Topological carbon allotropes: paradigm shift for materials innovation

TL;DR

This paper introduces a novel class of topological carbon allotropes, including a new 3D structure called Hopfene, which uses topological links to connect graphene sheets without traditional bonds, representing a paradigm shift in materials science.

Contribution

The authors propose new topological structures for carbon allotropes, including the invention of Hopfene, a 3D material with linked graphene sheets using Hopf-links, expanding the design space for materials.

Findings

Discovery of topologically linked carbon structures

Introduction of Hopfene as a new 3D carbon allotrope

Demonstration of topological robustness in material design

Abstract

Topology is a central concept of mathematics, which allows us to distinguish two isolated rings with linked ones. In material science, researchers discovered topologically different carbon allotropes in a form of a cage, a tube, and a sheet, which have unique translational and rotational symmetries, described by a crystallographic group theory, and the atoms are arranged at specific rigid positions in 3-dimensional ($D$) space. However, topological orders must be robust against deformations, so that we can make completely different families of topological materials. Here we propose various topological structures such as knots and links using covalent $\sigma$ bonds of carbon atoms, while allowing various topologically equivalent arrangements using weak $\pi$ bonds. By extending this idea, we invented a new 3D carbon allotrope, Hopfene, which has periodic arrays of Hopf-links to knit horizontal Graphene sheets into vertical ones without connecting by $\sigma$ bonds.

Figures

h-1o and Extended Fig. 2 ). We found a benzene ring can sustain its $\sigma$ bonds, while allowing another benzene ring to penetrate without making a proper $\sigma$ bond. The benzene ring can move freely without changing the topology of the Hopf-link, which will be a huge advantage to make flexible devices. However, the bond lengths of the benzene-based Hopf-link are significantly expanded, so that the high tensile strains of the order of 20% are accumulated (Extended Fig. 2a ). Therefore, it is not possible to introduce 2 benzene rings into 1 ring (Fig. 2b and Supplementary Video 2). This wi

c and Fig. 3). We also considered to open-up one of the benzene ring of Naphthalene (Fig. 1 b), which corresponds to use a larger ring (Cyclodecapentaene, C*10*H*10*) for the Hopf-link (Fig. 2d), whose bonds are significantly relaxed. In the larger ring, we could introduce as many as 4 rings (Fig. 2e), so that we can also construct the $2D$ 4-in-1 Chainmail (Fig. 3c) in addition to a standard Chain (Fig. 3b).

f), due to its rotational symmetry. One might think that polyacetylene would be more tolerant against the rotational disturbance, but no restoration force will work to prevent the rotation. On the other hand, the Hopf-linked chain can accumulate the global strain and the restoration force will be generated to protect itself against the structural deformation. This would be related to the reason why DNA with a double-stranded spiral structure was naturally selected due to its stronger tolerance in a competitive environment.

b). The valleys are degenerate at the same points, because the direction of the zig-zag edge ($z$) is the same for both layers due to sharing of the $1D$ chain. As we increase the number of inserted Graphene sheets, momentum space will be eventually filled by layers towards the complete $3D$ band structure. It is beyond the scope of this paper to predict how the $2D$ Dirac Fermions[[8](#bib.bib8), [31](#bib.bib31), [20](#bib.bib20), [27](#bib.bib27), [33](#bib.bib33)] are crossing over to the excitations in $3D$ Hopfene.

h-1o and Extended Fig. 2 ). We found a benzene ring can sustain its $\sigma$ bonds, while allowing another benzene ring to penetrate without making a proper $\sigma$ bond. The benzene ring can move freely without changing the topology of the Hopf-link, which will be a huge advantage to make flexible devices. However, the bond lengths of the benzene-based Hopf-link are significantly expanded, so that the high tensile strains of the order of 20% are accumulated (Extended Fig. 2a ). Therefore, it is not possible to introduce 2 benzene rings into 1 ring (Fig. 2b and Supplementary Video 2). This wi

c and Fig. 3). We also considered to open-up one of the benzene ring of Naphthalene (Fig. 1 b), which corresponds to use a larger ring (Cyclodecapentaene, C*10*H*10*) for the Hopf-link (Fig. 2d), whose bonds are significantly relaxed. In the larger ring, we could introduce as many as 4 rings (Fig. 2e), so that we can also construct the $2D$ 4-in-1 Chainmail (Fig. 3c) in addition to a standard Chain (Fig. 3b).

