Lambert W Function Framework for Graphene Nanoribbon Quantum Sensing: Theory, Verification, and Multi-Modal Applications

TL;DR

This work develops a rigorous framework that ties graphene nanoribbon quantum sensing to the Lambert W function via a finite square well analogy, revealing a branch-point-driven mechanism for universal sensitivity enhancement. By reformulating FSW bound-state conditions with $W_k(z)$ and validating band-gap theory against empirical scaling $E_g \approx 1.38/W$, the authors establish a predictive, analytically tractable pathway for designing GNR-based sensors. The universal sensitivity expression $S_X = \mathcal{G}_k \cdot \eta_{enh} \cdot \mathcal{P}_X$, with $\eta_{enh}\propto \delta^{-1/2}$ near $z_c=-1/e$, enables multi-modal performance gains across biomedical, environmental, and physical sensing, including LODs as low as 1 fg/mL (SARS-CoV-2) and 1 fM (PSA). This unified framework offers design principles, performance benchmarks, and a clear route toward experimental validation and integration of next-generation graphene quantum sensors with analytically predictable behavior.

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. 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.

Figures (10)

• Figure 1: Real branches of the Lambert W function. Left: Principal branch $W_0(x)$ and secondary branch $W_{-1}(x)$ coalescing at the branch point $(-1/e, -1)$. Right: Derivative magnitude $|dW_0/dx|$ and sensitivity enhancement factor $\eta_{\text{enh}}$ showing divergent behavior near the branch point, enabling orders-of-magnitude sensitivity enhancement for quantum sensors.
• Figure 2: Complex plane visualization of Lambert W function branches. Panels show $|W_0(z)|$, $|W_{-1}(z)|$, and $|W_1(z)|$ with the branch point at $z = -1/e$ marked (red $\times$). The distinct analytical structure of each branch enables selective utilization in different sensing regimes.
• Figure 3: Finite square well solutions and numerical verification. (a) Graphical solution showing constraint circles for $R = 2, 5, 10, 15$ intersecting transcendental curves (solid: $v\tan v$, dashed: $-v\cot v$). (b) Normalized energy levels $|E_n|/V_0 = (u_n/R)^2$ versus strength parameter for eight quantum states. (c) All seven wavefunctions for $R = 10$, showing alternating even/odd parity and increasing nodal structure. (d) Bound state count versus $R$: numerical results (blue steps) match the analytical formula $\lfloor 2R/\pi \rfloor + 1$ (red dashed) exactly.
• Figure 4: Graphene nanoribbon band structure and electronic properties. (a)--(c) Energy dispersion for AGNRs with $N = 40$ ($E_g \approx 287$ meV), $N = 41$ ($E_g \approx 392$ meV), and $N = 42$ (metallic), demonstrating the three AGNR families son2006energy. (d) Band gap versus width showing exact agreement between theory ($2\pi\hbar v_F/3W$, blue) and empirical formula ($1.38/W$, red dashed). Green markers indicate the calculated values for $N = 40, 41$. (e) Density of states comparing bulk graphene (linear, $D(E) \propto |E|$) with 5 nm GNR (van Hove singularities at subband edges). (f) Fermi level versus carrier density following $E_F = \hbar v_F\sqrt{\pi n}$neto2009graphene.
• Figure 5: Sensitivity enhancement analysis. (a) Enhancement factor $\eta_{\text{enh}}$ versus distance from branch point $\delta$, demonstrating $\delta^{-1/2}$ scaling (numerical: blue solid; theory: red dashed). (b) Relative sensitivity versus nanoribbon width for operating regimes $\epsilon = 0.1, 0.01, 0.001$. Narrower ribbons and smaller $\epsilon$ yield higher sensitivity. (c) Two-dimensional sensitivity map showing optimal design regions (small $L$, small $\delta$). Color scale spans four orders of magnitude.