Abstract
Topological constraint theory (TCT) has revealed itself to be a powerful tool in interpreting the behaviors of amorphous solids. The theory predicts a transition between a ‘rigid’ overconstrained network and a ‘floppy’ underconstrained network as a function of connectivity or average coordination number, . The predicted results have been shown experimentally for various glassy materials, the majority of these being based on four-fold-coordinate networks such as chalcogenide and oxide glasses. Here, we demonstrate the broader applicability of topological constraint theory to uniquely coordinated amorphous hydrogenated boron carbide (a-BC:H), based on six-fold-coordinate boron atoms arranged into partially hydrogenated interconnected twelve-vertex icosahedra. We have produced a substantial set of plasma-enhanced chemical vapor deposited a-BC:H films with a large range of densities and network coordination, and demonstrate a clear threshold in Young’s modulus as a function of , ascribed to a rigidity transition. We investigate constraint counting strategies in this material and show that by treating icosahedra as ‘superatoms,’ a rigidity transition is observed within the range of the theoretically predicted c value of 2.4 for covalent solids with bond-stretching and bond-bending forces. This experimental data set for a-BC:H is unique in that it represents a uniform change in connectivity with and demonstrates a distinct rigidity transition with data points both above and below the transition threshold. Finally, we discuss how TCT can be applied to explain and optimize mechanical and dielectric properties in a-BC:H and related materials in the context of microelectronics applications.
Highlights
Understanding the fundamental science of amorphous solids remains an essential problem in condensed matter physics (Billinge and Levin, 2007; Berthier and Biroli, 2010; Huang et al, 2013; Mauro, 2018)
The topological constraint theory (TCT) model predicts a rigidity transition at a critical average coordination value, r c, when the number of atomic degrees of freedom equals the number of atomic constraints −2.4 in 3D covalent networks—demarcating a threshold between an underconstrained or “floppy” network and an overconstrained or “rigid” network, with properties scaling above the rigidity threshold as a function of network coordination (He and Thorpe, 1985)
Topological constraint theory allows us to determine the number of zero-frequency modes or floppy modes, f, within a network— which can be related to its rigidity and glass properties— based on the difference between its atomic degrees of freedom, nd, and atomic constraints, nc
Summary
Understanding the fundamental science of amorphous solids remains an essential problem in condensed matter physics (Billinge and Levin, 2007; Berthier and Biroli, 2010; Huang et al, 2013; Mauro, 2018). As an extension to Maxwell’s work on the rigidity of structural trusses (Maxwell, 1864), TCT was advanced by Phillips and Thorpe as a model for understanding the mechanical properties of amorphous glasses as a function of average atomic constraints (Phillips, 1979; Thorpe, 1983; Phillips and Thorpe, 1985; Thorpe et al, 2002). The data span the “floppy,” “transition,” and “rigid” regimes, and—assuming no underlying phase transition exists—represent a direct change in network coordination and connectivity not obscured by additional chemical or structural ordering contributions This result is an essential contribution in supporting the generality and versatility of TCT for predicting and understanding the properties of amorphous solids
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