Sage Journals: Discover world-class research

Abstract

This study investigates the propagation of coupled waves in a nonlocal hygro-piezo-photo-thermoelastic semiconductor medium subjected to dual-moisture diffusion and fractional-order heat conduction. The mathematical model incorporates the effects of long-range nonlocal interactions, piezoelectric coupling, photothermal excitation, and dual-moisture transport mechanisms. A time-fractional Caputo derivative is introduced into the energy equation to account for memory-dependent heat conduction. The coupled system of partial differential equations governing displacement, temperature, carrier density, electric potential, and moisture concentration is analytically solved using the normal mode analysis. Numerical results are presented for cadmium selenide (CdSe) semiconductor material to evaluate the impact of fractional order, nonlocal parameter, pulsed laser excitation, and dual-diffusion effects on wave characteristics. The findings reveal that fractional and nonlocal effects significantly influence wave speed, attenuation, and coupling behavior, offering essential insights for the design of next-generation MEMS, sensors, and optoelectronic devices operating in humid or thermally transient environments.

Keywords

fractional thermoelasticity nonlocal dual-moisture diffusion piezo-photo-hygrothermoelasticity semiconductor carrier dynamics

Introduction

The analysis of wave propagation in semiconductor media under the combined influences of thermoelastic, piezoelectric, photothermal, plasma, and moisture diffusion mechanisms is of vital importance in the development of modern microelectronic, optoelectronic, and MEMS/NEMS technologies.^1–3 These devices often operate under multiphysical loading conditions, including thermal gradients, moisture absorption, mechanical vibrations, and photonic excitation. The dynamic behavior of such systems becomes particularly intricate when interactions among thermal, electrical, photothermal, piezoelectric, and hygrothermal fields are considered simultaneously. These systems typically operate in complex environments, where mechanical vibrations, heat loads, photonic excitations, and moisture exposure act simultaneously. The mutual coupling between mechanical, thermal, electrical, and diffusive processes in such media gives rise to intricate wave behaviors that directly affect device functionality, reliability, and service life. This complexity is especially pronounced under extreme conditions, including transient heating, high-frequency oscillations, and moisture infiltration. The accurate modeling of these interactions is essential for predicting device performance, enhancing reliability, and minimizing failure under real-world conditions.^4–6

Significant advancements have been made in the study of hygrothermal and thermoelastic interactions within solid media. Earlier investigations focused on the interplay between temperature and moisture transport in materials such as composites, polymers, and geomaterials, where the coupling of heat and moisture fluxes was found to play a critical role in stress generation and redistribution.^1–6 The foundational principles of thermoelasticity were first introduced by Biot,⁷ forming the basis for analyzing thermomechanical responses in solids. These principles were later extended through generalized thermoelastic theories proposed by Lord and Shulman⁸ and Green and Lindsay,⁹ which allowed for more accurate modeling of wave propagation involving thermal, electromagnetic, and mechanical fields.^10–14 The integration of piezoelectric effects into thermoelastic models was pioneered by Nowacki^15,16 and subsequently enhanced by Mindlin^17,18, leading to a broader understanding of electro-thermo-mechanical coupling in piezoelectric media. Chandrasekharaiah¹⁹ contributed further by introducing models with finite thermal wave speeds, facilitating more realistic representations of transient heat conduction. In the context of modern microscale applications, these classical theories are being revisited with the inclusion of fractional-order heat conduction models, which incorporate memory-dependent behavior into the energy transport process. This extension, along with the consideration of a thermal decay parameter, enables improved representation of non-Fourier and relaxation phenomena under high-frequency or pulsed thermal loading conditions.

The mechanical behavior of semiconductor materials has been widely investigated due to their critical role in electronic and optoelectronic applications. One of the early seminal works by Mindlin²⁰ emphasized the impact of piezoelectric coupling in semiconductors, which laid the foundation for subsequent analyses exploring the dynamic effects of photothermal excitation and plasma wave interactions on wave behavior in such materials.^21–25 Advances in this field were further made by Khamis et al.²² and Lotfy,²⁶ who analyzed wave propagation under dual-phase-lag heat conduction and memory-dependent thermal models,²⁷ emphasizing the need to account for non-Fourier effects. Building on this, Hobiny and Abbas²⁸ extended the theoretical framework by employing fractional-order thermoelasticity, allowing the description of heat conduction with memory effects and anomalous diffusion, which are particularly relevant at micro- and nanoscales. Despite the contributions of these studies, many of them relied on classical elasticity assumptions and did not address nonlocal elastic effects, which are critical for modeling wave dispersion and attenuation in miniature semiconductor structures.

To overcome these limitations, the present work introduces an advanced theoretical framework that integrates nonlocal elasticity into a fractional-order piezo-photo-hygrothermoelastic model for orthotropic semiconductors.^29–31 This model captures the intricate interplay between elastic deformation, piezoelectric coupling, photothermal excitation, plasma oscillations, and moisture transport, while incorporating the scale-sensitive features of nonlocal continuum mechanics.³² The addition of a nonlocal kernel in the stress-strain relations allows for more accurate prediction of dispersive wave modes, which is particularly relevant in the design of MEMS/NEMS devices with characteristic micro- or nanoscale dimensions.^33–35 Although prior models have addressed thermoelastic or hygrothermal effects in semiconducting or composite media, the full coupling of nonlocal elasticity with fractional-order heat conduction and moisture diffusion remains scarcely addressed in the literature.^36,37 The present study fills this critical research gap by formulating a unified model that combines nonlocality, thermal memory, and decay behavior to more accurately represent wave propagation in complex semiconductor environments.³⁸

In recent years, the propagation of waves in semiconductor and elastic media has been widely studied, particularly in the presence of multiphysics phenomena such as porosity, scale effects, nonlocal behavior, and thermo-piezoelectric coupling. Investigations into single- and double-porosity structures have revealed their substantial influence on shear-horizontal (SH) wave characteristics in periodic porous lattices, significantly modifying wave profiles through porosity-induced heterogeneity.³⁹ Similarly, the effect of dual pore connectivity in micropolar thermoelastic materials has been examined, where wave dispersion and attenuation were found to be strongly dependent on microscale geometry and material parameters.⁴⁰ Coupled moisture and thermal transport in orthotropic hygrothermo-piezoelastic media has also received attention, offering important insights into moisture-temperature interaction mechanisms in engineered materials.⁴¹ The role of nonlocal effects in shear wave propagation has been particularly emphasized in fiber-reinforced poroelastic composites, showing that interfacial discontinuities and nonlocality greatly impact stability and wave speed.⁴²

Further, the influence of thermal memory on wave dynamics in advanced semiconductors has been explored through memory-dependent frameworks such as the Moore–Gibson–Thompson photo-thermoelastic model, highlighting the delay characteristics in energy transport under rapid excitations.⁴³ Additional work on double-porosity magneto-thermoelastic media incorporating microtemperature effects and initial stress conditions has underscored the significance of relaxation time and structural non-uniformity in thermal stress prediction.⁴⁴ Rayleigh wave behavior in nonlocal piezo-thermoelastic solids with voids under thermal memory has been studied as well, revealing complex interactions between nonlocal stress distribution and delayed heat conduction.⁴⁵ Other studies have considered the role of flexoelectric phenomena and viscoelastic coatings on surface waves in geomaterials, demonstrating how surface-bound effects and material damping alter wave dispersion.⁴⁶ Notably, fractional-order and memory-based models have also been applied to evaluate wave reflection in pre-stressed solids with dual porosity, reinforcing the need to account for both material heterogeneity and thermal relaxation in wave modeling.⁴⁷

Although the present model assumes material homogeneity and ideal boundary conditions, these simplifications enable clear assessment of coupled field interactions, serving as a foundational framework for analyzing multiphysical behavior in realistic semiconductor applications.

The modeling of wave propagation has been greatly enhanced through the incorporation of micropolarity, microstretch elasticity, and moisture diffusion effects, enabling more comprehensive descriptions of wave behavior in complex elastic environments. Despite this progress, integrated models that consider the simultaneous impact of nonlocal elasticity, fractional-order heat conduction, decay behavior, and anisotropic piezoelectricity in semiconductor systems remain relatively scarce.⁴⁸ The current study addresses this gap by developing a hybrid theoretical framework that brings together nonlocal stress effects, memory-based thermal conduction, and dual-moisture diffusion mechanisms in an orthotropic semiconductor setting.⁴⁹ This formulation is well-suited for micro- and nanoscale applications where classical theories fail to accurately predict dynamic responses. While earlier efforts have examined piezo-photo-thermoelastic and hygrothermoelastic wave interactions, they have typically neglected the vital role of nonlocal stress and fractional-order thermal diffusion in shaping wave speed, damping, and field coupling.

