Addressing the isomer cataloguing problem for nanopores in two-dimensional materials


The presence of extended defects or nanopores in two-dimensional (2D) materials can change the electronic, magnetic and barrier membrane properties of the materials. However, the large number of possible lattice isomers of nanopores makes their quantitative study a seemingly intractable problem, confounding the interpretation of experimental and simulated data. Here we formulate a solution to this isomer cataloguing problem (ICP), combining electronic-structure calculations, kinetic Monte Carlo simulations, and chemical graph theory, to generate a catalogue of unique, most-probable isomers of 2D lattice nanopores. The results demonstrate remarkable agreement with precise nanopore shapes observed experimentally in graphene and show that the thermodynamic stability of a nanopore is distinct from its kinetic stability. Triangular nanopores prevalent in hexagonal boron nitride are also predicted, extending this approach to other 2D lattices. The proposed method should accelerate the application of nanoporous 2D materials by establishing specific links between experiment and theory/simulations, and by providing a much-needed connection between molecular design and fabrication.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Validation of calculated activation barriers for atomic rearrangements in the graphene lattice.
Fig. 2: Distinguishing between and counting nanopore isomers in graphene.
Fig. 3: Comparison of the results of the first in silico experiment (that is, without including the effect of edge diffusion) with experimental TEM images available in the literature.
Fig. 4: Solution of the ICP for nanopores in monolayer hBN.

Data availability

The datasets generated during and/or analysed during the current study, including the XYZ files of the MPIs, are available online at, under the directory ‘catalog’.


  1. 1.

    Yuan, W., Chen, J. & Shi, G. Nanoporous graphene materials. Mater. Today 17, 77–85 (2014).

    CAS  Article  Google Scholar 

  2. 2.

    Childres, I., Jauregui, L. A., Tian, J. & Chen, Y. P. Effect of oxygen plasma etching on graphene studied using Raman spectroscopy and electronic transport measurements. New J. Phys. 13, 025008 (2011).

    Article  Google Scholar 

  3. 3.

    Rao, C. N. R. & Sood, A. K. in Graphene: Synthesis, Properties, and Phenomena (ed. Enoki, T.) 131–157 (Wiley, 2012).

  4. 4.

    Zhu, Y. et al. Carbon-based supercapacitors produced by activation of graphene. Science 332, 1537–1541 (2011).

    CAS  Article  Google Scholar 

  5. 5.

    Surwade, S. P. et al. Water desalination using nanoporous single-layer graphene. Nat. Nanotech. 10, 459–464 (2015).

    CAS  Article  Google Scholar 

  6. 6.

    Nakada, K., Fujita, M., Dresselhaus, G. & Dresselhaus, M. S. Edge state in graphene ribbons: nanometer size effect and edge shape dependence. Phys. Rev. B 54, 17954–17961 (1996).

    CAS  Article  Google Scholar 

  7. 7.

    Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).

    CAS  Article  Google Scholar 

  8. 8.

    O’Hern, S. C. et al. Selective molecular transport through intrinsic defects in a single layer of CVD graphene. ACS Nano 6, 10130–10138 (2012).

    Article  Google Scholar 

  9. 9.

    Wang, L. et al. Molecular valves for controlling gas phase transport made from discrete ångström-sized pores in graphene. Nat. Nanotech. 10, 785–790 (2015).

    CAS  Article  Google Scholar 

  10. 10.

    O’Hern, S. C. et al. Selective ionic transport through tunable subnanometer pores in single-layer graphene membranes. Nano. Lett. 14, 1234–1241 (2014).

    Article  Google Scholar 

  11. 11.

    Branton, D. et al. The potential and challenges of nanopore sequencing. Nat. Biotechnol. 26, 1146–1153 (2008).

    CAS  Article  Google Scholar 

  12. 12.

    Kaplan, A. et al. Current and future directions in electron transfer chemistry of graphene. Chem. Soc. Rev. 46, 4530–4571 (2017).

  13. 13.

    Konstantinova, E. V. & Vidyuk, M. V. Discriminating tests of information and topological indices. Animals and trees. J. Chem. Inf. Comput. Sci. 43, 1860–1871 (2003).

