# Stone–Wales defects preserve hyperuniformity in amorphous two-dimensional networks

^{a}Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213;^{b}Department of Physics, Arizona State University, Tempe, AZ 85287;^{c}Materials Science and Engineering, Arizona State University, Tempe, AZ 85287;^{d}Department of Physics, University of Pennsylvania, Philadelphia, PA 19104;^{e}Center for Applied Physics and Technology, College of Engineering, Peking University, Beijing 100871, People’s Republic of China;^{f}Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287

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Edited by Salvatore Torquato, Princeton University, Princeton, NJ, and accepted by Editorial Board Member Peter J. Rossky November 30, 2020 (received for review August 8, 2020)

## Significance

We show that the Stone–Wales topological defects preserve hyperuniformity, a recently discovered novel state of many-body systems that suppresses large-scale density fluctuations, in defected hexagonal two-dimensional (2D) networks, spanning from perturbed crystalline structures to amorphous networks. Our findings have important implications for amorphous 2D materials since the Stone–Wales defects are well known to capture the salient feature of disorder in these materials. Our methods of building realistic hyperuniform 2D amorphous systems with large-scale electronic structure calculations are readily applicable for 2D materials discovery and design.

## Abstract

Disordered hyperuniformity (DHU) is a recently discovered novel state of many-body systems that possesses vanishing normalized infinite-wavelength density fluctuations similar to a perfect crystal and an amorphous structure like a liquid or glass. Here, we discover a hyperuniformity-preserving topological transformation in two-dimensional (2D) network structures that involves continuous introduction of Stone–Wales (SW) defects. Specifically, the static structure factor

Disordered hyperuniformity (DHU) is a recently discovered novel state of many-body systems (1, 2), possessing a hidden long-range order in between that of a perfect crystal and that of a totally disordered system (e.g., an ideal gas). DHU systems are statistically isotropic and possess no Bragg peaks similar to liquids or glasses, and yet, they suppress large-scale density fluctuations like crystals (1, 3). Specifically, DHU is manifested as the vanishing static structure factor

A wide spectrum of equilibrium and nonequilibrium physical and biological systems have been identified to possess the property of hyperuniformity (4⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–30). DHU materials are found to possess superior physical properties including large isotropic photonic band gaps (31, 32), optimized transport properties (33, 34), mechanical properties (35), and wave-propagation characteristics (34, 36, 37), as well as optimal multifunctionalities (38). It is noteworthy that the effects of imperfections/defects on hyperuniformity have also been extensively investigated in the context of perturbed lattices (e.g., refs. 39⇓⇓⇓–43 and references therein). Very recently, DHU patterns of electrons emerging from a quantum jamming transition of correlated many-electron state in two-dimensional (2D) materials, which leads to enhanced electronic transport, have been observed (44). In addition, it is found that DHU distribution of localized electrons in 2D amorphous silica results in an insulator-to-metal transition in the material (45). These exciting discoveries not only suggest the existence of a novel DHU state of electrons in low-dimensional materials but also shed light on novel device applications by exploring the unique emergent properties of the DHU electron states.

It is widely known that disorder can be introduced in ordered 2D network structures as topological defects, which are typically referred to as the Stone–Wales (SW) defects, via, for example, proton radiation (Fig. 1 *A*–*C*) (46). The resulting structures contain “flipped” bonds that change the local topology of the original honeycomb network, leading to, for example, clusters of two pentagons and two heptagons. The SW defects have been experimentally observed in many 2D materials as local defects (47⇓⇓⇓–51). There are also preliminary theoretical studies that investigated the effect of introducing SW defects into crystalline graphene systems on the structures and properties of the resulting materials (52, 53). However, comprehensive and systematic investigation of the large-scale structural features of amorphous 2D materials resulted from these local defects are still lacking. Recently, stand-alone single-layer truly amorphous graphene has been successfully synthesized (51). Subsequent detailed transmission electron microscopy characterization indicates that its structure is distinctly different from the random network model (51), a widely accepted structural model of amorphous 2D materials. Moreover, a recent study of amorphous 2D silica reveals that the distribution of silicon atoms possesses the remarkable property of DHU (45). In a wider context, amorphous carbon-based systems have been extensively studied, including graphene sheets (54⇓–56), cross-linked graphene network (57), and amorphous glassy carbon and carbon fibers (58, 59), to name a few.

