Lambert W Function Framework for Graphene Nanoribbon Quantum Sensing: Theory, Verification, and Multi-Modal Applications
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914258107760640 |
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| author | Chishtie, F. A. Roberts, K. Jisrawi, N. Valluri, S. R. Soni, A. Deshmukh, P. C. |
| author_facet | Chishtie, F. A. Roberts, K. Jisrawi, N. Valluri, S. R. Soni, A. Deshmukh, P. C. |
| contents | We establish a rigorous mathematical framework connecting graphene nanoribbon quantum sensing to the Lambert W function through the finite square well (FSW) analogy. The Lambert W function, defined as the inverse of $f(W) = We^W$, provides exact analytical solutions to transcendental equations governing quantum confinement. We demonstrate that operating near the branch point at $z = -1/e$ yields sensitivity enhancement factors scaling as $η_{\text{enh}} \propto (z - z_c)^{-1/2}$, achieving 35-fold enhancement at $δ= 0.001$. Comprehensive numerical verification confirms: (i) all seven bound states for strength parameter $R = 10$ satisfying the constraint $u^2 + v^2 = R^2$; (ii) exact agreement between theoretical band gap formula $E_g = 2π\hbar v_F/(3L)$ and empirical relation $E_g = 1.38/L$ eV$\cdot$nm; (iii) universal sensitivity scaling across biomedical (SARS-CoV-2, inflammatory markers, cancer biomarkers), environmental (CO$_2$, CH$_4$, NO$_2$, N$_2$O, H$_2$O), and physical (strain, magnetic field, temperature) sensing modalities. This unified framework provides design principles for next-generation graphene quantum sensors with analytically predictable performance. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_10767 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lambert W Function Framework for Graphene Nanoribbon Quantum Sensing: Theory, Verification, and Multi-Modal Applications Chishtie, F. A. Roberts, K. Jisrawi, N. Valluri, S. R. Soni, A. Deshmukh, P. C. Mesoscale and Nanoscale Physics Materials Science Quantum Physics We establish a rigorous mathematical framework connecting graphene nanoribbon quantum sensing to the Lambert W function through the finite square well (FSW) analogy. The Lambert W function, defined as the inverse of $f(W) = We^W$, provides exact analytical solutions to transcendental equations governing quantum confinement. We demonstrate that operating near the branch point at $z = -1/e$ yields sensitivity enhancement factors scaling as $η_{\text{enh}} \propto (z - z_c)^{-1/2}$, achieving 35-fold enhancement at $δ= 0.001$. Comprehensive numerical verification confirms: (i) all seven bound states for strength parameter $R = 10$ satisfying the constraint $u^2 + v^2 = R^2$; (ii) exact agreement between theoretical band gap formula $E_g = 2π\hbar v_F/(3L)$ and empirical relation $E_g = 1.38/L$ eV$\cdot$nm; (iii) universal sensitivity scaling across biomedical (SARS-CoV-2, inflammatory markers, cancer biomarkers), environmental (CO$_2$, CH$_4$, NO$_2$, N$_2$O, H$_2$O), and physical (strain, magnetic field, temperature) sensing modalities. This unified framework provides design principles for next-generation graphene quantum sensors with analytically predictable performance. |
| title | Lambert W Function Framework for Graphene Nanoribbon Quantum Sensing: Theory, Verification, and Multi-Modal Applications |
| topic | Mesoscale and Nanoscale Physics Materials Science Quantum Physics |
| url | https://arxiv.org/abs/2601.10767 |