By integrating these advanced effects into a unified model, the present work offers a more realistic representation of wave propagation behavior in semiconductor devices subjected to combined hygrothermal, photothermal, and piezoelectric excitations. This approach captures the inherent physics of microstructured media more effectively and provides refined tools for analyzing wave dispersion, attenuation, and field interactions under short-duration or high-frequency excitations. The present study aims to develop a comprehensive framework that captures the influence of fractional-order heat conduction and nonlocal elasticity on the dynamic behavior of semiconductor materials subjected to coupled thermal, photothermal, piezoelectric, and moisture diffusion effects. A key novelty of this work lies in the integration of fractional derivatives into the heat conduction equation, enabling the model to account for memory-dependent thermal relaxation and decay phenomena, critical under transient heating or laser excitation. Unlike previous models limited to classical Fourier or dual-phase-lag formulations, the current approach introduces both nonlocal spatial interactions and thermal memory effects, offering a more realistic depiction of wave propagation in micro- and nanoscale semiconductors. To analytically solve the coupled field equations governing displacement, temperature, carrier density, moisture concentration, stress, and electric potential, a normal mode technique is employed. Numerical simulations are performed using the material properties of cadmium selenide (CdSe) to examine the effects of the fractional order, nonlocality, and decay rate on wave profiles. One major advantage of the proposed model is its ability to capture dispersive and attenuated wave behavior that cannot be adequately explained by traditional theories. The results demonstrate that both nonlocal elasticity and the fractional heat decay parameter significantly influence wave amplitudes, propagation speeds, and energy dissipation across the coupled fields. Notably, as the fractional order decreases or the decay parameter increases, the system exhibits slower thermal diffusion, stronger attenuation, and enhanced coupling between thermo-mechanical and moisture effects. These findings underscore the importance of considering scale-dependent and memory-driven effects in the design and analysis of advanced semiconductor devices operating under multiphysical stimuli. The proposed model thus bridges a key gap in the literature by capturing physical behaviors overlooked in earlier studies, particularly in high-frequency and miniaturized applications.

Basic equation

This section presents the fundamental governing equations for the propagation of coupled piezo-photo-hygrothermoelastic disturbances in an orthotropic nonlocal semiconductor, incorporating the effects of fractional-order heat conduction and a thermal decay parameter. The analysis considers a semi-infinite, homogeneous orthotropic semiconductor medium that simultaneously responds to mechanical stress, thermal fields, electric charges, and moisture diffusion. The surface of the material is subjected to external photothermal stimulation and moisture flux, introducing complex multiphysical interactions. The medium is assumed to possess piezoelectric properties and to be both electrically and thermally conductive, with its thermal conductivity varying with temperature. To more accurately capture heat transport phenomena in such environments, the classical heat conduction law is generalized using a fractional time derivative, accounting for memory effects and nonlocal thermal responses. Additionally, a decay parameter is introduced to reflect the relaxation behavior of thermal disturbances over time. The coordinate system is defined such that the x-axis is oriented normal to the material surface and the z-axis lies along the plane of the surface, forming a semi-infinite geometry with (x ≥ 0) (see Figure 1). The main physical fields governing the behavior of the system include the displacement components $u (\vec{r}, t)$ , temperature increment $T (\vec{r}, t)$ , moisture concentration $m (\vec{r}, t)$ , and carrier (charge) density $N (\vec{r}, t)$ . These interdependent fields collectively describe the piezo-photo-hygrothermoelastic response of the nonlocal semiconductor and are formulated mathematically in the subsequent sections.^21,22

Figure 1.

Schematic of the problem.

The constitutive relationship linking stress, strain, temperature, and moisture in an orthotropic nonlocal semiconductor with piezo-hygrothermoelastic coupling is expressed as follows:

(1 - ε^{2} \nabla^{2}) σ_{i j} = σ_{i j}^{'} = c_{i j k l} e_{k l} - β_{i j}^{T} T δ_{i j} - δ_{i j}^{N} N^{*} δ_{i j} - β_{i j}^{m} m δ_{i j} - η_{i j k} E_{k},

(1)

D_{i} = η_{i j k} e_{j k} + ε_{i j} E_{j} + τ_{i} T + d_{i} m, e_{i j} = \frac{1}{2} (u_{i},_{j} + u_{j},_{i}), E_{j} = - φ,_{j} .

(2)

Nonlocal stress contribution modeling small-scale effects via the strain is $ε$ that named the nonlocal parameter. In the experimental findings,^29–31 it was observed that the classical Fourier and Fick laws do not fully capture the complex behavior of coupled heat and mass transport in hygroscopic materials. Their results reveal a bidirectional coupling between heat and moisture fluxes, leading to the manifestation of Dufour and Soret effects. In this coupled framework:

• The Dufour effect represents the influence of moisture concentration gradients on the heat flux.

• The Soret effect characterizes the influence of temperature gradients on the moisture flux.

Accordingly, the expressions for the total heat flux $q_{i}$ and total moisture flux $f_{i}$ are extended to include these cross-effects as follows³⁰:

Modified Heat Flux (Including Dufour Effect):

q_{i} = - D_{i j}^{T} T,_{i} - q_{i j}^{m} m,_{i},

(3)

Modified Moisture Flux (Including Soret Effect):

f_{i} = - D_{i j}^{m} m,_{i} - Q_{i j}^{T} T,_{i} .

(4)

These generalized flux laws indicate that temperature and moisture fields are interdependently coupled through the cross-diffusion phenomena. Incorporating these terms significantly improves the accuracy of predictive models for moisture and heat transport in semiconducting and porous hygroscopic media.

Here, $q^{F}$ denotes the conventional Fourier heat flux, which is determined by the thermal diffusivity $(D^{T})$ and typically represented as $q_{i}^{F} = - D_{i j}^{T} T,_{i}$ In parallel, $f^{F}$ refers to the classical Fickian moisture flux, governed by the moisture diffusivity $(D^{m})$ and is expressed as $f_{i}^{F} = - D_{i j}^{m} m,_{i}$ . The term $q^{D}$ accounts for the additional heat flux resulting from the Dufour effect, while $f^{s}$ represents the supplementary moisture flux attributed to the Soret effect. These thermodynamic cross-effects introduce new coupling parameters, specifically, the cross-diffusivities $q_{i j}^{d}$ and $Q_{i j}^{T}$ , which quantify the mutual interaction between thermal and moisture transport processes. The heat contribution due to moisture gradients is given by $q_{i}^{D} = - q_{i j}^{d} m,_{i}$ , whereas the Soret-induced moisture flux driven by temperature gradients is formulated $f_{i}^{s} = - Q_{i j}^{T} T,_{i}$ .⁵⁰ Orthotropic symmetry reduces the number of independent components in $c_{i j k l} = c_{i j k l},$ $β_{i j}^{T} = β_{i}^{T} δ_{i j}$ , $β_{i j}^{m} = β_{i}^{m} δ_{i j}$ , $δ_{i j}^{N} = δ_{i}^{N} δ_{i j}$ , $q_{i j}^{d} = q_{i}^{d} δ_{i j}$ , $Q_{i j}^{T} = Q_{i}^{T} δ_{i j}$ , $D_{i j}^{E} = D_{i}^{E} δ_{i j}$ , we get:

ρ T_{0} \dot{S} = - q_{i},_{i},

(5)

ρ T_{o} S = T + \frac{B_{i}^{T} T_{o}}{ρ C_{E}} e_{k k} - T_{0} τ_{i} \frac{\partial φ}{\partial z} .

(6)

Equations (5) and (6) lead to the following result:

- q_{i},_{i} = \dot{T} + \frac{B_{i}^{T} T_{o}}{ρ C_{E}} {\dot{e}}_{k k} - T_{0} τ_{k} \frac{\partial \dot{ϕ}}{\partial z} .

(7)

By incorporating the Thomson effect into Fourier’s heat conduction law and accounting for the impact of plasma on thermal transport, while neglecting the Peltier effect, the heat conduction model for semiconductors can be reformulated as follows^51,52:

q_{i} = - D_{i}^{T} T,_{i} - q_{i}^{d} m,_{i} - \int \frac{E_{g}}{τ} N^{*} d x_{i} .

(8)

Upon taking the derivative of equation (8) concerning $x_{i}$ , we arrive at the following relation⁵³:

q_{i, i} = - D_{i}^{T} T,_{i i} - q_{i}^{d} m,_{i i} - \frac{E_{g}}{τ} N^{*} .

(9)

The thermal equation is influenced by both plasma and moisture content, while the plasma transport is affected by temperature and moisture gradients, suitable for piezo-hygro-photo-semiconductor systems. The combination of equations (9) and (10) provides the fundamental relation for thermal and plasma transport in the piezoelectric hygrothermal semiconductor framework or the fractional-order heat conduction equation with Caputo derivative is⁵⁴:

D_{i}^{T} T,_{i i} + q_{i}^{d} m,_{i i} + \frac{E_{g}}{τ} N^{*} - (1 + τ_{0}^{α} {{}^{C}D}_{t}^{α}) (T + \frac{B_{i}^{T} T_{0}}{ρ C_{E}} e_{k k}) + T_{0} τ_{k} \frac{\partial \dot{φ}}{\partial z} = 0,

(10)

where

{{}^{C}D}_{t}^{α}

represents the fractional time derivative (Caputo derivative of order

α \in [0, 1]

) that introduces the thermal memory into the piezoelectric hygrothermal.

Similarly, the equation governing moisture and plasma diffusion within the piezoelectric hygrothermal semiconductor medium can be formulated as follows or fractional-order moisture-plasma diffusion equation³¹:

D_{i}^{m} m,_{i i} + Q_{i}^{T} T,_{i i} + \frac{E_{g}}{τ} N^{*} - (1 + τ_{0}^{α} {{}^{C}D}_{t}^{α}) (m + \frac{B_{i}^{m} m_{0} D_{i}^{m}}{k_{m}} u_{j},_{j}) + m_{0} d_{k} \frac{\partial \dot{ϕ}}{\partial z} = 0 .