f), due to its rotational symmetry. One might think that polyacetylene would be more tolerant against the rotational disturbance, but no restoration force will work to prevent the rotation. On the other hand, the Hopf-linked chain can accumulate the global strain and the restoration force will be generated to protect itself against the structural deformation. This would be related to the reason why DNA with a double-stranded spiral structure was naturally selected due to its stronger tolerance in a competitive environment.

b). The valleys are degenerate at the same points, because the direction of the zig-zag edge ($z$) is the same for both layers due to sharing of the $1D$ chain. As we increase the number of inserted Graphene sheets, momentum space will be eventually filled by layers towards the complete $3D$ band structure. It is beyond the scope of this paper to predict how the $2D$ Dirac Fermions[[8](#bib.bib8), [31](#bib.bib31), [20](#bib.bib20), [27](#bib.bib27), [33](#bib.bib33)] are crossing over to the excitations in $3D$ Hopfene.

Full text

Introduction

Research areas on topology are spanning from pure mathematics[1, 2, 3, 4], physics[5, 6, 7, 8, 9], to chemistry[10, 11, 12, 13, 14]. Mathematicians could even classify the shape of our universe[3] and proved the Poincar$\rm\acute{e}$ conjecture successfully[4]. In condensed-matter physics, vortex and anti-vortex pairs in superconductors exhibit unusual topological phase transitions due to their binding and dissociations across the transition temperature[5, 6]. More recently, a topological insulator was proposed theoretically[8] and discovered experimentally[9], which showed insulating character in the bulk regardless of its conducting surface. In these materials, quantum states exhibit inherent topological nature[5, 6, 8] upon the appropriate choice of materials[9]. On the other hand, nontrivial topological structures themselves were also chemically synthesised using polymers[15, 16, 17], macromolecules[10, 11, 12, 13], and proteins[14]. These pioneering works are well-established in soft materials[15, 16, 17, 10, 11, 12, 13, 14], and the scope of our present paper is to expand the concept to apply for hard materials made by strong covalent bonds while keeping the flexibility.

Topological molecules ($0D$)

To start with, we considered many topological molecules (Methods, Figs. 1 and 2, Supplementary Videos 1 and 2). The most direct applications of topological materials would be Nano-Electro-Mechanical-Systems (NEMS)[18], towards developing a molecular motor and associated advanced chemical technologies[10, 7, 11, 12, 13, 14]. Figure 1a shows an ultimate example of trapped 1 benzene molecule using a straight-chain alkane of 1D atomistic carbon atoms connected to a Graphene sheet (Supplementary Video 1). It is technologically accessible to make such a molecular motor by combining patterning technologies of a free-standing Graphene sheet by ion or electron beam irradiations and growth technologies such as Chemical-Vapour-Deposition[19]. The combination is very important to construct a topologically complex structures, because we must close the bond (in this case, the benzene ring) without affecting the other bonds (straight-chain). Obviously, larger molecules and wider chain will be more suitable to construct a practical NEMS device.

In order to realise directional rotation of a molecular motor, it is important to control chirality of a molecule. The most simplest nontrivial knot is the Trefoil knot, which has a chiral counter part (Fig. 1c). The left (right) Trefoil knot has always left (right) spiral circulation as climbing up the strand (Supplementary Video 2). In other words, handedness is memorised in the molecule, which will be useful for applications in molecular memories and polarisation rotators.

It is not possible to memorise the handedness solely by twisting a 1D carbon ring to the left or the right, since it merely change the geometry without changing the topology (Fig. 1b). In order to memorise the handedness, we need to lock the status of the twisted ring, by inserting the other molecule into the ring and close the bond. Then, we can make left and right topological molecules of Whitehead-links, whose handedness will be protected against the deformations of molecules as far as the bonds are sustained.

