first electric field.

11. The method of any one of Claims 9 or 10, wherein the magnetic field perpendicular to the first electric field and the plane of the monolayer.UCI-00825 12. A method of reading a qubit, the method comprising: providing a qubit initialized according to any one of Claim 4-9, the qubit characterized by a spin; and measuring the spin of the qubit.

13. A device, comprising: a graphene ribbon of any one of Claims 1 through 3; a means for generating an electric field, the means configured to generate: a first electric field, the first electric field being homogenous in a first region that comprises a first portion of the graphene ribbon, the first electric field having a component perpendicular to the direction of periodicity ^⃗^; a second electric field, the second electric field being homogenous in a second region that comprises a second portion of the graphene ribbon, the second electric field having a component that is antiparallel to the first electric field, a third electric field, the third electric field being homogenous in a third region that comprises a third portion of the graphene ribbon, the third electric field having a component that is parallel to the first electric field, wherein the second region is disposed next to the first region, and the third region is disposed next to the second region, wherein the means for generating the electric field is configured to translate the first region, the second region, and the third region, each along the plane of the monolayer; and a means for applying a magnetic field to the first, the second, and the third regions, the magnetic field having a component that is perpendicular to the first electric field and / or second electric field and the plane of the monolayer.

14. The device of Claim 13, wherein: the first electric field is perpendicular to the direction of periodicity ^⃗^; the second electric field is antiparallel to the first electric field; and the third electric field is parallel to the first electric field.

15. The device of any one of Claims 13 or 14, wherein the magnetic field is perpendicular to the first electric field and the plane of the monolayer.

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