(11)

These equations generalize classical heat, moisture, and carrier transport by incorporating fractional-order time derivatives, which capture the memory effects relevant in nanoscale, heterogeneous, or nonlocal media. In modern semiconductor technologies, especially those involving MEMS/NEMS, flexible electronics, and optoelectronic systems, environmental factors such as humidity play a critical role in device behavior and reliability. Moisture ingress can alter material properties, induce hygrothermal stresses, and degrade electronic performance by affecting carrier mobility and dielectric integrity. These effects become even more pronounced under localized heating, such as laser pulse excitation, which enhances moisture diffusion and thermo-mechanical coupling. Therefore, incorporating dual-moisture diffusion into the theoretical framework is essential for accurately capturing the complex interactions among thermal, photothermal, piezoelectric, and hygrothermal fields. This addition ensures that the proposed model remains applicable to real-world operating conditions, particularly in high-precision micro/nanoscale semiconductor environments where environmental sensitivity is a major concern.

When $α = 1$ , the equation simplifies to the traditional Fourier heat conduction model (Lord–Shulman model with one relaxation time), which implies an immediate thermal response. However, when $0 < α < 1$ , the formulation accounts for anomalous or sub-diffusive behavior, which is particularly relevant in nanoscale or structurally heterogeneous piezoelectric semiconductor materials where classical diffusion assumptions fail due to low thermal conductivity. The Caputo fractional derivative of order $α$ for time, applied to a function $n (x, t)$ , is defined as

{{}^{C}D}_{t}^{α} n (x, t) = \frac{\partial^{α}}{\partial t^{α}} n (x, t) = I^{α - 1} \frac{\partial n (x, t)}{\partial t}, 0 \leq α < 1,

(12)

\lim_{α \to 1} \frac{d^{α}}{d t^{α}} n (t) = n^{'} (t), I^{α + 1} n (t) = \frac{1}{Γ (α)} \int_{0}^{t} {(t - ξ)}^{α} n (ξ) d ξ,

(13)

where

Γ (α + 1) = α!

expresses the Gamma function.

Equation of motion for heterogeneous piezoelectric nonlocal semiconductor material is^32,33:

ρ (1 - ξ^{2} \nabla^{2}) \frac{\partial^{2} u_{i} (r, t)}{\partial t^{2}} = σ_{i j, j}^{'} .

(14)

Problem formulation

To reformulate the governing equation of motion equation (14) for the fractional-order nonlocal wave propagation in hygro-piezo-photo-thermoelastic semiconductors with dual-moisture diffusion under 2D deformation when $\overset{⇀}{u} (x, y, z, t) = (u, 0, w)$ , according to equations (1) and (2), we incorporate⁵⁵:

c_{11} \frac{\partial^{2} u}{\partial x^{2}} + c_{55} \frac{\partial^{2} u}{\partial z^{2}} + (c_{13} + c_{55}) \frac{\partial^{2} w}{\partial x \partial z} - δ_{1}^{N} \frac{\partial N^{*}}{\partial x} - β_{1}^{T} \frac{\partial T}{\partial x} - β_{1}^{m} \frac{\partial m}{\partial x} + (η_{31} + η_{15}) \frac{\partial^{2} φ}{\partial x \partial z} = ρ (1 - ε^{2} \nabla^{2}) \frac{\partial^{2} u}{\partial t^{2}},

(15)

c_{55} \frac{\partial^{2} w}{\partial x^{2}} + c_{33} \frac{\partial^{2} w}{\partial z^{2}} + (c_{13} + c_{55}) \frac{\partial^{2} u}{\partial x \partial z} - δ_{3}^{N} \frac{\partial N^{*}}{\partial z} - B_{3}^{T} \frac{\partial T}{\partial z} - β_{3}^{m} \frac{\partial m}{\partial z} + η_{15} \frac{\partial^{2} φ}{\partial x^{2}} + η_{33} \frac{\partial^{2} φ}{\partial z^{2}} = ρ (1 - ε^{2} \nabla^{2}) \frac{\partial^{2} w}{\partial t^{2}},

(16)

where

\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial z^{2}}

Photoexcited carriers’ equation in 2D is⁵⁶:

\frac{\partial N^{*}}{\partial t} = (D_{1}^{E} \frac{\partial^{2} N^{*}}{\partial x^{2}} + D_{3}^{E} \frac{\partial^{2} N^{*}}{\partial z^{2}}) - \frac{N^{*}}{τ} + κ T .

(17)

Fractional-order heat conduction (Caputo form) equation (10) in 2D is

(D_{1}^{T} \frac{\partial^{2} T}{\partial x^{2}} + D_{3}^{T} \frac{\partial^{2} T}{\partial z^{2}}) + (q_{1}^{m} \frac{\partial^{2} m}{\partial x^{2}} + q_{3}^{m} \frac{\partial^{2} m}{\partial z^{2}}) + \frac{E_{g}}{τ} N^{*} - (1 + \frac{τ_{0}^{α}}{α!} \frac{\partial^{α}}{\partial t^{α}}) [T - \frac{T_{0}}{ρ C_{E}} (β_{1}^{T} \frac{\partial u}{\partial x} + β_{3}^{T} \frac{\partial w}{\partial z})] + T_{0} τ_{3} \frac{\partial^{2} φ}{\partial z \partial t} = 0 .

(18)

The moisture-plasma diffusion equation that characterizes the coupled response in a piezoelectric hygrothermal semiconductor medium, derived from equation (11), can be expressed as follows:

(D_{1}^{m} \frac{\partial^{2} m}{\partial x^{2}} + D_{3}^{m} \frac{\partial^{2} m}{\partial z^{2}}) + (Q_{1}^{T} \frac{\partial^{2} T}{\partial x^{2}} + Q_{3}^{T} \frac{\partial^{2} T}{\partial z^{2}}) + \frac{E_{g}}{τ} N^{*} - (1 + \frac{τ_{0}^{α}}{α!} \frac{\partial^{α}}{\partial t^{α}}) [m + \frac{m_{0}}{k_{m}} (β_{1}^{m} D_{1}^{m} \frac{\partial u}{\partial x} + β_{3}^{m} D_{3}^{m} \frac{\partial w}{\partial z})] + m_{0} d_{3} \frac{\partial^{2} φ}{\partial z \partial t} = 0,

(19)

η_{15} \frac{\partial^{2} w}{\partial x^{2}} + η_{33} \frac{\partial^{2} w}{\partial z^{2}} + (η_{31} + η_{15}) \frac{\partial^{2} u}{\partial x \partial z} - ε_{11} \frac{\partial^{2} φ}{\partial x^{2}} - ε_{33} \frac{\partial^{2} φ}{\partial z^{2}} + τ_{3} \frac{\partial T}{\partial z} + d_{3} \frac{\partial m}{\partial z} = 0,

(20)

where

β_{1}^{T} = (c_{11} + c_{13}) α_{11}^{T} + c_{13} α_{11}^{T}

β_{3}^{T} = 2 c_{13} α_{33}^{T} + c_{33} α_{33}^{T}

β_{1}^{m} = (c_{11} + c_{13}) α_{11}^{m} + c_{13} α_{11}^{m}

β_{3}^{T} = 2 c_{13} α_{33}^{m} + c_{33} α_{33}^{m}

$δ_{1}^{N} = (c_{11} + c_{13}) d_{11}^{N} + c_{13} d_{11}^{N},$ $δ_{3}^{N} = 2 c_{13} d_{33}^{N} + c_{33} d_{33}^{N},$ $α_{11}^{T}, α_{33}^{T}$ express the linear thermal expansion constants, $α_{11}^{m}, α_{33}^{m}$ is the moisture expansion coefficients, $d_{11}^{N},$ and $d_{33}^{N}$ represent the electronic deformation coefficient.

Solution of the problem

The analytical solution in this study is obtained using the normal mode method, which assumes a linear, homogeneous, and semi-infinite domain. While this method effectively captures the core multiphysics coupling phenomena, it is limited in handling complex geometries, nonlinear field dependencies, or material heterogeneities that are prevalent in modern device architectures. To investigate the behavior of harmonic wave propagation in the medium under consideration, attention is given to waves traveling within the xz-plane. In order to derive analytical solutions for the coupled system described by equations (15)–(20), the normal mode approach is adopted. This technique involves expressing the physical field variables in a specific functional form, simplifying the analysis. The normal mode method is particularly effective for addressing wave propagation in Multiphysics settings, as it transforms the system of partial differential equations into a more manageable algebraic form. This transformation facilitates a direct examination of wave dispersion, damping, and interaction between the thermoelastic, piezoelectric, photothermal, and plasma-related phenomena, making it an ideal tool for complex semiconductor problems that incorporate nonlocal elasticity and hygrothermal coupling effects^57–59:

[T, N^{*}, u, w, m, φ] (x, z, t) = [\tilde{T} (x), \tilde{N^{*}} (x), \tilde{u} (x), \tilde{w} (x), \tilde{m} (x), \tilde{φ} (x)] e^{ω t + i b z},

(21)

where the angular frequency in this equation is

ω

, the wave number is

b

, and

i = \sqrt{- 1}

. On the other hand, the quantities

\tilde{T} (x), \tilde{N^{*}} (x), \tilde{u} (x), \tilde{w} (x), \tilde{m} (x)

and

\tilde{φ} (x)

represent the amplitude of the main functions. By substituting harmonic wave equation (21) into equations (15)–(20), the system simplifies to a set of six interrelated homogeneous equations that govern the behavior of the field variables:

