Lambert W function framework for graphene nanoribbon quantum sensing: Theory, verification, and multi-modal applications

2026/01/31 by F. A. Chishtie, K. Roberts, N. Jisrawi +5
Engineering · Physics and Astronomy · #Experimental and Theoretical Physics Studies #Mechanical and Optical Resonators #Sports Dynamics and Biomechanics #cond-mat.mes-hall #cond-mat.mtrl-sci #quant-ph

paper · pdf · doi:10.1016/j.rineng.2026.112238

published as Results in Engineering, Volume 32, 2026, 112238 · 21 pages, 10 figures, published version at Results in Engineering journal

openalex publication_date 2026/07/29 · openalex created_date 2026/07/30 · arxiv created 2026/07/31 · openalex updated_date 2026/08/01 · arxiv updated 2026/08/04

Abstract

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. Operating near the branch point singularity at z=-1/e yields sensitivity enhancement factors scaling as (z-z_c)^-1/2, achieving 35-fold enhancement when the operating point lies within δ=0.001 of the branch point. Comprehensive numerical verification confirms: (i) all seven bound states for strength parameter R=10 satisfy the constraint u²+v²=R² to machine precision; (ii) the theoretical band gap formula E_g=2πℏ v_F/(3W) is analytically equivalent to the independently determined empirical relation E_g=1.38/W~eV⋅nm, establishing the validity of the FSW-GNR analogy; (iii) a universal sensitivity factorization S_X = G_k ⋅ η_{\rm enh} ⋅ P_X applies across biomedical (SARS-CoV-2, inflammatory markers, cancer biomarkers), environmental (CO₂, CH₄, NO₂, N₂O, H₂O), and physical (strain, magnetic field, temperature) sensing modalities. This unified framework provides analytically predictable design principles for next-generation graphene quantum sensors. The framework is analytic and predictive rather than microscopic or experimental: band-structure and adsorption parameters are taken as inputs from tight-binding, first-principles, and experimental sources, and the framework returns closed-form sensitivity and design relations built upon them. Reported detection limits are labelled throughout as either literature-demonstrated device values or values predicted by the present framework.

Citations

• Tutorial: The quantum finite square well and the Lambert W function
• THE ADSORPTION OF GASES ON PLANE SURFACES OF GLASS, MICA AND PLATINUM.
• Significance of Electromagnetic Potentials in the Quantum Theory
• Do Clusters Generate Greater Innovation and Growth? An Analysis of European Regions. Bruges European Economic Research (BEER) Papers 21/October 2011
• The Grotthuss mechanism
• Edge state in graphene ribbons: Nanometer size effect and edge shape dependence
• Quantal phase factors accompanying adiabatic changes
• The electronic properties of graphene
• Introduction to Quantum Noise, Measurement and Amplification
• Nontoxic concentrations of PEGylated graphene nanoribbons for selective cancer cell imaging and photothermal therapy
• SciPy 1.0: fundamental algorithms for scientific computing in Python
• Controllable terahertz cross-shaped three-dimensional graphene intrinsically chiral metastructure and its biosensing application
• C-reactive protein: a critical update
• “Clickable” graphene nanoribbons for biosensor interfaces
• Climate Change 2021 – The Physical Science Basis
• Three Dimensional Liquid Gated Graphene Field Effect Strain Sensor
• Strain effects on optical properties of linearly polarized resonant modes in the presence of monolayer graphene
• The Role of Density Functional Theory Methods in the Prediction of Nanostructured Gas-Adsorbent Materials
• Band Gap of Strained Graphene Nanoribbons
• Uniaxial strain on graphene: Raman spectroscopy study and bandgap opening
• Strain Effect on Energy Gaps of Armchair Graphene Nanoribbons
• Electric Field Effect in Atomically Thin Carbon Films