The situation is slightly changed, if we employ a Graphene nano-ribbon[20]. Even if it consists of just several benzene rings, the ring will define a $2D$ plane to confine the ribbon so that it is different from a pure 1D chain. As a result, the Möbius strip[7, 21] can memorise the handedness whether the Graphene nano-ribbons were twisted to the left or the right before bonding without using another Graphene nano-ribbon (Fig. 1d). Moreover, the number of twisted rotations is also a topologically protected valuable as a winding number, which is also robust against the deformation. We also made even more complex structure such as Borromean rings (Fig. 1e). To our surprise, the converged structure is rather beautiful with a proper symmetry upon exchanging rings, which was energetically favourable rather than keeping random deformations.

Besides knots, the simplest nontrivial link is the Hopf-link[1, 11, 13, 14] (Fig. 1h-1o and Extended Fig. 2 ). We found a benzene ring can sustain its $\sigma$ bonds, while allowing another benzene ring to penetrate without making a proper $\sigma$ bond. The benzene ring can move freely without changing the topology of the Hopf-link, which will be a huge advantage to make flexible devices. However, the bond lengths of the benzene-based Hopf-link are significantly expanded, so that the high tensile strains of the order of 20% are accumulated (Extended Fig. 2a ). Therefore, it is not possible to introduce 2 benzene rings into 1 ring (Fig. 2b and Supplementary Video 2). This will certainly limit the use of the benzene-based Hopf-link towards the construction of the $1D$ chain.

However, if we use 2 connected benzene rings (Naphthalene, C10H8), we can overcome this problem, since the insertion of just 1 benzene ring into the Naphthalene is enough for extending the length of the $1D$ Carbon-Nano-Chain (Fig. 2c and Fig. 3). We also considered to open-up one of the benzene ring of Naphthalene (Fig. 1 b), which corresponds to use a larger ring (Cyclodecapentaene, C10H10) for the Hopf-link (Fig. 2d), whose bonds are significantly relaxed. In the larger ring, we could introduce as many as 4 rings (Fig. 2e), so that we can also construct the $2D$ 4-in-1 Chainmail (Fig. 3c) in addition to a standard Chain (Fig. 3b).

Topological Chains ($1D$) and Chainmail ($2D$)

The unique features of these Chains and Chainmail are their flexibility with the rigidity ensured by the $\sigma$ bonds (Fig. 3 and Supplementary Video 3). Our Chains can be aligned to be straight-lines or to be rolled-up without breaking the topological links (Fig. 4). The lengths can be changed to be several times, while it will be difficult to change the lengths of Carbon-Nano-Tubes[22] due to its strong rigidity. The $\sigma$ bonds using SP2 orbitals are considered to be even stronger than SP3 bonds of diamond, so that our chains will be much stronger than flexible polyacetylene[23]. It is interesting to consider a similarity of our proposed Chain to a macroscopic metallic chain and chainmail, against the Fullerene[24] to a football. We think that it would not be mere coincidence that we came up to the same structure that the human race is already using for a long time. The idea behind the football shape would be its high spherical symmetry. It is the compatibility of flexibility with its rigidity why people made a chain, by introducing the Hopf-links[1] even without being aware of mathematical rigidity. The argument would also be true in the molecular level (Fig. 4).

We have simulated various configurations of Chains by changing the boundary conditions. For example, by introducing a twisted boundary conditions between one end of the chain to the other for the $1D$ Hopf-linked Chain, we could introduce a global kink as a form of a soliton, which is a topologically protected excitation (Fig. 4). Upon rotating twice, we found the chain was broken due to the high tensile strains accumulated in the Chain (Fig. 4e).

The chain can memorise the global winding number and the chirality (the left or the right rotation) upon twisting the Chain (Fig. 4). This situation is completely different from a $1D$ atomic chain like a polyacetylene (Fig. 4f), due to its rotational symmetry. One might think that polyacetylene would be more tolerant against the rotational disturbance, but no restoration force will work to prevent the rotation. On the other hand, the Hopf-linked chain can accumulate the global strain and the restoration force will be generated to protect itself against the structural deformation. This would be related to the reason why DNA with a double-stranded spiral structure was naturally selected due to its stronger tolerance in a competitive environment.