(a_{11} D^{2} + a_{12}) \tilde{u} + a_{13} D \tilde{w} + a_{14} D \tilde{N^{*}} + a_{15} D \tilde{T} + a_{16} D \tilde{m} + a_{17} D \tilde{φ} = 0,

(22)

(a_{21} D^{2} + a_{22}) \tilde{w} + a_{23} D \tilde{u} + a_{24} \tilde{N^{*}} + a_{25} \tilde{T} + a_{26} \tilde{m} + (a_{27} D^{2} + a_{28}) \tilde{φ} = 0,

(23)

(a_{31} D^{2} + a_{32}) \tilde{N^{*}} + a_{33} \tilde{T} = 0,

(24)

(a_{41} D^{2} + a_{42}) \tilde{T} + (a_{43} D^{2} + a_{44}) \tilde{m} + a_{45} \tilde{N^{*}} + a_{46} \tilde{u} + a_{47} \tilde{w} + a_{48} \tilde{φ} = 0,

(25)

(a_{51} D^{2} + a_{52}) \tilde{m} + (a_{53} D^{2} + a_{54}) \tilde{T} + a_{55} \tilde{N^{*}} + a_{56} D \tilde{u} + a_{57} \tilde{w} + a_{58} \tilde{φ} = 0,

(26)

(a_{61} D^{2} + a_{62}) \tilde{w} D + a_{63} \tilde{u} - (a_{64} D^{2} + a_{65}) \tilde{φ} + a_{66} \tilde{T} + a_{67} \tilde{m} = 0 .

(27)

Where

D = \frac{d}{d x}

a_{11} = c_{11} + ρ ω^{2} ε^{2}, a_{12} = - (c_{55} b^{2} + ρ ω^{2} (1 + ε^{2} b^{2})), a_{13} = i b (c_{13} + c_{55}),

a_{14} = - δ_{1}^{N}, a_{15} = - β_{1}^{T}, a_{16} = - β_{1}^{m}, a_{17} = i b (η_{31} + η_{15}), a_{32} = (b^{2} D_{3}^{E} + ω + \frac{1}{τ}), a_{33} = κ

a_{21} = c_{55} + ρ ω^{2} ε^{2}, a_{22} = - (b^{2} c_{33} + ρ ω^{2} (1 + ε^{2} b^{2})), a_{23} = i b (c_{13} + c_{55})

a_{24} = - i b δ_{3}^{N}, a_{25} = - i b B_{3}^{T}, a_{26} = - i b β_{3}^{m}, a_{27} = η_{15}, a_{28} = - η_{33} b^{2},

a_{31} = D_{1}^{E},

a_{41} = D_{1}^{T}, a_{42} = b^{2} D_{3}^{T} + (1 + \frac{τ_{0}^{α}}{α!} ω^{α}), a_{43} = q_{1}^{m}, a_{44} = - b^{2} q_{3}^{m}, a_{45} = \frac{E_{g}}{τ},

a_{46} = - \frac{T_{0}}{ρ C_{E}} β_{1}^{T} (1 + \frac{τ_{0}^{α}}{α!} ω^{α}), a_{47} = - i b (1 + \frac{τ_{0}^{α}}{α!} ω^{α}) \frac{T_{0}}{ρ C_{E}} β_{3}^{T}, a_{48} = i b ω T_{0} τ_{3},

S $a_{51} = D_{1}^{m}, a_{52} = - (b^{2} D_{3}^{m} + (1 + \frac{τ_{0}^{α}}{α!} ω^{α})), a_{57} = - \frac{i b m_{0}}{k_{m}} (1 + \frac{τ_{0}^{α}}{α!} ω^{α}) β_{3}^{m} D_{3}^{m}, a_{53} = Q_{1}^{T}, a_{54} = - b^{2} Q_{3}^{T}, a_{55} = a_{45}, a_{56} = - \frac{m_{0}}{k_{m}} β_{1}^{m} D_{1}^{m} (1 + \frac{τ_{0}^{α}}{α!} ω^{α}), a_{58} = - i b ω m_{0} d_{3} a_{61} = η_{15}, a_{62} = - η_{33} b^{2}, a_{63} = - i b (η_{31} + η_{15}), a_{64} = ε_{11}, a_{65} = b^{2} ε_{33}, a_{66} = - i b τ_{3}, a_{67} = - i b d_{3} .$ Applying the elimination technique to the system of equations (20)–(25) yields the following outcomes:

(Δ_{1} D^{12} + Δ_{2} D^{10} + Δ_{3} D^{8} + Δ_{4} D^{6} + Δ_{5} D^{4} + Δ_{6} D^{2} + Δ_{7}) {\tilde{T}, \tilde{N^{*}}, \tilde{u}, \tilde{w}, \tilde{m}, \tilde{φ}} (x) = 0 .

(28)

The detailed forms of the coefficients $Δ_{i}, i = 1, . . . . ., 7$ are presented in Appendix A. The factorized solution of equation (28) is given as follows:

(D^{2} - k_{1}^{2}) (D^{2} - k_{2}^{2}) (D^{2} - k_{3}^{2}) (D^{2} - k_{4}^{2}) (D^{2} - k_{5}^{2}) (D^{2} - k_{6}^{2}) {\tilde{T}, \tilde{N^{*}}, \tilde{u}, \tilde{w}, \tilde{m}, \tilde{φ}} (x) = 0 .

(29)

Here, $k_{i}^{2} (i = 1 \to 7)$ represent the roots of equation (29), selected so that their real parts are positive, ensuring the solution remains bounded as $x \to \infty$ . Due to the linear nature of the problem, the general solution of equation (29) can be written as

T^{*} (x) = \sum_{i = 1}^{6} M_{i} (b, ω) e^{- k_{i} x} .

(30)

Likewise, the corresponding solutions for the remaining physical variables can be expressed in the following form:

N^{*} (x) = \sum_{i = 1}^{6} M_{i}^{'} (b, ω) e^{- k_{i} x} = \sum_{i = 1}^{6} h_{1 i} M_{i} (b, ω) e^{- k_{i} x},

(31)

u^{*} (x) = \sum_{i = 1}^{6} {M_{i}}^{″} (b, ω) e^{- k_{i} x} = \sum_{i = 1}^{6} h_{2 i} M_{i} (b, ω) e^{- k_{i} x},

(32)

w^{*} (x) = \sum_{i = 1}^{6} M_{i}^{‴} (b, ω) e^{- k_{i} x} = \sum_{i = 1}^{6} h_{3 i} M_{i} (b, ω) e^{- k_{i} x},

(33)

φ^{*} (x) = \sum_{i = 1}^{6} M_{i}^{(4)} (b, ω) e^{- k_{i} x} = \sum_{i = 1}^{6} h_{4 i} M_{i} (b, ω) e^{- k_{i} x},

(34)

m^{*} (x) = \sum_{i = 1}^{6} M_{i}^{(5)} (b, ω) e^{- k_{i} x} = \sum_{i = 1}^{6} h_{5 i} M_{i} (b, ω) e^{- k_{i} x},

(35)

σ_{x x}^{*} (x) = \sum_{i = 1}^{6} M_{i}^{(6)} (b, ω) e^{- k_{i} x} = \sum_{i = 1}^{6} h_{6 i} M_{i} (b, ω) e^{- k_{i} x},

(36)

σ_{z z}^{*} (x) = \sum_{i = 1}^{6} {M_{i}}^{(7)} (b, ω) e^{- k_{i} x} = \sum_{i = 1}^{6} h_{7 i} M_{i} (b, ω) e^{- k_{i} x},

(37)

σ_{x z}^{*} (x) = \sum_{i = 1}^{6} M_{i}^{(8)} (b, ω) e^{- k_{i} x} = \sum_{i = 1}^{6} h_{8 i} M_{i} (b, ω) e^{- k_{i} x},

(38)

{D_{x}}^{*} (x) = \sum_{i = 1}^{6} M_{i}^{(9)} (b, ω) e^{- k_{i} x} = \sum_{i = 1}^{6} h_{9 i} M_{i} (b, ω) e^{- k_{i} x},

(39)

{D_{z}}^{*} (x) = \sum_{i = 1}^{6} M_{i}^{(10)} (b, ω) e^{- k_{i} x} = \sum_{i = 1}^{6} h_{10 i} M_{i} (b, ω) e^{- k_{i} x} .

