<p>This paper develops a distributed-order fractional photo-thermoelastic diffusion model for semiconductor plates subjected to ultrafast laser heating with variable thermal conductivity. The present formulation introduces continuous weight functions into both phase lags of the dual-phase-lag heat conduction law, so that each fractional derivative is integrated over a distribution of orders rather than evaluated at a single fixed order. This construction encodes the full spectrum of thermal relaxation mechanisms from ultrafast carrier–lattice energy exchange to slow moisture-assisted redistribution within a single unified framework that contains all classical, single-fractional, and dual-fractional thermoelastic theories as recoverable limiting cases. The generalized heat equation is coupled with carrier diffusion–recombination, moisture transport, and linear thermoelasticity, and the variable thermal conductivity is incorporated through the Kirchhoff transformation. Five physically motivated weight functions are investigated: Dirac delta (recovering the dual-fractional model as a special case), Gamma-type, Uniform, Gaussian, and Bi-modal distributions. The resulting system of distributed-order fractional integro-differential equations is discretized using Gauss–Legendre quadrature for the order-domain integrals, the L1/L1-2 formulas for the Caputo fractional derivatives, selected according to whether the quadrature-node order lies below or above unity, and second-order central finite differences in space; a rigorous convergence analysis establishes the overall truncation error as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$O(h_\alpha ^{2G} + \Delta t^{2-\alpha _{\max }} + h^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msubsup> <mi>h</mi> <mi>α</mi> <mrow> <mn>2</mn> <mi>G</mi> </mrow> </msubsup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow> <mn>2</mn> <mo>-</mo> <msub> <mi>α</mi> <mo movablelimits="true">max</mo> </msub> </mrow> </msup> <mo>+</mo> <msup> <mi>h</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which is confirmed numerically. Parametric studies on a silicon plate reveal three principal effects of the distributed-order formulation: (i) a memory spectrum smoothing mechanism in which the superposition of multiple relaxation scales attenuates the thermoelastic wave amplitude by 50–<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$60\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>60</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> relative to the dual-fractional reference; (ii) a phase modification effect that elongates the oscillation wavelength of the displacement and stress fields due to dispersive spreading across the memory spectrum; and (iii) a non-trivial interaction between variable thermal conductivity and the memory spectrum, whereby the continuous distribution partially masks the VTC-induced spatial redistribution of thermal energy. These findings demonstrate that distributed-order fractional models capture qualitatively different physics from their fixed-order counterparts and provide a more faithful description of multiscale thermal memory in laser-excited semiconductors.</p>

Memory-spectrum photo-thermoelastic diffusion in ultrafast laser-heated semiconductors: a distributed-order dual-phase-lag model with variable thermal conductivity

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发表日期：

2026年8月3日

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Mumtaz Khan,
M. S. Anwar,
Xiaoting Gan

摘要

This paper develops a distributed-order fractional photo-thermoelastic diffusion model for semiconductor plates subjected to ultrafast laser heating with variable thermal conductivity. The present formulation introduces continuous weight functions into both phase lags of the dual-phase-lag heat conduction law, so that each fractional derivative is integrated over a distribution of orders rather than evaluated at a single fixed order. This construction encodes the full spectrum of thermal relaxation mechanisms from ultrafast carrier–lattice energy exchange to slow moisture-assisted redistribution within a single unified framework that contains all classical, single-fractional, and dual-fractional thermoelastic theories as recoverable limiting cases. The generalized heat equation is coupled with carrier diffusion–recombination, moisture transport, and linear thermoelasticity, and the variable thermal conductivity is incorporated through the Kirchhoff transformation. Five physically motivated weight functions are investigated: Dirac delta (recovering the dual-fractional model as a special case), Gamma-type, Uniform, Gaussian, and Bi-modal distributions. The resulting system of distributed-order fractional integro-differential equations is discretized using Gauss–Legendre quadrature for the order-domain integrals, the L1/L1-2 formulas for the Caputo fractional derivatives, selected according to whether the quadrature-node order lies below or above unity, and second-order central finite differences in space; a rigorous convergence analysis establishes the overall truncation error as $O(h_\alpha ^{2G} + \Delta t^{2-\alpha _{\max }} + h^2)$ $O (h_{α}^{2 G} + Δ t^{2 - α_{max}} + h^{2})$ , which is confirmed numerically. Parametric studies on a silicon plate reveal three principal effects of the distributed-order formulation: (i) a memory spectrum smoothing mechanism in which the superposition of multiple relaxation scales attenuates the thermoelastic wave amplitude by 50– $60\%$ $60 %$ relative to the dual-fractional reference; (ii) a phase modification effect that elongates the oscillation wavelength of the displacement and stress fields due to dispersive spreading across the memory spectrum; and (iii) a non-trivial interaction between variable thermal conductivity and the memory spectrum, whereby the continuous distribution partially masks the VTC-induced spatial redistribution of thermal energy. These findings demonstrate that distributed-order fractional models capture qualitatively different physics from their fixed-order counterparts and provide a more faithful description of multiscale thermal memory in laser-excited semiconductors.