In this work, we provide strong numerical evidence that the SW defects preserve hyperuniformity in hexagonal 2D network structures for all of the defect concentrations p up to values beyond percolation of SW defects. Specifically, the static structure factor

## SW Defects Preserve Hyperuniformity in Hexagonal 2D Network Structures

We first construct structural models to generate amorphous 2D network structures, which consist of two steps: 1) SW transformation and 2) structural relaxation. Specifically, we start from the perfect honeycomb lattice and randomly introduce SW defects until a specific defect concentration p is achieved. Here, we define p as the fraction of bonds in the network that undergoes the SW transformation (i.e., *Materials and Methods* has detail). Fig. 2 shows examples of obtained amorphous 2D network models at selected p.

We now investigate the effects of the SW defects on structural correlations across length scales in our generic structural models. We generate ensembles of network configurations with *A*–*D*) and compute ensemble-averaged *E* and *F* and 3*B*). In particular, to quantify the degree of hyperuniformity, we compute the hyperuniformity index *SI Appendix* has details). These results indicate that the SW transformation preserves hyperuniformity in the resulting defected materials. This is consistent with the observation that SW defects are local perturbations and thus, do not fundamentally change the nature of density fluctuations on large length scales compared with the original honeycomb lattice, which is hyperuniform.

We also note that there are clear wiggles in

Visual examination of the obtained configurations suggests that as p increases, the defects initially appear as isolated clusters at low p, then gradually become more connected, and form chains of defects at large p. This behavior appears to be similar to those in 2D colloidal systems during melting, in which a two-step process, involving crystal–hexatic–liquid phase transitions, occurs (63). We thus compute the bond-orientational order metric *Materials and Methods* has definitions), similar to those often used in 2D melting analysis to investigate our systems. We note that

In our systems, we find that

Importantly, the static structure factor *C*). The initial decrease of α as p increases is driven by the increasing disorder associated with the defects introduced to the system. This behavior is also manifested by the change of scaling from *F*).

A closer examination of the network configurations indicates that a percolation of SW defect bonds might occur as p increases. To demonstrate this, we first identify all of the SW defect bonds, which are bonds that belong to the nonhexagons in the networks as highlighted in Fig. 2 *A*–*D*. We then compute the normalized largest cluster size *D*. Note that

## Modeling Amorphous 2D Materials Using Disordered Hyperuniform Networks

We can convert the generic structural networks into 2D amorphous material models by decorating each vertex and/or the midpoint of each bond in the network with an atom of a particular type. Examples of resulting 2D materials include graphene and graphene-like materials such as boron nitride, molydenum disulphide, and silica, to name a few. Visual examination of the experimental samples (48, 51) indicates that not all amorphous 2D materials are created alike, and the stable state of different amorphous 2D materials may be close to our structural models at different defect concentrations p. In particular, as demonstrated by the pair correlation *D*, structure factor *E*, and polygonal shape distributions in Fig. 1*F*, experimentally obtained stable amorphous 2D graphene (51) appears to resemble the network structure at a low defect concentration of

We note that in our model, the SW defects are randomly introduced into the hexagonal network, subject to the only constraint that the number of bonds that each particle possesses should stay the same before and after the transformations, a reflection of the underlying chemical constraint that each carbon atom should possess three bonds. However, as mentioned above, we do not consider other couplings between neighboring SW-type defects during their formation, which may play a role in actual amorphous 2D materials, in particular at high defect concentrations. Moreover, other types of defects such as vacancies and interstitials that are present in many amorphous 2D materials are not taken into account in our model. In addition, it is well known that at finite temperatures, the defects can lead to local “ripples” in 2D materials, and the resulting structures are no longer planar, which is not explicitly considered in our current geometrical network models. Nonetheless, we expect that our network models capture the salient features of the projected structures of amorphous 2D materials (as evidenced by our analysis in this section) and could offer an excellent starting point for numerical investigation of a wide spectrum of amorphous 2D materials with relatively low SW defect concentration at low temperatures.