(40)

Where

[\begin{array}{c} h_{1 i} \\ h_{2 i} \\ h_{3 i} \\ \begin{array}{c} h_{4 i} \\ h_{5 i} \end{array} \end{array}] = {[\begin{array}{c} a_{14} k_{i} & (a_{11} {k_{i}}^{2} + a_{12}) & a_{13} k_{i} & \begin{array}{c} a_{17} k_{i} & a_{16} k_{i} \end{array} \\ a_{24} & a_{23} k_{i} & (a_{21} {k_{i}}^{2} + a_{22}) & \begin{array}{c} (a_{27} {k_{i}}^{2} + a_{28}) & a_{26} \end{array} \\ (a_{31} {k_{i}}^{2} + a_{32}) & 0 & 0 & \begin{array}{c} 0 & 0 \end{array} \\ \begin{array}{c} a_{45} \\ a_{55} \end{array} & \begin{array}{c} a_{46} \\ a_{56} k_{i} \end{array} & \begin{array}{c} a_{47} \\ a_{57} \end{array} & \begin{array}{c} \begin{array}{c} a_{48} \\ a_{58} \end{array} & \begin{array}{c} (a_{43} {k_{i}}^{2} + a_{44}) \\ (a_{51} {k_{i}}^{2} + a_{52}) \end{array} \end{array} \end{array}]}^{- 1} [\begin{array}{c} - a_{15} k_{i} \\ - a_{25} \\ - a_{33} \\ \begin{array}{c} - (a_{41} {k_{i}}^{2} + a_{42}) \\ - (a_{53} {k_{i}}^{2} + a_{54}) \end{array} \end{array}],

h_{6 i} = c_{11} k_{i} h_{2 i} + i c_{13} b h_{3 i} - δ_{1}^{N} h_{1 i} - β_{1}^{T} - β_{1}^{m} h_{5 i} - i b η_{31} h_{4 i},

h_{7 i} = c_{55} (i b h_{2 i} + k_{i} h_{3 i}) + η_{15} k_{i} h_{4 i},

h_{8 i} = c_{13} k_{i} h_{2 i} + i b c_{33} h_{3 i} - δ_{3}^{N} h_{1 i} - β_{3}^{T} - β_{3}^{m} m + i b η_{33} h_{4 i},

h_{9 i} = η_{15} (i b h_{2 i} + k_{i} h_{3 i}) - ε_{11} k_{i} h_{4 i} + d_{1} h_{5 i}, h_{10 i} = η_{31} k_{i} h_{2 i} + i b η_{33} h_{3 i} + τ_{3} - i b ε_{33} h_{4 i} + d_{3} h_{5 i} .

Here

M_{i}, M_{i}^{'}, M_{i}^{″}, {M_{i}}^{(3)}, {M_{i}}^{(4)}, {M_{i}}^{(5)}, {M_{i}}^{(6)}, {M_{i}}^{(7)}, {M_{i}}^{(8)}, {M_{i}}^{(9)},

and

{M_{i}}^{(6)}, i = 1, 2, 3, 4, 5

are unknown constants, such that each expression satisfies the corresponding governing equations and boundary conditions defined within the model framework.

Boundary conditions

To solve the governing equations, it is essential to impose boundary conditions that accurately represent the physical scenario under investigation. In this study, the surface of the piezoelectric hygrothermal semiconductor medium is influenced by combined mechanical, thermal, plasma, moisture, and electrical effects. As a result, the boundary conditions are carefully designed to incorporate external influences such as heat and plasma excitation, moisture fluctuations, and mechanical loads. These conditions are fundamental in capturing the interplay among temperature, moisture concentration, charge carrier density, electric potential, and stress, critical factors in the analysis of real-world applications like thermal regulation, optoelectronic systems, and humidity-sensitive semiconductor technologies. The selected boundary conditions ensure the model’s physical validity and the attenuation of wave disturbances away from the excitation zone. To determine the integration constants $M_{i}$ , we assume that:

(I) Pulsed laser excitation induces rapid temperature variations in the target medium, often resulting in negligible heat loss to the surroundings due to the short interaction time. This makes pulsed laser stimulation particularly advantageous for absorption-based investigations. When a laser beam strikes the surface of a solid, it can initiate various energy-dependent physical phenomena. A fraction of the laser energy is absorbed and converted into heat, leading to localized thermal expansion and the generation of photothermal and photoacoustic effects. These thermal disturbances propagate through the material, causing measurable responses across multiple coupled fields. In this study, it is assumed that laser pulses are applied along the plane surface of the medium (e.g., at $x = 0$ ), producing a time-dependent heat input. Accordingly, the temperature boundary condition at the irradiated surface takes the form:

T^{*} (0, z, t) = G (z, t) = \frac{16 \tilde{E} χ (1 - \tilde{R})}{t_{n}^{2} R_{G} \sqrt{2 π}} t^{3} \exp (- 2 z^{2} / R_{G}) \exp (- 2 t^{2} / t_{n}^{2}) .

(41)

The laser pulse energy per unit length is denoted by $\tilde{E}$ , where $\tilde{R}$ represents the surface reflectivity, $R_{G}$ is the Gaussian beam radius, $t_{0}$ indicates the rise time of the laser pulse, and $χ$ denotes the optical extinction coefficient of the material. This boundary condition represents the application of an initial thermal input at the free surface, which decays exponentially over time. It simulates scenarios involving rapid thermal shocks, such as those caused by laser pulses or transient heating events, typically observed in semiconductor fabrication and thermal diagnostics. In this context, it results in:

\sum_{i = 1}^{6} M_{i} (b, ω) = G (z, t) \exp (ω t + i b z) = G_{0} .

(42)

Where the function

G_{0}

governs the laser-induced heating on the boundary.

(II) In the case of an electrically insulated surface, no electric flux is allowed to pass through the boundary. This condition is particularly relevant in semiconductor and piezoelectric devices where the surface is not in electrical contact with an external circuit or electrode. Physically, it implies that the normal component of the electric displacement field vanishes at the surface. Mathematically, this translates to a zero-gradient condition for the electric potential at the boundary, expressed as

\frac{\partial φ^{*} (0, b, ω)}{\partial x} = 0 .

(43)

Therefore,

\sum_{i = 1}^{6} \frac{h_{4 i}}{k_{i}} M_{i} (b, ω) = 0 .

(44)

This boundary condition ensures that the electric field remains tangential to the surface, preserving the charge neutrality of the boundary and accurately modeling insulated configurations commonly encountered in MEMS and microelectronic systems.

(III) Under laser excitation, a significant number of charge carriers (electrons and holes) can be generated near the surface of the semiconductor due to photon absorption. This phenomenon is especially prominent in optoelectronic and photoresponsive materials where the energy of the incident laser exceeds the semiconductor band gap. To accurately represent this effect, a boundary condition is imposed on the carrier density at the irradiated surface. It assumes a spatially and temporally localized carrier injection, typically modeled using a Gaussian pulse:

N^{*} (0, b, ω) = n_{0} \exp (- z^{2} / R_{G}) \exp (- t^{2} / t_{n}^{2}) = N_{0},

(45)

where

n_{0}

is the peak carrier density. This condition captures the rapid generation and diffusion of carriers near the surface, which significantly influences the coupled electro-thermo-mechanical response of the semiconductor. It is crucial in modeling short-duration excitations such as pulsed laser irradiation in photonic and thermal sensors. Hence,

\sum_{i = 1}^{6} h_{1 i} M_{i} (b, ω) = N_{0} .

(46)

(IV) Traction-free surface condition is typical in many semiconductor and MEMS applications where the top surface is free-standing or exposed to air. In this scenario, the mechanical stress components normal and tangential to the surface must vanish at the boundary. Mathematically, this is expressed as

σ_{x x}^{*} (0, z, t) = 0 .

(47)

Or,

\sum_{i = 1}^{6} h_{6 i} M_{i} (b, ω) = 0 .

(48)

These boundary conditions ensure that no normal or shear forces act on the surface, allowing the material to deform naturally in response to internal thermal, piezoelectric, and diffusive effects. This assumption is critical for accurately capturing surface wave propagation, stress redistribution, and field interactions in the absence of external mechanical confinement.

(IIV) On the other hand, the surface tangent mechanical condition is

σ_{x z}^{*} (0, b, ω) = 0 .

(49)

\sum_{i = 1}^{6} h_{8 i} M_{i} (b, ω) = 0 .

(50)

(IIIV) In many practical applications, especially in humid or controlled environments, the surface of the semiconductor medium may be exposed to a predefined moisture level. This is particularly relevant in devices sensitive to environmental humidity, such as hygrothermal sensors and moisture-responsive semiconductors. To simulate such conditions, a prescribed boundary condition is imposed on the moisture concentration at the surface. Prescribed moisture concentration (controlled environment) condition at

x = 0

, is

m^{*} (0, b, ω) = m_{0} .

(51)

So,

\sum_{i = 1}^{6} h_{5 i} M_{i} (b, ω) = m_{0} .

(52)

where

m_{0}

represents the imposed moisture concentration on the surface. This condition assumes that the surface is in equilibrium with the surrounding environment, maintaining a constant moisture level due to either saturation or externally controlled humidity. Alternatively, for moisture flux-controlled scenarios, a Neumann-type condition may be applied to represent diffusive exchange. The use of a prescribed moisture concentration is essential for modeling diffusion-driven stresses and field couplings in hygro-piezo-thermoelastic systems operating under environmental exposure or material encapsulation.^60,61

Numerical results and discussions

In this section, numerical simulations are performed using the physical parameters of cadmium selenide (CdSe) as the reference semiconductor material. The resulting computational outcomes serve to demonstrate the effects and predictive capabilities of the proposed theoretical model, highlighting its relevance to physics-based technologies. These findings have potential implications for a wide range of applications in contemporary mechanical systems and semiconductor devices. The specific material constants employed in the simulations are provided in Table 1, with all values presented in SI units.^36,37

Table 1.

The CdSe medium physical constants.

