Fractional Thermal Memory Reshapes Light-Induced Deformation in Fiber-Reinforced Silicon

When a laser strikes the surface of a semiconductor, the material does not respond along a single physical pathway. Light generates heat and excess charge carriers at once, the heat drives mechanical deformation, the carriers add their own strain, and any magnetic field present pushes back through the Lorentz force. A new theoretical study published in Frontiers in Physics on 23 October 2026 has assembled all of these interactions, plus rotation and fiber reinforcement, into a single mathematical model of a silicon half-space, and it finds that the material’s thermal memory is the dominant lever controlling how strongly the whole coupled system responds.

The research team, led by Adel Emam of the Arab Open University in Egypt together with M. Yusuf and A. El-Dali of Capital University in Helwan, Egypt, and Zaki Mrzog Alaofi of King Khalid University in Saudi Arabia, developed what they call a fractional modified photo-thermoelastic model. Their central innovation is the incorporation of a Liouville–Weyl fractional-order time derivative into the Lord–Shulman heat-conduction equation. In plain terms, the classical heat equation assumes that the temperature at any instant depends only on the current state of the material. The fractional version lets the thermal response depend on the material’s accumulated thermal history, which is precisely what happens in real semiconductors under rapid laser loading, where heat does not simply diffuse away but propagates as a wave with finite speed while leaving a hereditary imprint. The fractional order, denoted by the parameter beta, controls the strength of this memory: the value one recovers the classical Lord–Shulman relaxation operator, and smaller values encode progressively stronger hereditary behavior.

The physical setting is deliberately complex. The model describes a semi-infinite silicon medium whose surface is irradiated by an optical source, rotating at a uniform angular velocity about the axis perpendicular to the surface, and immersed in a constant external magnetic field applied normal to the plane of deformation. Continuous reinforcing fibers are assumed to run parallel to the surface along a preferred direction, giving the silicon an anisotropic stiffness that varies with loading direction. Each of these features enters the governing equations in a specific way: the fibers modify the elastic constants through reinforcement coefficients and a longitudinal shear modulus associated with the fiber direction; the magnetic field generates induced electric and magnetic perturbations via linearized Maxwell equations, producing Lorentz-force body forces; and rotation contributes both Coriolis and centrifugal accelerations to the equations of motion, the first coupling the displacement components through velocity and the second adding displacement-dependent inertia.

The coupling runs in every direction. The stress tensor is tied not only to strain and temperature but also to the excess carrier concentration, through carrier-expansion parameters that capture how photo-generated charge carriers deform the lattice. The carrier-density equation, in turn, is governed by diffusion, recombination, and a thermally driven generation term, so carriers cannot be treated independently of the temperature field. Heat conduction itself is anisotropic and coupled to displacement gradients and carrier density. The result is a tightly interwoven system in which temperature, carrier density, displacement, and stress each feed back into the others, all under rotation, magnetic loading, and fractional thermal memory, with the physical properties entering the equations through compact dimensionless coefficient groups rather than as isolated constants.

To solve the system analytically, the authors employed a normal-mode technique in which every field quantity is expressed as an exponential in time and a harmonic wave in the transverse direction. Because the fractional operator acts on exponential functions as an eigenfunction, the fractional time derivative reduces to a complex fractional power of the temporal parameter, converting the partial differential equations into ordinary differential equations in depth. A state-space reformulation then turned these into a first-order matrix problem, whose eighth-order characteristic polynomial yields four physically admissible attenuation eigenvalues, all satisfying the boundedness condition required for a semi-infinite domain. Eigenvectors follow from the characteristic equation, boundary conditions at the irradiated surface determine the modal amplitudes, and the superposition of eigenmodes gives closed-form expressions for every field. The authors verified their numerical procedure, carried out in Wolfram Mathematica, with residual norms of the eigenvalue–eigenvector pairs and of the assembled boundary-condition system, all reported at very small magnitudes.

The numerical implementation uses the material properties of silicon in SI units as dimensional input for the dimensionless coefficients. The authors deliberately stress that the optical excitation enters as an idealized boundary condition, a prescribed harmonic surface temperature and a prescribed photo-generated carrier concentration, rather than a fully resolved absorption and photogeneration model, so optical absorption depth, reflectivity, and photon-energy-dependent generation are not explicitly computed. Two limiting-case checks anchor the formulation: suppressing rotation, magnetism, and fractional memory recovers an established classical fiber-reinforced photo-thermoelastic formulation, and suppressing only the fractional term while retaining rotation and magnetism recovers a rotating magneto-photo-thermoelastic limit reported earlier in the literature. These comparisons demonstrate that the proposed formulation consistently recovers established formulations when the additional physical effects are removed.

