Novel multi-temperature photoacoustic dynamics in nanoscale hydro-semiconductors under variable thermal conductivity

Abstract

This study develops a novel theoretical framework to model photoacoustic wave dynamics in such media, accounting for the interplay between photothermal, nonlocal thermomechanical, and hydrodynamic interactions. The effects of thermal conductivity variations on wave behavior are rigorously analyzed by integrating the multi-temperature theory with a hydrodynamic semiconductor model. The proposed model employs advanced mathematical techniques, including normal mode analysis and numerical simulations, to derive and solve coupled governing equations for thermal, acoustic, and optical waves. Graphical representations highlight the sensitivity of wave propagation characteristics to changes in thermal conductivity and multi-temperature interactions. Comparative analyses with single-temperature models demonstrate enhanced accuracy and relevance of the multi-temperature approach, especially in predicting wave dispersion and thermal gradients at the nanoscale. The findings offer critical insights into optimizing thermal and acoustic behavior in semiconductor materials, paving the way for advancements in nano-electronic and photonic device design. 

Introduction

The study of semiconductor materials has become increasingly vital in modern industrial applications due to their indispensable role in optoelectronics, thermoelectric devices, and nanotechnology. With their exceptional electrical, thermal, and optical properties, semiconductors form the foundation of cutting-edge technologies, such as high-speed electronics, renewable energy systems, and sensors for precision applications. Among these, porous semiconductors have garnered significant attention due to their high surface area, tunable porosity, and enhanced photothermal characteristics, making them essential for energy storage, catalysis, and advanced photonic devices. Understanding the behavior of these materials under complex thermal and mechanical conditions is crucial for optimizing their performance in high-demand industrial applications.

The theory of thermal elasticity has undergone significant evolution to address the limitations of classical elasticity, particularly under conditions involving rapid thermal and mechanical changes. Biot’s pioneering work on coupled thermoelasticity¹ introduced the interdependence of thermal and elastic effects, laying the groundwork for studying dynamic thermal behaviors. This was followed by the Lord and Shulman model², which incorporated thermal relaxation time to account for non-Fourier heat conduction, and the Green-Lindsay theory³, which extended the framework by including dual-phase lag effects. These advancements were pivotal in analyzing the thermal and elastic wave interactions in hydrodynamic elastic materials, where tangential forces and fluid-structure coupling significantly influence wave propagation^4–7. The application of these theories to tangential hydrodynamic materials has opened new avenues for understanding complex thermal behaviors in industrial systems^8–12.

The development of semiconductor materials has seen remarkable progress since the early 20th century, transitioning from bulk materials to nanoscale and porous structures^13,14. Porous semiconductors (or hydro-semiconductors), in particular, offer unique advantages such as low density, high thermal conductivity, and enhanced mechanical strength, making them ideal for advanced industrial applications¹⁵. Recent studies have explored their use in thermoelectric converters, optoelectronic devices, and biosensors, demonstrating their versatility^16,17. The interplay of porosity and semiconductor properties, coupled with advancements in nano-fabrication techniques, has further expanded the potential of these materials in modern technology^18–22. Nonlocal semiconductor materials have added another layer of complexity and opportunity to the field. Unlike classical semiconductors, nonlocal materials account for long-range interactions, which become significant at the nanoscale^23,24. The incorporation of variable thermal conductivity into nonlocal models has enhanced our understanding of heat transfer and wave propagation under extreme conditions^25,26. Variable thermal conductivity, influenced by temperature gradients and material properties, has been shown to profoundly affect the behavior of thermal, acoustic, and optical waves, making it an essential factor in designing efficient nanoscale devices^27,28.

The theory of multiple temperatures has emerged as a transformative framework for addressing thermal behaviors where different components of a system exhibit distinct thermal responses²⁹. Initially developed to overcome the limitations of one-temperature models, the multi-temperature theory allows for the representation of thermal nonequilibrium, particularly in systems subjected to high-frequency or high-gradient heat fluxes^30,31. Over time, this theory has been extended to include hyperbolic heat conduction, phase-lag effects, and anisotropic materials, enabling more accurate modeling of thermal wave propagation in complex systems. Recent advancements have applied multi-temperature frameworks to semiconductors and porous materials, enhancing their utility in predicting energy transfer and mechanical deformation under rapid heating conditions³².