$λ$ $μ$	$6.4 \times 1 0^{10}$ , $6.5 \times 1 0^{10}$	$c_{11}$	$74.1 \times 1 0^{9}$
$ρ$	$5504$	$c_{13}$	$39.3 \times 1 0^{9}$
$T_{0}$	$298$	$c_{33}$	$83.6 \times 1 0^{9}$
$τ$	$5 x 1 0^{- 5}$	$c_{55}$	$13.2 \times 1 0^{9}$
$d_{n}$	$- 9 x 1 0^{- 31}$	$q_{1}^{m}$	$2.1 \times 1 0^{- 7}$
$D_{E}$	$2.5 \times 1 0^{- 3}$	$q_{3}^{m}$	$2.1 \times 1 0^{- 7}$
$E_{g}$	$1.11$	$α_{m}$	$2.68 \times 1 0^{- 3}$
$α_{t}$	$4.14 x 1 0^{- 6}$	$β_{1}^{T}$	$6.21 \times 1 0^{5}$
$k$	$150$	$β_{3}^{T}$	$5.51 \times 1 0^{5}$
$D_{3}^{T}$	$\frac{k}{ρ C_{e}}$	$D_{1}^{T}$	$\frac{k}{ρ C_{e}}$
$D_{3}^{m}$	$0.648 \times 1 0^{- 6}$	$C_{e}$	$260$
$Q_{1}^{T}$	$0.648 \times 1 0^{- 6}$	$m_{0}$	$20 %$
$Q_{3}^{T}$	$0.648 \times 1 0^{- 6}$	$D_{1}^{m}$	$2.1 \times 1 0^{- 7}$
$k_{m} k_{m}$	$2.2 \times 1 0^{- 8}$

The laser parameters used in this problem can be represented by $\overset{̑}{R} = 91 %$ , $χ = 0.001 m^{- 1}$ , $R_{G} = 0.45 mm$ , $t_{n} = 9 p s$ , $\tilde{E} = 10 J$ .⁶²

Recent advancements in thermoacoustic and piezoelectric energy harvesting have demonstrated promising strategies for converting waste heat into usable electrical energy. Zhao⁶³ introduced a convection-driven Rijke–Zhao thermoacoustic-piezo system for efficient thermal energy harvesting, highlighting the potential of integrating thermal and acoustic fields in closed-loop devices. Further extending this concept, Zhao and Chew⁶⁵ developed a thermoacoustic engine that harnesses convective flow to amplify energy conversion performance, offering a foundation for compact and scalable energy systems. In a related study, Liu et al.⁶⁴ presented a theoretical model for a two-degree-of-freedom piezoelectric harvester with mechanical stoppers, illustrating how nonlinear constraints can enhance output stability and energy capture across varying frequencies. Collectively, these studies emphasize the value of multiphysics coupling—particularly thermo-piezoacoustic interactions, in advancing the design of efficient and responsive energy-harvesting devices.

Impact of fractional-order parameter

Figure 2 presents the spatial distribution of key physical fields in a nonlocal piezo-hygrothermal semiconductor medium exposed to pulsed laser excitation, illustrating the effect of the fractional-order parameter α\alphaα on wave propagation. Three cases are compared: $α = 0$ (classical Fourier heat conduction), $α = 0.5$ (fractional sub-diffusion), and $α = 1$ (Lord–Shulman thermal relaxation model). To explore the sensitivity of the model to nonlocal effects, the nonlocal parameter was varied across a representative range consistent with previous studies,^32,33 revealing that higher values notably enhance wave dispersion and dampening, emphasizing the importance of scale-dependent modeling in microscale semiconductors. The subfigures display the distributions of temperature, moisture concentration, carrier density, current density, displacement, and electric potential versus normalized distance. The results demonstrate the influence of fractional-order heat and thermal memory effects on wave dispersion, attenuation, and coupling behavior across the thermoelastic, photothermal, piezoelectric, and moisture diffusion fields. In the top-left subplot, the temperature distribution shows a more rapid decay for $α = 0$ due to instant thermal diffusion, whereas higher $α$ values introduce thermal memory, slowing heat conduction and resulting in greater temperature retention near the surface. Moisture concentration (top-right) displays damped oscillations whose amplitudes and frequencies are significantly affected by $α$ ; fractional models ( $α = 0.5$ and 1) reveal stronger coupling between thermal and hygro fields, indicating that thermal relaxation enhances moisture interaction. The second row illustrates the carrier density and current density profiles, both of which decay with distance from the laser-excited boundary. As $α$ increases, the profiles show a slower decline, reflecting prolonged carrier mobility and delayed current decay due to moderated heat flow and enhanced charge retention, vital in optoelectronic and photothermal transport modeling. The third-row plots display mechanical displacement, where fractional and thermal relaxation models exhibit greater oscillatory behavior and higher wave amplitudes, revealing that both nonlocal elasticity and fractional-order heat conduction significantly contribute to energy preservation and mechanical wave persistence. This trend continues in the electric potential profiles (bottom row), where increasing $α$ leads to stronger and more extended field amplitudes, highlighting the importance of thermal memory in photo-induced electro-mechanical responses. Overall, the presence of the fractional-order parameter modulates the dispersive and attenuative characteristics of all physical fields, thermal, moisture, mechanical, electrical, and plasma, leading to a more accurate depiction of multiphysics wave behavior. The results emphasize that classical models ( $α = 0$ ) underestimate coupling effects and dissipate energy too rapidly, whereas fractional ( $α = 0.5$ ) and Lord–Shulman ( $α = 1$ ) models provide improved physical realism, especially for micro- and nanoscale applications where thermal relaxation, scale-dependent effects, and energy memory mechanisms become crucial. These findings underscore the necessity of incorporating fractional heat conduction and decay parameters in designing advanced semiconductor systems that operate under laser heating, humidity variation, and electro-mechanical coupling.

Figure 2.

Spatial profiles of the primary physical fields in a nonlocal piezo-hygrothermal semiconductor medium subjected to pulsed laser excitation for different values of the fractional-order heat conduction parameter α\alphaα (α = 0: classical theory, α = 0.5: fractional case, α = 1: Lord–Shulman model).

Impact of laser excitation

The provided Figure 3 illustrates the spatial variation of the primary physical fields in a nonlocal piezo-hygrothermal semiconductor medium under fractional-order heat conduction (α = 0.5), comparing cases with and without laser pulse excitation. Each subplot represents a key field quantity, temperature, moisture concentration, carrier density, current density, displacement, and electric potential, plotted against the normalized distance x. The subfigures illustrate the responses of temperature, moisture concentration, carrier density, current density, displacement, and electric potential with respect to normalized distance. The results reveal that laser excitation significantly enhances wave amplitudes, alters decay behavior, and strengthens the coupling between thermal, electrical, mechanical, and moisture fields due to the combined effects of photothermal input, nonlocal elasticity, and thermal memory. In the top-left subplot, the temperature distribution shows a significant rise when laser excitation is introduced, indicating localized heating near the surface and slower decay due to the thermal memory effect introduced by the fractional order. The moisture concentration (top-right) responds similarly; laser heating intensifies thermo-hydro coupling, leading to stronger oscillations and more pronounced diffusion behavior near the excitation zone. In the second row, carrier density and current density reveal the direct impact of photo-excitation, laser energy generates additional electron-hole pairs, raising both the carrier population and current magnitude. With laser input, the carrier density increases sharply near the surface and sustains a higher value over a greater depth, while the current density shows a steeper initial peak and more gradual decay, reflecting enhanced plasma mobility. The displacement field (third row, right) experiences more pronounced oscillations in the presence of laser excitation due to the additional photothermal stress and charge-driven deformation, which enhances elastic wave propagation in the medium. Without laser, the mechanical response is much milder, indicating reduced coupling among thermal, piezoelectric, and elastic components. Finally, the electric potential (bottom-right) is significantly amplified under laser excitation. The higher field amplitude and extended wave reach reflect increased carrier activity and electric field generation due to photothermal and piezoelectric effects. These observations affirm that laser excitation introduces energy that strengthens multiphysical coupling and wave dispersion in all fields. The thermal memory incorporated via the fractional parameter (α = 0.5) allows for non-instantaneous heat diffusion, which in turn prolongs thermal–mechanical–electrical interactions. Overall, the figure demonstrates that laser excitation enhances wave amplitudes, alters decay rates, and intensifies the coupling among the governing physical fields, all of which are essential for modeling realistic semiconductor behavior under dynamic thermal and optical loading.

Figure 3.

Spatial distributions of the main physical field variables in a nonlocal piezo-hygrothermal semiconductor medium under fractional-order heat conduction (α = 0.5), comparing the effects with and without laser pulse excitation.

Comparison between the present study and existing literature

The present study offers a comprehensive framework that integrates fractional-order heat conduction, nonlocal elasticity, and dual-moisture diffusion effects in a piezo-photo-hygrothermoelastic semiconductor medium under laser pulse excitation. Compared to earlier works, which often considered only classical or dual-phase-lag heat theories and neglected size-dependent behaviors, this model provides an improved understanding of wave propagation, attenuation, and field coupling, particularly at micro- and nanoscales. While previous models by Lotfy,²⁶ Hobiny and Abbas,²⁸ and others incorporated photothermal and fractional effects independently, they did not simultaneously account for the interplay between fractional thermal memory, decay parameters, and nonlocal stress interactions. Moreover, earlier works lacked detailed consideration of moisture diffusion coupling with plasma and piezoelectric fields. The present approach advances the literature by capturing the synergistic effects of thermal memory, nonlocal behavior, and moisture interaction, key for accurately modeling semiconductor responses under transient and optically driven environments (see Table 2).

Table 2.

Comparison between the present study and existing works.