The headline finding concerns the fractional-order parameter. The computations show that decreasing the parameter substantially lowers the amplitude of the temperature field, the two displacement components, and all three stress components. Strong memory delays and spreads the transport of thermal energy over longer intervals, so less energy reaches the elastic subsystem at any given moment. Conversely, the classical case produces the highest temperature, the largest deformation, and the sharpest stress concentrations, including a compressive region near the illuminated surface followed by a tensile overshoot before the fields decay to zero. The excess carrier density, by contrast, is nearly insensitive to the fractional parameter; its curves for different memory strengths almost overlap, indicating that carrier transport is governed primarily by diffusion and recombination rather than by thermal history. In the fiber-reinforced medium, the shear stress alternates between positive and negative values near the surface, and these amplitudes grow as the fractional parameter approaches unity, reflecting stronger shear interactions between the thermal and elastic fields.

Rotation tells a different story. Increasing the angular velocity leaves the temperature and carrier-density profiles essentially unchanged, with their curves nearly overlapping for all rotational values considered, but it noticeably reshapes the mechanical response. The longitudinal displacement and the corresponding normal stress both shrink as rotation increases, while the transverse displacement and the shear stress grow. The authors attribute this to Coriolis coupling, which redirects part of the mechanical energy from the longitudinal direction into transverse motion, while centrifugal inertia restrains the deformation along the wave-propagation direction. Rotation thus acts as a redistributor of stress and energy between the principal directions rather than a blanket amplifier or suppressor of the coupled fields, and the authors note that rotational effects can be relevant when analyzing semiconductor and microelectronic systems subjected to rotational conditions.

The magnetic field behaves in yet another way. Temperature and carrier density again show negligible sensitivity to changes in field intensity, but the displacement and stress components respond clearly. Raising the magnetic field reduces the longitudinal and transverse displacement amplitudes and dampens both the compressive and shear stress peaks. The physical explanation is a damping-like electromagnetic resistance: motion of the conducting medium in the magnetic field induces currents, and the resulting Lorentz forces oppose that motion, converting a portion of the mechanical energy into electromagnetic energy. In effect, the magnetic field stiffens the medium and suppresses excessive deformation, most visibly near the illuminated boundary where the coupled interactions are strongest, while the thermal and carrier fields remain governed primarily by optical excitation, heat diffusion, and carrier recombination.

The authors position the framework as a theoretical foundation rather than a design prescription. They suggest it could inform future investigations of semiconductor devices, photonic structures, microelectromechanical and nanoelectromechanical systems, and optical sensors, particularly where rapid thermal loading, rotation, or magnetic fields are present and classical thermoelastic models would miss the memory-dependent behavior that the fractional operator captures. The study’s limitations are explicit: the results are analytical predictions for an idealized silicon half-space with prescribed boundary excitations, two-dimensional deformation, linearized field equations, and small deformations, and no experimental validation is presented within the paper. Even so, the work demonstrates that in a strongly coupled magneto-photo-thermoelastic system, the choice of heat-conduction model, specifically how much thermal memory the mathematics allows, changes nearly every predicted field amplitude, while rotation and magnetic loading modulate the shape of the mechanical response rather than the carrier physics. For anyone modeling laser-excited semiconductor structures, that hierarchy of effects is the study’s most consequential takeaway.

Subject of Research: A fractional modified photo-thermoelasticity model of a rotating magnetized fiber-reinforced silicon semiconductor half-space under laser excitation

Article Title: Fractional thermal memory effects in a rotating magneto-photo-thermoelastic anisotropic fiber-reinforced semiconductor half-space within the framework of modified photo-thermoelasticity

Article References: Emam, A., Yusuf, M., El-Dali, A., & Alaofi, Z. M. (2026). Fractional thermal memory effects in a rotating magneto-photo-thermoelastic anisotropic fiber-reinforced semiconductor half-space within the framework of modified photo-thermoelasticity. Frontiers in Physics, 14, Article 1964122. https://doi.org/10.3389/fphy.2026.1964122

Image Credits: AI Generated

DOI: 10.3389/fphy.2026.1964122

Keywords: photo-thermoelasticity, fractional calculus, thermal memory, semiconductors, fiber reinforcement, magnetoelastic coupling, rotation, silicon, laser excitation, carrier diffusion, wave propagation, analytical modeling