The geometrical coordinates of straight and rolled Carbon-Nano-Chains are apparently different, while both structures are topologically equivalent. Therefore, we can regard these chains as possessing Topological-Long-Range-Order (TLRO) even without having translational nor rotational symmetries. We can construct whole new varieties of materials with TLRO, as topological materials, which will not be categorised as a conventional crystal nor a completely-randomised amorphous material. Examples of Chains and Chainmail are clear examples of topological materials.

Topological Crystals ($3D$)

The concept of Topological-Long-Range-Order (TLRO) is also applicable to a $3D$ structure, and the idea of a topological crystal[15, 16, 17] would be as exciting as a time crystal[25]. We have conceived to the idea of a new $3D$ carbon allotrope by considering nested structures with stacked Graphene sheets both vertically and horizontally, while the intersections of these sheets are made of Hopf-links (Figa. 5 and 6, Supplementary Figs. 1-5, and Supplementary Video 4). To our surprise, the optimised structures have almost perfect translational symmetries, so that these will be realised as a conventional single crystal.

We propose to call this new structure, as Hopfene, named after a mathematician, Heinz Hopf, because of the importance of Hopf-links[1] for this crystal. Similar to Carbon-Nano-Tubes[22], which has a family of allotropes depending on how to roll-up a Graphene sheet, Hopfene also has a family depending on how to insert Graphene sheets (Methods, Supplementary Figs. 1-5, and Supplementary Video 4). Graphene sheets were inserted perpendicular to the other stacks, and it was important to align the directions of the zig-zag edges in parallel to match the periodicity of the Graphene lattice (Supplementary Fig. 2). Let’s assume that we inserted the first sheet to the bottom available slot, labelled as slot 0 (Supplementary Fig. 2). If we insert next sheets into the most adjacent available slot, slot 1, for both horizontal ($x$) and vertical ($y$) directions, the structure is called as (1,1) Hopfene (Figs. 5a, 5d, and Supplementary Fig. 2). In this case, the adjacent stacks are half-lattice constant shifted ($c/2$) along zig-zag edge ($z$) direction, so that the stack is called as AB-stack. On the other hand, if we insert Graphene sheets into slot 2, while making the slot 1 empty, the structure is called as (2,2) Hopfene (Figs. 5b, 5e, and Supplementary Fig. 2). In this case, the nearest adjacent sheets are aligned so that the stack is called as AA-stack. We considered more complicated insertions, such as the insertion into slot 1 for $x$-direction, while the insertion along $y$-direction is from slot 2, whose structure is (1,2) Hopfene (Figs. 5c, 5f, and Supplementary Fig. 3). In general, there exists $(n,m)$ Hopfene, where $n$ and $m$ are integers to describe the insertion of Graphene sheets, and if $n$ and $m$ are odd (even), the sheets are AB (AA) stack. If $n=m$, the Hopfene structure is tetragonal with the lattice constants $a=b\neq c$, while it is orthogonal ($a\neq b\neq c$) for $n\neq m$. Therefore, we can recognise rectangular holes if we see the structure from the bottom (Fig. 5). Therese holes would allow carrier doping by intercalation, which may lead superconductivity similar to alkali-metal doped Fullerenes[26]. If $n$ and $m$ are large, the size of the hole will be large, so that the Graphene sheets and the crystal itself can be deformed substantially, breaking the translational symmetry. It is also expected that the actual experimental structures would be different due to the huge strains accumulated in rings, as theoretically predicted in a highly-strained Graphene, which stabilised to be a dimerised Kekul$\rm\acute{e}$-like structure[27]. Even in that case, TLRO would be kept as far as $\sigma$ bonds survived to keep Hopf-links.

It is straightforward to extend this concept[15, 16, 17] to make heterogeneous topological structures by introducing different materials such as boron-nitride and molybdenum-disulfide. The unique aspect of this approach is a topological link to bind various sheets strongly together without forming a proper chemical bond. This configuration is topologically different from the simple stacking weakly bound together by van der Waals force[28].