## Correlations and Interactions among SW Defects in Amorphous Graphene

It is noteworthy that we have demonstrated the ability to generate a wide spectrum of amorphous 2D materials by continuously varying the defect concentration p in our generic model, which is not limited to experimental samples that have been currently synthesized. In particular, we can tune the degree of disorder and even the class of hyperuniformity of the resulting materials by tuning the value of p. Nonetheless, any 2D amorphous materials that can be described by our generic model at a specific concentration p possess the remarkable property of hyperuniformity, as demonstrated by our analysis.

As a proof of concept, we perform density functional theory-based tight-binding (DFTB) calculations (67) on graphene supercells containing N = 2,500 atoms with different concentrations of SW defects ranging from 0 to 0.14 at an incremental step of 0.02. These structures correspond to eight DHU systems from our structure simulations. We choose amorphous graphene as our examples here for two reasons: 1) stand-alone truly amorphous graphene has recently been successfully synthesized experimentally (51), allowing us to validate our simulations, and 2) the computational tools (e.g., DFTB) for these materials are well developed and calibrated to produce accurate calculations of electronic structures.

We first examine the energetics of these eight systems. We apply two methods to compute p-dependent energy increase *Materials and Methods* has details of DFTB and DFT simulations). With the number of flipped bonds and the energy per flipped bond known, we are able to obtain the variation of

We notice from Fig. 4*A* that the energy increase calculated with the DFTB method exhibits distinct behaviors as p increases. For small p, both BFCM and DFTB methods show that the energies of DHU graphene increase linearly with the increasing concentrations of SW defects. The increased energies result from flipped C–C bonds that lead to the molecular orbitals deviating from the energetically more stable

## Mechanical Properties, Binding Energy, and Electronic Structure of Amorphous Graphene

We perform molecular dynamic (MD) simulations to further study the mechanical properties of the defected graphene using the structural models constructed above. Fig. 4*B* displays the averaged fracture strength *SI Appendix*, Fig. S4*A* shows typical tensile stress–strain curves of the defected graphene with different p values). It can be seen that *SI Appendix*, Fig. S4*B* reveals that the emerging plasticity is due to the increasing number of C–C bonds that remain connected after the overall fracture strength is achieved. 3) The fracture strength of defected graphene remains finite even for relatively large defect concentrations.

We also carry out DFTB calculations to obtain insights into the reactivity of the defected graphene with p = 0.06 by studying the binding of an H atom to different C atoms with distinct local environments. In the perfect graphene, each C atom corresponds to the common vertex of three neighboring hexagons denoted by (6,6,6), while in the defected honeycomb networks, a C atom can correspond to the common vertex of different polygon configurations, such as (5,6,6) and (5,6,8), etc. We have enumerated these distinct local environments and calculated the associated binding energies. Interestingly, we observe from Fig. 4*C* that the hydrogen binding energy (HBE) associated with the local environment for perfect graphene (6,6,6) is not the lowest among all possible environments found in the defected graphene. For example, the environments associated with (5,5,6), (5,6,6), and (5,7,8), which are highly populated in defected graphene, possess binding energies significantly lower than or similar to that of (6,6,6). This is because, in the local environments such as (5,6,6), the C atom that binds with the H atom may have donated most of its electrons to its neighboring C atoms before forming the C–H bond. These calculations provide insights on the relatively low reactivity of defected graphene.

To illustrate the effect of SW defects on the electronic structure of graphene, Fig. 5*A* shows the density of states (DOS) of the DHU systems. As can be seen, our DFTB calculations reproduce the Dirac cone of perfect graphene associated with zero and linear DOS at and near the Fermi level, respectively. The Dirac cone in DHU graphene disappears (i.e., the semimetal nature of crystalline graphene is destructed, and the DHU graphene possesses increasingly higher DOS at the Fermi level as p increases). These results are consistent with the calculations based on experimentally obtained amorphous graphene (51). We also extract the DOS values *B*, exhibiting monotonically increasing behavior as p increases.

The increased DOS at the Fermi level is also manifested in the other two aspects: energies and charge densities. In particular, we observe that the carbon atoms at the flipped C–C bonds and their adjacent regions exhibit higher energies. This can be seen in Fig. 5 *C* and *D*, showing the atom-resolved total energies for two representative systems with two distinct hyperuniform classes with p = 0.02 (class I) and 0.12 (class II), respectively. Fig. 5 *E* and *F* shows the charge density at the Fermi level for these two systems. The complete sets of charge density maps for *SI Appendix*. It can be seen that the electrons in class I DHU graphene spread out in the entire system, while the electrons in class II DHU graphene are localized in separate islands. These patches are similar to the localization regions found by Van Tuan et al. (68) and shown to degrade the electrical transport of graphene.