Reference	Heat model	Nonlocal effect	Moisture diffusion	Laser excitation	Novel contribution
Lotfy²⁶	Dual-phase-lag (DPL)	No	No	Yes	Photothermal elasticity with memory effect
Hobiny & Abbas²⁸	Fractional order	No	No	No	Fractional thermoelasticity in semiconductors
Khamis et al.²²	Memory-dependent	No	No	Yes	Photothermal stress under memory heat flux
Present study	Fractional order and decay laser excitation	Yes	Yes (dual)	Yes	Full coupling of nonlocal elasticity, fractional heat, piezoelectricity, photothermal effects, and dual-moisture diffusion under pulsed laser excitation

Although numerical simulations in this study focus on CdSe, the theoretical model is material-agnostic and can be extended to other semiconductors by adjusting the material constants; differences in piezoelectric, thermal, and moisture properties would quantitatively alter field responses, but the underlying coupled behavior remains consistent.

Limitations and future work

Despite the comprehensive nature of the proposed fractional-order nonlocal piezo-photo-hygrothermoelastic model, certain idealizations have been adopted to facilitate analytical tractability and focus on the core multiphysical interactions. For instance, the semiconductor material is considered homogeneous and orthotropic, which omits the effects of grain boundaries, doping heterogeneities, or anisotropies beyond the orthotropic level. Similarly, boundary conditions, such as traction-free surfaces, prescribed moisture levels, and laser-induced heat flux, are formulated in a controlled and idealized manner to mimic common experimental setups but may not capture all the intricacies of practical device geometries and interfaces.

Furthermore, the nonlocal elasticity parameter is treated as a tunable quantity, with values guided by theoretical and simulation-based precedent rather than direct experimental measurement for CdSe. While this does not compromise the physical validity of the results, it highlights the need for experimental calibration or high-resolution simulation (e.g., molecular dynamics or finite element analysis) to quantitatively fine-tune the model for specific materials and devices.

In future work, the model may be extended to include material heterogeneity, porosity, anisotropic moisture diffusivity, or complex geometrical domains such as thin films and multilayered structures. Additionally, the influence of boundary contact conditions, temperature-dependent material properties, and time-varying environmental loads may be explored to improve alignment with real-world operational scenarios. Numerical methods, such as the finite element or spectral approaches, could also be employed to validate and generalize the findings beyond the assumptions imposed in the current normal mode analysis framework.

The current analysis assumes linear material behavior and harmonic excitation, which may not capture strongly nonlinear, transient, or damage-driven phenomena such as plastic deformation, carrier recombination nonlinearity, or moisture-induced interface failure. Future research may incorporate nonlinear constitutive laws, rate-dependent thermal transport, or progressive failure models to extend the applicability of the current theory to conditions involving laser ablation, thermal shock, or mechanical fatigue in high-performance micro/nanoscale devices.

The moisture diffusion model used in this study is based on established coefficients and assumes idealized boundary conditions representative of equilibrium-controlled environments. While this is appropriate for early-stage analysis, micro/nanoscale devices may exhibit additional phenomena such as interface resistance, diffusion hysteresis, or surface-driven transport that require refined modeling. Incorporating nonlinear or humidity-dependent diffusion laws, as well as validating predictions with experimental or molecular simulation data, remains an important direction for future work.

Extending the current framework using numerical methods, such as FEM or spectral Galerkin approaches, would enable application to more realistic device configurations and allow for the inclusion of geometric constraints, boundary discontinuities, or interface effects.

The simulations in this work are limited to an idealized 2D geometry with simplified boundary conditions that reflect common experimental setups. While these assumptions facilitate analytical modeling, they may not fully capture the geometric complexity, boundary interactions, and heterogeneities present in practical device designs. To extend the applicability of the results, future studies may employ numerical approaches to simulate finite, layered, or patterned geometries and incorporate realistic surface constraints and material interfaces encountered in semiconductor fabrication and packaging.

Conclusion

This study has developed a generalized fractional-order, nonlocal model to analyze wave propagation in piezo-photo-hygrothermoelastic semiconductor media subjected to pulsed laser excitation. The governing equations incorporate the effects of thermal memory via fractional-order heat conduction, nonlocal elasticity for size-dependent behavior, and dual-moisture diffusion for hygrothermal coupling. Through normal mode analysis and numerical simulation, the influence of the fractional parameter α, and laser excitation on various field variables, temperature, moisture concentration, carrier density, current density, displacement, and electric potential, was thoroughly examined. The results presented in the two figures highlight several key findings. First, fractional heat conduction significantly alters the diffusion behavior, where lower values of α result in enhanced wave attenuation, delayed thermal response, and stronger coupling among the physical fields. Second, laser excitation introduces an additional energy source, dramatically increasing field amplitudes and wave penetration, particularly in the temperature, displacement, and electric potential profiles. These enhancements are evident in both steady-state and transient responses and are amplified by the interaction of laser-induced heat with nonlocal and memory-dependent effects. This shows the critical role of fractional thermal modeling in accurately describing semiconductor behavior under optical, electrical, and mechanical stimuli.

While the present work is primarily theoretical, the model’s parameters are selected based on experimentally reported physical constants for CdSe semiconductors,^36,37 ensuring realistic simulation outcomes. The predicted trends are consistent with previously validated behaviors in simplified limiting cases of thermoelasticity, photo-thermoelastic wave propagation, and fractional-order heat conduction.^26,28,43 As such, this model serves as a predictive platform to guide future experimental and computational validation efforts, especially for micro/nanoscale semiconductor systems subjected to coupled thermal, electrical, and environmental stimuli.

Applications

The proposed model and findings have direct applications in the design and analysis of advanced micro/nano-electronic and optoelectronic devices, particularly those operating under high-frequency, short-pulse, or thermally dynamic environments. Potential applications include thermal management in microchips, performance optimization of piezoelectric sensors and actuators, laser-assisted fabrication of semiconductor devices, and the development of moisture-sensitive components such as MEMS/NEMS-based humidity sensors. Furthermore, the model serves as a predictive tool for reliability assessment in laser-irradiated semiconductor materials, where understanding the coupled multiphysical interactions is essential for durability and functionality in modern device engineering.

The results provide guidance for tailoring material selection and geometric scaling in photothermal and humidity-sensitive semiconductor devices, highlighting that optimizing fractional thermal parameters and nonlocal effects can improve energy efficiency, wave control, and device stability under multiphysical loading.

Supplemental Material

Supplemental Material - Fractional-order nonlocal wave propagation in hygro-piezo-photo-thermoelastic semiconductors with dual-moisture diffusion and laser excitation

Supplemental Material for Fractional-order nonlocal wave propagation in hygro-piezo-photo-thermoelastic semiconductors with dual-moisture diffusion and laser excitation by Ahmed M. Alshehri1 and Khaled Lotfy in Journal of Low Frequency Noise, Vibration and Active Control.

Footnotes

Author’s Note

Khaled Lotfy: National Committee for Mathematics, Academy of Scientific Research and Technology, Cairo, Egypt.

ORCID iD

Khaled Lotfy

Author contributions

All authors have equally participated in the preparation of the manuscript during the implementation of ideas, findings, result, and writing of the manuscript.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Current submission does not contain the pool data of the manuscript, but the data used in the manuscript will be provided on request.*

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Supplemental Material

Supplemental material for this article is available online.

Appendix

References

Sih

Michopoulos

Chou

. Hygro-thermoelasticity. Martinus Niijhoof Publishing, 1986.

Chang

. Transient hygrothermal responses in a solid cylinder by linear theory of coupled heat and moisture. Appl Math Model 1994; 18(8): 467–473.

Benkhedda

Tounsi

Addabedia

. Effect of temperature and humidity on transient hygrothermal stresses during moisture desorption in laminated composite plates. Compos Struct 2008; 82(4): 629–635.

Zenkour

. Hygrothermoelastic responses of inhomogeneous piezoelectric and exponentially graded cylinders. Int J Pres Ves Pip 2014; 119: 8–18.

Chiba

Sugano

. Transient hygrothermoelastic analysis of layered plates with one-dimensional temperature and moisture variations through the thickness. Compos Struct 2011; 93(9): 2260–2268.

Ishihara

Ootao

Kameo

. Hygrothermal field considering nonlinear coupling between heat and binary moisture diffusion in porous media. J Therm Stress 2014; 37(10): 1173–1200.

Biot

. Theory of deformation of a porous viscoelastic anisotropic solid. J Appl Phys 1956; 27(5): 459–467.

Lord

Shulman

. A generalized dynamical theory of thermoelasticity. J Mech Phys Solid 1967; 15: 299–309.

Green

Lindsay

. Thermoelasticity. J Elasticity 1972; 2: 1–7.

10.

Chandrasekharaiah

. Thermoelasticity with second sound: a review. Appl Mech Rev 1986; 39: 355–376.

11.

Chandrasekharaiah

. Hyperbolic thermoelasticity: a review of recent literature. Appl Mech Rev 1998; 51: 705–729.

12.

Sharma

Kumar

Dayal

. Reflection of generalized thermoelastic waves from the boundary of a half-space. J Therm Stress 2003; 26: 925–942.

13.

Lotfy

Hassan

. Normal mode method for two-temperature generalized thermoelasticity under thermal shock problem. J Therm Stress 2014; 37(5): 545–560.

14.

Abo-Dahab

Lotfy

Gohaly

. Rotation and magnetic field effect on surface waves propagation in an elastic layer lying over a generalized thermoelastic diffusive half-space with imperfect boundary. Math Probl Eng 2015; 2015: 1–15.

15.

Nowacki W

. Some general theorems of thermo-piezoelectricity. J Therm Stress 1978; 1: 171–182.

