TO DEVELOP QUANTUM TUNNELLING ON CONDUCTION SURFACE FOR HEAT TRANSFERS BY MATRIX LABORATORY PROGRAMMING

Quantum-scale effects increasingly influence heat transfer in modern nanoscale devices. Yet, existing engineering models often rely on classical conduction theory, which cannot accurately capture nonlocal and wave-like thermal behaviours. In particular, quantum tunnelling—where energy carriers such as electrons or phonons penetrate potential barriers—remains insufficiently represented in thermal modelling frameworks. This research addresses four key gaps: the limited integration of quantum heat-transfer theory into engineering computation, the inadequate modelling of tunnelling as a dominant thermal mechanism, the lack of accessible numerical tools for practitioners, and the scarcity of validated MATLAB simulations suited for experimental comparison. This study develops a MATLAB-based computational model for analysing quantum tunnelling on conduction surfaces, with emphasis on nanoscale interfaces where conventional Fourier-based laws break down. The model incorporates transmission coefficients derived from the Schrödinger Equation and integrates them into a modified heat-flux formulation consistent with ballistic–diffusive transport principles. Parametric simulations explore how barrier width, material properties, temperature gradients, and surface potential profiles influence tunnelling-driven heat transfer. Results are compared against analytical benchmarks and available nanoscale thermal measurements to assess accuracy and applicability. Ultimately, the work helps bridge the gap between fundamental quantum heat-transfer theory and practical computational engineering analysis at the nanoscale.