From incommensurate bilayer heterostructures to Allen-Cahn: An exact thermodynamic limit
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910503083704320 |
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| author | Hott, Michael Watson, Alexander B. Luskin, Mitchell |
| author_facet | Hott, Michael Watson, Alexander B. Luskin, Mitchell |
| contents | We give a complete and rigorous derivation of the mechanical energy for twisted 2D bilayer heterostructures without any approximation beyond the existence of an empirical many-body site energy. Our results apply to both the continuous and discontinuous continuum limit. Approximating the intralayer Cauchy-Born energy by linear elasticity theory and assuming an interlayer coupling via pair potentials, our model reduces to a modified Allen-Cahn functional. We rigorously control the error, and, in the case of sufficiently smooth lattice displacements, provide a rate of convergence for twist angles satisfying a Diophantine condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_18186 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | From incommensurate bilayer heterostructures to Allen-Cahn: An exact thermodynamic limit Hott, Michael Watson, Alexander B. Luskin, Mitchell Mathematical Physics Mesoscale and Nanoscale Physics Materials Science Analysis of PDEs Dynamical Systems 82-10, 74-10, 70-10, 37A44, 37A46, 35Qxx, 35Q56 We give a complete and rigorous derivation of the mechanical energy for twisted 2D bilayer heterostructures without any approximation beyond the existence of an empirical many-body site energy. Our results apply to both the continuous and discontinuous continuum limit. Approximating the intralayer Cauchy-Born energy by linear elasticity theory and assuming an interlayer coupling via pair potentials, our model reduces to a modified Allen-Cahn functional. We rigorously control the error, and, in the case of sufficiently smooth lattice displacements, provide a rate of convergence for twist angles satisfying a Diophantine condition. |
| title | From incommensurate bilayer heterostructures to Allen-Cahn: An exact thermodynamic limit |
| topic | Mathematical Physics Mesoscale and Nanoscale Physics Materials Science Analysis of PDEs Dynamical Systems 82-10, 74-10, 70-10, 37A44, 37A46, 35Qxx, 35Q56 |
| url | https://arxiv.org/abs/2305.18186 |