## Conclusions and Discussion

In summary, we have shown numerically that the SW topological transformations preserve hyperuniformity in hexagonal 2D networks models, which have important implications for real 2D amorphous materials. As the SW defect concentration increases, we also observed a percolation transition of SW defects. With the increasing interest in 2D amorphous materials, we expect our methods of building realistic DHU structural models of 2D amorphous material systems along with large-scale electronic structure and mechanical property calculations to be applicable to a wide range of other 2D materials such as graphene (68) and transition-metal dichalcogenides (50) in the amorphous form. Our analysis indicates that experimentally obtained amorphous graphene (51) belongs to the class I hyperuniform materials. It is interesting to see whether it would be possible to experimentally realize hyperuniform amorphous graphene with varying degrees of disorder. In future works, we will also systematically study the various mechanical, transport, and chemical properties of these amorphous 2D materials, which may shed light on the discovery and design of functional materials.

In addition, we note that in this work we discovered a hyperuniformity-preserving operation in two dimensions: SW transformation. Previously, it was found that operations such as “uncorrelated” stochastic displacements in perfect lattices preserve hyperuniformity but could change the hyperuniformity class of the systems (40, 42, 43), similar to what we observed in this work. Moreover, hyperuniformity-generating operations that can convert nonhyperuniform point patterns into hyperuniform materials have also been discovered (28, 69). For example, Klatt et al. (28) showed that by applying centroidal voronoi tessellation, one can convert a wide spectrum of nonhyperuniform and hyperuniform point patterns into effectively stealthy hyperuniform patterns. Kim and Torquato (69) have discovered that by decorating each point in a point pattern with spheres of different sizes in a delicate manner, one can convert a variety of point patterns into perfectly hyperuniform materials. This operation shared some similarities with the “equal-volume tessellation” operation that can be used to obtain hyperuniform point patterns (70).

With all of these discovered hyperuniformity-preserving and -generating operations, perhaps it is time to ask general questions such as these (to name a few). 1) Are there more examples of such hyperuniformity-preserving and -generating operations? 2) Can general conditions be imposed on such operations? 3) Do such operations form groups? 4) What is the effect of dimensionality? We will address these questions in our future works, which we believe will greatly strengthen our fundamental understanding of hyperuniformity and the physics underlying this exotic state of matter. In addition, we note that there are also known operations that will destroy hyperuniformity (e.g., by introducing uncorrelated point defects such as vacancies and interstitials) or certain “correlated” displacements such as thermal excitation in the classical harmonic regime (42).

## Materials and Methods

### Generation of Hyperuniform Amorphous 2D Network Structures.

Here, we briefly describe the procedures that we employ to generate hyperuniform amorphous 2D networks. For more illustration and detailed description, the readers are referred to *SI Appendix*. As mentioned here, our procedure consists of the two steps of 1) SW transformation and 2) structural relaxation, which are schematically shown in *SI Appendix*, Fig. S1. Specifically, we start from the perfect honeycomb lattice and continuously introduce SW defects at randomly picked sites in the network until the specified defect fraction p is reached. Here, we define p as the fraction of bonds in the network that undergoes the SW transformation. A SW transformation involves the rotation of a bond by 90° with respect to the midpoint of the bond and the change of connectivity of the vertices in the network. We further require a successful transformation to respect the bonding (topology) constraints in the original lattice (i.e., the number of bonds that each vertex possesses should remain unchanged [equal to three] before and after a transformation).

Subsequently, we allow the transformed structures to undergo structural relaxation by translationally perturbing the positions of the vertices in a way that drives the bond lengths and bond angles in the network toward values associated with the honeycomb lattice. In particular, this involves local minimization of the energy function E defined as follows:*p* = 0.14 because of the technical difficulty of dealing with an appreciable number of concave polygons in our analysis within this regime.