16.

Nowacki

. Foundations of linear piezoelectricity. In: Parkus

(ed). Electromagnetic interactions in ElasticSolids. Springer, Wein, 1979. Chapter 1.

17.

Mindlin

. On the equations of motion of piezoelectric crystals. Prob Contin Mech 1961; 1: 282–290.

18.

Mindlin

. Equations of high-frequency vibrations of thermo piezoelectric crystal plates. Int J Solid Struct 1964; 10: 625–637.

19.

Chandrasekharaiah

. A generalized linear thermoelasticity theory of piezoelectric media. Acta Mech 1984; 71: 293–349.

20.

Mindlin

. Piezoelectricity and anisotropy of crystals. Phys Rev 1962; 126(5): 1511–1524.

21.

Lotfy

Elidy

Tantawi

. Piezo-photo-thermoelasticity transport process for hyperbolic two-temperature theory of semiconductor material. Int J Mod Phys C 2021; 32(7): 2150088.

22.

Khamis

Lotfy

El-Bary

, et al. Thermal-piezoelectric problem of a semiconductor medium during photo-thermal excitation. Waves Random Complex Media 2021; 31(6): 2499–2513.

23.

Tam

. Photothermal investigations in solids and fluids. Academic Press, 1989.

24.

Todorovic

Nikolic

Bojicic

. Photoacoustic frequency transmission technique: electronic deformation mechanism in semiconductors. J Appl Phys 1999; 85: 7716.

25.

Song

Todorovic

Cretin

, et al. Study on the generalized thermoelastic vibration of the optically excited semiconducting microcantilevers. Int J Solid Struct 2010; 47: 1871–1875.

26.

Lotfy

. The elastic wave motions for a photothermal medium of a dual-phase-lag model with an internal heat source and gravitational field. Can J Phys 2016; 94: 400–409.

27.

Lotfy

Sarkar

. Memory-dependent derivatives for photothermal semiconducting medium in generalized thermoelasticity with two-temperature. Mech Time-Dependent Mater 2017; 21: 519–534.

28.

Hobiny

Abbas

. Fractional order G.N. model on photothermal interaction in a semiconductor plane. Silicon 2020; 12: 1957–1964.

29.

Eringen

. Nonlocal polar elastic continua. Int J Eng Sci 1972; 10: 1–16.

30.

Eringen

Edelen

. On nonlocal elasticity. Int J Eng Sci 1972; 10: 233–248.

31.

Eringen

. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 1983; 54: 4703–4710.

32.

Bachher

Sarkar

Lahiri

. Generalized thermoelastic infinite medium with voids subjected to an instantaneous heat sources with fractional derivative heat transfer. Int J Mech Sci 2014; 89: 84–91.

33.

Abbas

Kumar

. Response of thermal source in initially stressed generalized thermoelastic half-space with voids. J Comput Theor Nanosci 2014; 11(6): 1472–1479.

34.

Abbas

Marin

. Analytical solutions of a two-dimensional generalized thermoelastic diffusions problem due to laser pulse. Iran J Sci Technol Trans Mech Eng 2018; 42: 57–71.

35.

Marin

Öchsner

Bhatti

. Some results in Moore‐Gibson‐Thompson thermoelasticity of dipolar bodies, (ZAMM) Zfur Angew. Math. Mech 2020; 100(12): e202000090.

36.

Liu

Wang

. A novel semi-analytical meshless method for the thickness optimization of porous material distributed on sound barriers. Appl Math Lett 2024; 147: 108844.

37.

Hosseini

Zhang

. Plasma-affected photo-thermoelastic wave propagation in a semiconductor Love–Bishop nanorod using strain-gradient Moore–Gibson–Thompson theories. Thin-Walled Struct 2022; 179: 109480.

38.

Abouelregal

Sedighi

Megahid

. Photothermal-induced interactions in a semiconductor solid with a cylindrical gap due to laser pulse duration using a fractional MGT heat conduction model. Arch Appl Mech 2023; 93: 2287–2305.

39.

Dutta

Das

Ahmad

, et al. Comparative analysis of double and single porosity effects on SH-wave induced vibrations in periodic porous lattices. Soil Dynam Earthq Eng 2024; 186: 108919.

40.

Das

Dutta

Craciun

, et al. Size-dependent effect on the interaction of surface waves in micropolar thermoelastic medium with dual pore connectivity. Phys Scripta 2024; 99(6): 065232.

41.

Gupta

Barak

Ahmad

, et al. Response of moisture and temperature diffusivity on an orthotropic hygro-thermo-piezo-elastic medium. J Nonlinear Math Phys 2024; 31(26): 26.

42.

Dutta

Das

Bhengra

, et al. Nonlocal effect on shear wave propagation in a fiber-reinforced poroelastic layered structure subjected to interfacial impulsive disturbance. Soil Dynam Earthq Eng 2024; 176: 108307.

43.

Gupta

Barak

Das

. Vibrational analysis of size-dependent thermo-piezo-photo-electric semiconductor medium under memory-dependent Moore–Gibson–Thompson photo-thermoelasticity theory. Mech Adv Mater Struct 2023; 31(28): 10543–10559.

44.

Gupta

Dutta

Das

, et al. Double poro-magneto-thermoelastic model with microtemperatures and initial stress under memory-dependent heat transfer. J Therm Stress 2023; 46(8): 743–774.

45.

Gupta

Barak

Das

. Impact of memory-dependent heat transfer on Rayleigh waves propagation in nonlocal piezo-thermo-elastic medium with voids. Int J Numer Methods Heat Fluid Flow 2024; 34(4): 1902–1926.

46.

Seema

Singhal

Tiwari

, et al. Rheology-dependent surface wave characteristics in an advanced geomaterial flexoelectric plate with viscoelastic coating. Phys Scripta 2025; 100: 015232.

47.

Das

Dutta

Gupta

, et al. Fractional and memory effects on wave reflection in pre-stressed microstructured solids with dual porosity. Eur J Mech Solid 2025; 111: 105565.

48.

Tochhawng

Singh

Debnath

. Specific heat and magnetic effects in orthotropic piezoelectric micropolar medium under three-phase-lag thermoelastic model. Int J Comput Mater Sci Eng 2023; 12(04): 2350010.

49.

Debnath

Singh

. Propagation of lamb wave in the plate of microstretch thermoelastic diffusion materials. J Braz Soc Mech Sci Eng 2024; 46: 218.

50.

Yadav

Barak

Gupta

. Reflection at the free surface of the orthotropic piezo-hygro-thermoelastic medium. Int J Numer Methods Heat Fluid Flow 2023; 33(10): 3535–3560.

51.

Yadav

. Reflection of plane waves from the free surface of a rotating orthotropic magneto-thermoelastic solid half-space with diffusion. J Therm Stress 2021; 44(1): 86–106.

52.

Yadav

. Reflection of plane waves in a fraction-order generalized magneto-thermo-elasticity in a rotating triclinic solid half-space. Mech Adv Mater Struct 2021; 29(18): 4273–4290.

53.

Marin

Hobiny

Abbas

. The effects of fractional time derivatives in porothermoelastic materials using finite element method. Mathematics 2021; 9(14): 1606.

54.

Marin

Seadawy

Vlase

, et al. On mixed problem in thermoelasticity of type III for Cosserat media. J Taibah Univ Sci 2022; 16(1): 1264–1274.

55.

Mahdy

Mohamed

Lotfy

, et al. Numerical solution and dynamical behaviors for solving fractional nonlinear Rubella ailment disease model. Results Phys 2021; 24: 104091.

56.

Mahdy

Gepreel

Lotfy

, et al. A numerical method for solving the Rubella ailment disease model. Int J Mod Phys C 2021; 32(7): 2150097.

57.

El-Sapa

Lotfy

Elidy

, et al. Photothermal excitation process in semiconductor materials under the effect moisture diffusivity. Silicon 2023; 15: 4171–4182.

58.

Lotfy

Elidy

Tantawi

. Photothermal excitation process during hyperbolic two-temperature theory for magneto-thermo-elastic semiconducting medium. Silicon 2021; 13: 2275–2288.

59.

Mahdy

. Stability, existence, and uniqueness for solving fractional glioblastoma multiforme using a Caputo–Fabrizio derivative. Math Methods Appl Sci 2023; 48: 7360–7377.

60.

El-Mesady

Elsadany

Mahdy

, et al. Nonlinear dynamics and optimal control strategies of a novel fractional-order lumpy skin disease model. Journal of Computational Science 2024; 79: 102286.

61.

Mahdy

Mohamed

. Approximate solution of Cauchy integral equations by using Lucas polynomials. Comput Appl Math 2022; 41: 403.

62.

Lotfy

Elshazly

Halouani

, et al. Piezo-photothermal wave dynamics in an orthotropic hygrothermal semiconductor exposed to heat and moisture flux. Eur Phys J B 2025; 98: 9.

63.

Zhao

. Waste thermal energy harvesting from a convection-driven Rijke–Zhao thermo-acoustic-piezo system. Energy Convers Manag 2013; 66: 87–97.

64.

Liu

Cheng

Zhao

, et al. Theoretical modeling and analysis of two-degree-of-freedom piezoelectric energy harvester with stopper. Sensor Actuator Phys 2016; 245: 97–105.

65.

Zhao

Chew

. Energy harvesting from a convection-driven Rijke-Zhao thermoacoustic engine. J Appl Phys 2012; 112(11): 114507.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.20 MB