    CAS  Article  Google Scholar 

  14. 14.

    Aleksandrowicz, G. & Barequet, G. Counting d-dimensional polycubes and nonrectangular planar polyominoes. Int. J. Comput. Geom. Appl. 19, 215–229 (2009).

    Article  Google Scholar 

  15. 15.

    Yuan, Z. et al. Mechanism and prediction of gas permeation through sub-nanometer graphene pores: comparison of theory and simulation. ACS Nano 11, 7974–7987 (2017).

    CAS  Article  Google Scholar 

  16. 16.

    Sint, K., Wang, B. & Král, P. Selective ion passage through functionalized graphene nanopores. J. Am. Chem. Soc. 130, 16448–16449 (2008).

    CAS  Article  Google Scholar 

  17. 17.

    Siria, A. et al. Giant osmotic energy conversion measured in a single transmembrane boron nitride nanotube. Nature 494, 455–458 (2013).

    CAS  Article  Google Scholar 

  18. 18.

    Feng, J. et al. Single-layer MoS2 nanopores as nanopower generators. Nature 536, 197–200 (2016).

    CAS  Article  Google Scholar 

  19. 19.

    Cui, X. Y. et al. Magic numbers of nanoholes in graphene: tunable magnetism and semiconductivity. Phys. Rev. B 84, 125410 (2011).

    Article  Google Scholar 

  20. 20.

    Carlsson, J. M. & Scheffler, M. Structural, electronic, and chemical properties of nanoporous carbon. Phys. Rev. Lett. 96, 046806 (2006).

    Article  Google Scholar 

  21. 21.

    Cohen-Tanugi, D. & Grossman, J. C. Water desalination across nanoporous graphene. Nano. Lett. 12, 3602–3608 (2012).

    CAS  Article  Google Scholar 

  22. 22.

    Sun, C. et al. Mechanisms of molecular permeation through nanoporous graphene membranes. Langmuir 30, 675–682 (2014).

    CAS  Article  Google Scholar 

  23. 23.

    Drahushuk, L. W. & Strano, M. S. Mechanisms of gas permeation through single layer graphene membranes. Langmuir 28, 16671–16678 (2012).

    CAS  Article  Google Scholar 

  24. 24.

    Robertson, A. W. et al. Atomic structure of graphene subnanometer pores. ACS Nano 9, 11599–11607 (2015).

    CAS  Article  Google Scholar 

  25. 25.

    Pham, T. et al. Formation and dynamics of electron-irradiation-induced defects in hexagonal boron nitride at elevated temperatures. Nano. Lett. 16, 7142–7147 (2016).

    CAS  Article  Google Scholar 

  26. 26.

    Girit, C. O. et al. Graphene at the edge: stability and dynamics. Science 323, 1705–1708 (2009).

    CAS  Article  Google Scholar 

  27. 27.

    Russo, C. J. & Golovchenko, J. A. Atom-by-atom nucleation and growth of graphene nanopores. Proc. Natl Acad. Sci. USA 109, 5953–5957 (2012).

    CAS  Article  Google Scholar 

  28. 28.

    Yoon, K. et al. Atomistic-scale simulations of defect formation in graphene under noble gas ion irradiation. ACS Nano 10, 8376–8384 (2016).

    CAS  Article  Google Scholar 

  29. 29.

    Saito, M., Yamashita, K. & Oda, T. Magic numbers of graphene multivacancies. Jpn J. Appl. Phys. 46, L1185–L1187 (2007).

    CAS  Article  Google Scholar 

  30. 30.

    Baskin, A. & Král, P. Electronic structures of porous nanocarbons. Sci. Rep. 1, 36 (2011).

    Article  Google Scholar 

  31. 31.

    Voter, A. F. in Radiation Effects in Solids (eds Sickafus, K. E., Kotomin, E. A. & Uberuaga, B. P.) 1–23 (Springer, Dordrecht, 2007).

  32. 32.

    Govind Rajan, A., Warner, J. H., Blankschtein, D. & Strano, M. S. Generalized mechanistic model for the chemical vapor deposition of 2D transition metal dichalcogenide monolayers. ACS Nano 10, 4330–4344 (2016).