As the first step towards making Hopfene, we propose to make Hopf-linked bilayer-Graphene (Fig. 7 and Supplementary Fig. 6). Unlike to the stacked bilayer-Graphene[29, 30, 31, 32], the Graphene sheets are linked only at the $1D$ chain, so that the impacts of the coupling on the band structure would be limited. Reflecting the double layers, momentum space will also be double, leading to the crossed layers even in momentum space (Fig. 7b). The valleys are degenerate at the same points, because the direction of the zig-zag edge ($z$) is the same for both layers due to sharing of the $1D$ chain. As we increase the number of inserted Graphene sheets, momentum space will be eventually filled by layers towards the complete $3D$ band structure. It is beyond the scope of this paper to predict how the $2D$ Dirac Fermions[8, 31, 20, 27, 33] are crossing over to the excitations in $3D$ Hopfene.

Conclusion

We have validated the concept of topological materials[15, 16, 17] in materials with hard covalent bonds, which we believe, will be discovered for the future. $1D$ Carbon-Nano-Chains, Hopf-linked bilayer-Graphene ($2D$), and $3D$ Hopfene have been proposed as examples of new topological carbon allotropes, which will be useful to examine fundamental physics of massless Dirac Fermions[8, 31, 20, 27, 33] in topologically nontrivial geometries. We envisage various practical applications (Extended Fig. 12) of these hard and flexible topological materials for DNA-sensing, functional materials, and NEMS.

Methods

We have used Chem3D, a molecular editor, for all our simulations[34].

For the simulations of the rolled Chains (Figs. 3 and 4) and Chainmail (Fig. 3), we have gradually changed the initial coordination of atoms. If we changed the initial condition significantly different from the previously converged geometries, Hopf-links were completely broken and the structures were topologically changed. This will also happen in the real situation, where the external force exceeded the threshold enough to break one of the $\sigma$ bonds for the Hopf-links. Conversely, the proposed Chains and Chainmail will be topologically robust as far as the $\sigma$ bonds are maintained. Thus, we expect a topologically flexible material, which would be as flexible as a rubber, while it is very stiff as hard as or even harder than a diamond due to its SP2 nature of the rings.

Hopfene was made by preparing AA-stacked Graphene sheets, separated by the distance of the lattice constant of a Graphene, which is the adjacent distance between the sides of the ring (Supplementary Figs. 4 and 5). Depending on the targeted stacking, described by $(n,m)$ where $n$ and $m$ are integers for the available slots along horizontal ($x$) and vertical ($y$) directions, the number of inserted Graphene sheets and separations were adjusted. We have prepared the Graphene sheets by hands using the GUI (Graphical User Interface) of the software, so that the position and the distance were not perfect. This uncertainty resulted in the formation of topological defects in the final structure (Supplementary Fig. 1), but we kept as it is to highlight new types of defects without broken bonds. Then, we optimised the structure relying on the algorithm of the software to minimise the energy. After the optimisation, for (1,1) Hopfene, AB-stack was automatically chosen for both $x$ and $y$ directions by shifting the Graphene sheets with the amount of the half of the lattice constant along $z$ direction, regardless of the initial condition of AA-stack. Here, please note that AB-stack of Hopfene is different from AB-stack of Graphite. AB-stack of Hopfene means the parallel-shift of the Graphene sheet along the zig-zag edge (Fig. 6), while AB-stack of Graphite is the parallel-shift along the arm-chair edge, so that the directions are perpendicular to that of Hopfene. The process of the convergence of the structure would be similar to the growth process in experiments for the future, so that we have recorded as a video (Supplementary Video 4).

Acknowledgements

This work is supported by EPSRC Manufacturing Fellowship (EP/M008975/1). We would like to thank Prof. H. Mizuta, Dr. M. Muruganathan, Prof. Y. Oshima, Prof. S. Matsui, Prof. S. Ogawa, Prof. S. Kurihara, and Prof. H. N. Rutt for stimulating discussions. S.S also would like to thank JAIST for their hospitalities during his stay at the Center for Single Nanoscale Innovative Devices.

Author contributions statement

S.S. and I.T. conceived the idea of topological carbon allotropes, together. S.S. conceived the simulations, and I.T. analysed the results. S.S. wrote the draft. S.S. and I.T. reviewed the manuscript.

Additional information

Accession codes (The data from the paper can be obtained from the University of Southampton ePrint research repository:

http://dx.doi.org/10.5258/SOTON/D0884); Competing interests (The authors declare no competing interests both financially and non-financially.).