### Conversion from Network Structures to Amorphous 2D Materials.

To generate various amorphous 2D materials from our generic network models, we decorate each vertex in the network with an atom of a particular type or a set of atoms. For example, if we decorate each vertex with a carbon atom, we obtain an amorphous graphene material. On the other hand, if we place a silicon atom centered at each vertex and an oxygen atom at the midpoint of every pair of connected silicon atoms, we convert our transformed structure into an amorphous silica material. In addition, we note that the stable state of different amorphous 2D materials may be associated with different defect concentrations p in our structural model.

### Bond-Orientational Order Metric and Correlation Function.

Here, we provide the definitions for the aforementioned

### Determination of Percolation of Defect Bonds.

We first replicate a network structure along the longest dimension of the system (e.g., the vertical direction for the configurations shown in Fig. 2 *A*–*D*) and obtain a

### DFT Calculations.

We apply the Vienna Ab initio Simulation Package (73, 74) to compute the energy cost of flipping a C–C bond by

### DFTB Calculations.

We use the DFTB+ package (67, 76) to perform DFTB calculations. The C–C and C–H Slater–Koster parameters are from ref. 77. Periodic boundary conditions are applied in all of the three directions. Typical in-plane lattice constants of DHU graphene (e.g., p = 0.02) are 106.9 and 61.7 Å in the x and y directions, respectively. We also add a vacuum spacing of 18 Å in the z direction to separate image interactions. We compute the hydrogen binding energy (HBE) following the equation (78) HBE =

### MD Simulations.

We perform MD simulations using LAMMPS (79) to obtain tensile stress–strain curves of amorphous graphene using the modified reactive empirical bond order potential (80). We focus on the mechanical behavior of amorphous graphene at low temperature (set to 1 K). Each simulation cell has 10,000 C atoms, and we equilibrate each system for 30 ps with a time step of 1 fs. After the equilibrium process, we apply tensile strains at a strain rate of 0.0005 per 10 ps and compute the average tensile stress (normalized by the thickness of 3.4 Å for graphene) every 10 ps. The NPT (isothermal–isobaric) ensemble is used throughout the MD simulations. Using these parameters, we obtain the fracture strength of perfect graphene as 109 GPa, close to the results (107 and 118 GPa along two different directions) from DFT calculations (81).

## Data and Code Availability.

All study data are included in the article and *SI Appendix*.

## Acknowledgments

L.L. and H.Z. thank Arizona State University (ASU) for the start-up funds. Y.J. thanks ASU for support and Peking University for hospitality during his sabbatical leave. M.C. is supported by the National Science Foundation of China under Grant 12074007. This research used computational resources of the Agave Research Computer Cluster of ASU and the Texas Advanced Computing Center under Contract TG-DMR170070. We thank Dr. Salvatore Torquato, Dr. Michael Klatt, and Dr. Jaeuk Kim for helpful discussion, as well as anonymous reviewers for constructive and thought-provoking comments. We are grateful to the Barbaros Oezyilmaz group at National University of Singapore for providing the atomic coordinates of the amorphous graphene sample obtained in experiments.

## Footnotes

↵

^{1}D.C., Y.Z., and L.L. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: duyu{at}alumni.princeton.edu, mohanchen{at}pku.edu.cn, yang.jiao.2{at}asu.edu, or hzhuang7{at}asu.edu. ↵

^{3}Present address: Materials Research Laboratory, University of California, Santa Barbara, CA 93106.

Author contributions: D.C., M.C., Y.J., and H.Z. designed research; D.C., Y.Z., L.L., G.Z., M.C., Y.J., and H.Z. performed research; D.C., Y.J., and H.Z. contributed new reagents/analytic tools; D.C., Y.Z., L.L., G.Z., M.C., Y.J., and H.Z. analyzed data; and D.C., M.C., Y.J., and H.Z. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission. S.T. is a guest editor invited by the Editorial Board.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2016862118/-/DCSupplemental.

Published under the PNAS license.

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- Abstract
- SW Defects Preserve Hyperuniformity in Hexagonal 2D Network Structures
- Modeling Amorphous 2D Materials Using Disordered Hyperuniform Networks
- Correlations and Interactions among SW Defects in Amorphous Graphene
- Mechanical Properties, Binding Energy, and Electronic Structure of Amorphous Graphene
- Conclusions and Discussion
- Materials and Methods
- Data and Code Availability.
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