    CAS  Article  Google Scholar 

  33. 33.

    Masel, R. I. Chemical Kinetics and Catalysis (Wiley, New York, 2001).

  34. 34.

    Marcus, R. A. Theoretical relations among rate constants, barriers, and Broensted slopes of chemical reactions. J. Phys. Chem. 72, 891–899 (1968).

    CAS  Article  Google Scholar 

  35. 35.

    Evans, M. G. & Polanyi, M. Inertia and driving force of chemical reactions. Trans. Faraday Soc. 34, 11 (1938).

    CAS  Article  Google Scholar 

  36. 36.

    Singh, A. K., Penev, E. S. & Yakobson, B. I. Armchair or zigzag? A tool for characterizing graphene edge. Comput. Phys. Commun. 182, 804–807 (2011).

    CAS  Article  Google Scholar 

  37. 37.

    Wang, W. L. et al. Direct observation of a long-lived single-atom catalyst chiseling atomic structures in graphene. Nano. Lett. 14, 450–455 (2014).

    CAS  Article  Google Scholar 

  38. 38.

    Lisi, N. et al. Contamination-free graphene by chemical vapor deposition in quartz furnaces. Sci. Rep. 7, 9927 (2017).

    Article  Google Scholar 

  39. 39.

    Markov, I. V. Crystal Growth for Beginners (World Scientific, Singapore, 1995).

  40. 40.

    Jónsson, H., Mills, G. & Jacobsen, K. W. in Classical and Quantum Dynamics in Condensed Phase Simulations 385–404 (World Scientific, Singapore, 1998).

  41. 41.

    Meyer, J. C. et al. Accurate measurement of electron beam induced displacement cross sections for single-layer graphene. Phys. Rev. Lett. 108, 196102 (2012).

    Article  Google Scholar 

  42. 42.

    Bonchev, D. & Rouvray, D. H. (eds) Chemical Graph Theory: Introduction and Fundamentals (Abacus, New York, 1991).

  43. 43.

    Skowron, S. T., Lebedeva, I. V., Popov, A. M. & Bichoutskaia, E. Energetics of atomic scale structure changes in graphene. Chem. Soc. Rev. 44, 3143–3176 (2015).

    CAS  Article  Google Scholar 

  44. 44.

    Robertson, A. W. et al. Spatial control of defect creation in graphene at the nanoscale. Nat. Commun. 3, 1144 (2012).

    Article  Google Scholar 

  45. 45.

    Togo, A. & Tanaka, I. First principles phonon calculations in materials science. Scr. Mater. 108, 1–5 (2015).

    CAS  Article  Google Scholar 

  46. 46.

    Meyer, J. C., Chuvilin, A., Algara-Siller, G., Biskupek, J. & Kaiser, U. Selective sputtering and atomic resolution imaging of atomically thin boron nitride membranes. Nano. Lett. 9, 2683–2689 (2009).

    CAS  Article  Google Scholar 

  47. 47.

    Ryu, G. H. et al. Atomic-scale dynamics of triangular hole growth in monolayer hexagonal boron nitride under electron irradiation. Nanoscale 7, 10600–10605 (2015).

    CAS  Article  Google Scholar 

  48. 48.

    Kotakoski, J., Jin, C. H., Lehtinen, O., Suenaga, K. & Krasheninnikov, A. V. Electron knock-on damage in hexagonal boron nitride monolayers. Phys. Rev. B 82, 113404 (2010).

    Article  Google Scholar 

  49. 49.

    Gilbert, S. M. et al. Fabrication of subnanometer-precision nanopores in hexagonal boron nitride. Sci. Rep. 7, 15096 (2017).

    Article  Google Scholar 

  50. 50.

    VandeVondele, J. et al. Quickstep: fast and accurate density functional calculations using a mixed Gaussian and plane waves approach. Comput. Phys. Commun. 167, 103–128 (2005).

    CAS  Article  Google Scholar 

  51. 51.

