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Theoretical Properties of Materials Formed as Wire Network Graphs from Triply Periodic CMC Surfaces, Especially the Gyroid

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The Role of Topology in Materials

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 189))

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Abstract

We report on our recent results from a mathematical study of wire network graphs that are complements to triply periodic CMC surfaces and can be synthesized in the lab on the nanoscale. Here, we studied all three cases in which the graphs corresponding to the networks are symmetric and self-dual. These are the cubic, diamond and gyroid surfaces. The gyroid is the most interesting case in its geometry and properties as it exhibits Dirac points (in 3d). It can be seen as a generalization of the honeycomb lattice in 2d that models graphene. Indeed, our theory works in more general cases, such as periodic networks in any dimension and even more abstract settings. After presenting our theoretical results, we aim to invite an experimental study of these Dirac points and a possible quantum Hall effect. The general theory also allows to find local symmetry groups which force degeneracies aka level crossings from a finite graph encoding the elementary cell structure. Vice-versa one could hope to start with graphs and then construct matching materials that will then exhibit the properties dictated by such graphs.

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Notes

  1. 1.

    Here \(I4_132\) and \(Ia\bar{3}d\) are given in the international or Hermann–Mauguin notation for symmetry groups, see e.g. [4].

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Acknowledgements

BK thankfully acknowledges support from the NSF under the grants PHY-0969689 and PHY-1255409. RK thanks the Simons Foundation for support under the collaboration grant #317149.

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN, 47907, USA

    Ralph M. Kaufmann

  2. Department of Mathematics and Department of Physics and Astronomy, Purdue University, 150 N. University Street, West Lafayette, IN, 47907, USA

    Birgit Wehefritz-Kaufmann

Authors
  1. Ralph M. Kaufmann
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  2. Birgit Wehefritz-Kaufmann
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Correspondence to Birgit Wehefritz-Kaufmann .

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Editors and Affiliations

  1. Department of Physics and Astronomy, Western Kentucky University, Bowling Green, Kentucky, USA

    Sanju Gupta

  2. Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA

    Avadh Saxena

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Kaufmann, R.M., Wehefritz-Kaufmann, B. (2018). Theoretical Properties of Materials Formed as Wire Network Graphs from Triply Periodic CMC Surfaces, Especially the Gyroid. In: Gupta, S., Saxena, A. (eds) The Role of Topology in Materials. Springer Series in Solid-State Sciences, vol 189. Springer, Cham. https://doi.org/10.1007/978-3-319-76596-9_7

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