    Hutter, J., Iannuzzi, M., Schiffmann, F. & VandeVondele, J. cp2k: atomistic simulations of condensed matter systems. Wiley Interdiscip. Rev. Comput. Mol. Sci. 4, 15–25 (2014).

    CAS  Article  Google Scholar 

  52. 52.

    Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).

    Article  Google Scholar 

  53. 53.

    Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758–1775 (1999).

    CAS  Article  Google Scholar 

  54. 54.

    Kresse, G. & Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, 558–561 (1993).

    CAS  Article  Google Scholar 

  55. 55.

    Kresse, G. & Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15–50 (1996).

    CAS  Article  Google Scholar 

  56. 56.

    Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

    CAS  Article  Google Scholar 

  57. 57.

    Henkelman, G., Uberuaga, B. P. & Jónsson, H. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J. Chem. Phys. 113, 9901 (2000).

    CAS  Article  Google Scholar 

  58. 58.

    VandeVondele, J. & Hutter, J. Gaussian basis sets for accurate calculations on molecular systems in gas and condensed phases. J. Chem. Phys. 127, 114105 (2007).

    Article  Google Scholar 

  59. 59.

    Goedecker, S., Teter, M. & Hutter, J. Separable dual-space Gaussian pseudopotentials. Phys. Rev. B 54, 1703–1710 (1996).

    CAS  Article  Google Scholar 

  60. 60.

    Grimme, S., Antony, J., Ehrlich, S. & Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 132, 154104 (2010).

    Article  Google Scholar 

  61. 61.

    Grimme, S., Ehrlich, S. & Goerigk, L. Effect of the damping function in dispersion corrected density functional theory. J. Comput. Chem. 32, 1456–1465 (2011).

    CAS  Article  Google Scholar 

  62. 62.

    Gillespie, D. T. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434 (1976).

    CAS  Article  Google Scholar 

  63. 63.

    Wu, Y. A. et al. Large single crystals of graphene on melted copper using chemical vapor deposition. ACS Nano 6, 5010–5017 (2012).

    CAS  Article  Google Scholar 

  64. 64.

    Fan, Y., He, K., Tan, H., Speller, S. & Warner, J. H. Crack-free growth and transfer of continuous monolayer graphene grown on melted copper. Chem. Mater. 26, 4984–4991 (2014).

    CAS  Article  Google Scholar 

Download references


We acknowledge the Army Research Office (grant 64655-CH-ISN to M.S.S. via the Institute for Soldier Nanotechnologies) for the work on graphene, US Department of Energy (DOE), Office of Science, Basic Energy Sciences (grant DE-FG02-08ER46488 Mod 0008, to M.S.S. and A.G.R.) for the work on hBN, the National Science Foundation (NSF) (grant CBET-1511526, to D.B. and A.G.R.) for modelling the interactions of etchant atoms with 2D materials and the DOE CSGF (grant DE-FG02-97ER25308, to K.S.S.). This work used the XSEDE supercomputing resources, which are supported using NSF grant ACI-1053575. Sample preparation/imaging (Fig. 3c) was conducted at the Center for Nanophase Materials Sciences, by P. Bedworth, S. Heise and D. Cullen. We thank Z. Yuan, R. P. Misra, A. Cardellini and D. Kozawa for discussions.

Author information




A.G.R., D.B. and M.S.S. formulated the solution to the ICP, including the isomer-distinguishing methodology. A.G.R. carried out ab initio and KMC simulations and performed data analysis. K.S.S. assisted in formulating the isomer distinguishing methodology. J.S. prepared the graphene nanopore sample depicted in Fig. 3c. A.W.R. and J.H.W. contributed to understanding the kinetics of silicon-catalysed etching of graphene nanopores and provided TEM images of graphene nanopores depicted as Fig. 3b. A.G.R., D.B. and M.S.S. wrote the manuscript. All authors commented on the final version of the manuscript.

Corresponding author

Correspondence to Michael S. Strano.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Figures 1–18, Supplementary Tables 1–7, Supplementary References 1–23.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Govind Rajan, A., Silmore, K.S., Swett, J. et al. Addressing the isomer cataloguing problem for nanopores in two-dimensional materials. Nature Mater 18, 129–135 (2019).

Download citation

Further reading