Porous thermoelastic materials have gained significant attention due to their unique ability to combine lightweight structures with excellent thermal and mechanical properties³³. Characterized by interconnected voids or pores, these materials exhibit enhanced thermal conductivity, lower density, and increased mechanical flexibility compared to their dense counterparts, making them highly suitable for applications in energy storage, thermal insulation, and mechanical damping systems^34,35. Early studies on porous thermoelastic materials focused primarily on understanding their stress-strain behavior and thermal responses, with emphasis on how porosity and pore distribution influence their overall performance under mechanical and thermal loads^36,37. Over time, advancements in material science and fabrication techniques enabled the integration of semiconductor properties into porous structures, giving rise to a new class of materials known as porous semiconductors^38,39. These poro-semiconductor materials retain the mechanical and thermal advantages of porous systems while incorporating tunable electronic and optical properties, making them ideal for advanced applications such as thermoelectric devices, photonic systems, and energy-efficient sensors^40,41. The ability of semiconductor moisture materials to support coupled thermoelastic and photoelectronic interactions has expanded their utility in nano-optoelectronics and renewable energy technologies, offering a versatile platform for addressing complex industrial challenges⁴².

This work introduces a novel approach to modeling photoacoustic wave propagation in nonlocal hydro-semiconductors under the influence of variable thermal conductivity. Unlike previous studies, which primarily relied on single-temperature models, this research adopts a multi-temperature theoretical framework to capture the intricate thermal and mechanical interactions more comprehensively. The governing equations for thermal, optical, and elastic waves are derived and solved using normal mode analysis. Numerical results are graphically represented to highlight the impact of variable thermal conductivity and multi-temperature interactions on wave propagation. Comparative analyses demonstrate the superiority of this model in predicting complex behaviors, offering new insights into optimizing semiconductor materials for energy, photonic, and high-performance industrial applications.

Mathematical model and basic equations

The mathematical formulation of wave propagation in nonlocal hydro-semiconductor media requires a comprehensive approach that accounts for both the material’s intrinsic properties and the influence of external factors. One of the critical aspects influencing wave behavior in such systems is the variable thermal conductivity, which varies with temperature and plays a pivotal role in thermal energy transport. Unlike constant thermal conductivity models, incorporating its variability provides a more accurate representation of heat transfer, especially in nano-scale systems where temperature gradients are pronounced.

• (i)In the context of photo-excited processes of poro-semiconductor elastic media, particularly during thermal activation parameter effect, the interaction between thermal and plasma phenomena can be expressed in the following dimensional form^45–47:

1

• (ii)The nonlocal motion equation accounts for internal forces influenced by displacements at distant points within the medium, rather than just immediate neighbors. This is particularly important at the nanoscale, where (nonlocal scale parameter) modulates how strongly such long-range interactions affect stress and strain fields. In the context of hydro-semiconductors, where carrier motion and elastic deformations are coupled, the governing equations must capture the intricate interplay of thermal, optical, and mechanical effects⁴⁵:

2

The nonlocal parameter  (nonlocal scale parameter), a dimensionless factor, is a fundamental component in nanoscale modeling. It accounts for spatial interactions that extend beyond local points, reflecting microscale behaviors in hydrodynamic semiconductors, l represents the external length scale typically which expresses the macroscopic spatial dimensions of the system or structure under consideration, a is the internal length scale corresponds to the microscopic properties intrinsic to the material and is a dimensionless factor, characterizes the material’s sensitivity to nonlocal interactions.

• (iii)The heat equation of hydro-semiconductors under the effects of thermal conductivity variability (which depends on the temperature) can be expressed for distinct two-temperature as^45,46:

3

• (iv)This conservation equation is fundamental for analyzing fluid flow, water transport, and interactions within porous media, such as poro-elastic or poro-semiconductor systems, under various hydrodynamic and thermal conditions. The water mass conservation equation for a porous medium can be expressed according to the excess pore water pressure P as³⁶:

4

where and .

• (xxii)In the context of the two-temperature theory (Youssef⁵), the model includes two distinct temperature fields:

• T: the thermodynamic temperature, typically associated with the lattice or solid matrix, and

• : the conductive temperature, associated with the temperature of the heat carriers (e.g., electrons or phonons).

The inclusion of these two temperatures allows for the modeling of non-equilibrium thermal behavior, where thermal energy exchange between the two subsystems occurs over finite time scales. This is particularly important for nano-structured semiconductors, ultrafast laser excitation, or high-frequency thermal processes. The model assumes that both temperatures evolve dynamically and are coupled via a relaxation mechanism. The multi-temperature equation describes the evolution of distinct temperature fields in a medium (thermal equilibrium between different components). It takes the following general form⁴⁴:

5

• (vi)The nonlocal constitutive equations are inherently coupled, with thermal effects influencing mechanical stresses, and hydraulic flow modifying pore pressure and deformation. Such a formulation is crucial for modeling poro-semiconductor materials under dynamic thermal, electrical, and mechanical loading, particularly for applications in optoelectronics and energy harvesting systems^32,36:

6

In practice, the thermal conductivity of semiconductors and porous materials generally exhibits nonlinear dependence on temperature due to phonon scattering and microstructural effects. However, for small to moderate temperature variations, a first-order Taylor expansion around a reference temperature ​ provides a useful approximation:

7

This linearized form is widely used to facilitate analytical treatment of coupled thermal and mechanical fields, particularly in nanoscale applications. The thermal conductivity , models the variation of conductivity with temperature. Here ​ is the reference conductivity, and is the temperature sensitivity coefficient (thermal conductivity temperature coefficient (dimensionless)), which is a small coefficient reflecting how sensitive the material’s thermal response is to heating. In semiconductors or certain porous materials, is often negative (implying reduced conductivity at higher temperatures), reflecting that thermal conductivity typically decreases with increasing temperature (with neglecting the minimal values).

The linear approximation is commonly used in modeling semiconductor and porous materials where temperature variation is moderate and the dominant change in thermal conductivity can be captured by a first-order term. This approach simplifies analysis while remaining accurate for many nanoscale applications^28,43,45. The coefficient ​ (in dimensionless form) is typically small and negative, reflecting the decrease in conductivity with increasing temperature due to enhanced phonon scattering, as observed in porous silicon [43; 48]. Higher-order nonlinear models may be necessary at extreme temperatures, but are often computationally intractable and less essential for short-time, small-scale responses considered here. This linear approximation is essential in theoretical studies where temperature-dependent properties must be accounted for without introducing complex, nonlinear expressions. When thermal conductivity varies linearly with temperature, The Kirchhoff transformation is applied to simplify the heat conduction equation when thermal conductivity varies with temperature. By defining a new variable , this transformation integrates the effects of over temperature, enabling easier analytical treatment of thermal wave behavior. The transformation is defined as:

8

 

The modified equations, after replacing the thermal conductivity with the temperature-dependent function derived from the Kirchhoff transformation and ignoring the nonlinear term, can be written as follows:

9

10

where ​ denotes the partial derivative of temperature T with respect to the spatial coordinate ​, i.e., ​, and  represents the material (total or substantial) time derivative of T, i.e., ​. These terms account for the spatial and temporal evolution of thermal fields in porous and semiconductor media. On the other hand, Eqs. (9) and (10) are satisfied for . In 2D deformation problems for a hydro-semiconductor medium, the main fields typically include displacements, stresses, temperature, and carrier (electron) density. These fields are formulated to account for both spatial and temporal variations. The displacement components along the x and z-directions are represented as:

11

The following expressions describe the 2D (Fig. 1, schematic diagram of the problem setup: a nonlocal hydro-semiconductor half-space subjected to a time-decaying thermal shock at the boundary.) mechanical equilibrium and energy conservation in the hydro-semiconductor medium after applying the Kirchhoff transformation and accounting for variable thermal conductivity Eqs. (7)-(10) as:

Fig. 1.

The schematic of the problem.

12

13

14

15

16

17

18

19

To simplify and streamline the analysis, the governing equations are often rewritten using dimensionless quantities. Below are typical dimensionless quantities used in such a study:

20

Substituting the nondimensional variables (Eq. 20) and removing the primes for simplicity, Eqs. (12)-(19) become:

21

22

23

24

25

26

27

,

28

where.

By differentiating and rearranging terms, Eqs. (15) and (16) can be expressed as:

29

Solution to the problem

In the context of the current study, normal mode analysis facilitates the reduction of variables such as temperature, stress, and displacement into harmonic forms, simplifying the resolution of the governing equations. The application of normal modes allows the primary fields in 2D of interest to be expressed in terms of exponential functions, capturing how disturbances propagate and oscillate through the medium^31,36:

30

where is the frequency (representing the temporal variation), b is the wave number, and are the amplitude of the main fields. Each field is assumed to vary sinusoidally in both time and space, reflecting wave-like propagation behavior. All variables are expressed in harmonic forms as shown in Eq. (23), to the system of equations (21) to (28), the partial differential equations transform into algebraic equations as follow:

31

32

33

34

35

36

37

where ,, , ,, .

Solving Eqs. (31)-(35), yields:

38

where

39

40

41

42

and .

Equation (38) can be reformulated in the following factorization form:

43

​ represent the positive-real roots of Eq. (38). when , the amplitudes can be expressed in the following linear solution form:

44

where 

The quantities M_n express the unknown parameters. To obtain the horizontal and vertical displacement components (Eqs. (23) and (24)) using the normal mode method, we can be expressed in an exponential form as:

45

46

To reformulate the components of mechanical stresses, the general approach involves expressing stress components in terms of displacement, temperature, and other physical parameters using the Eqs. (36) and (37), which yields:

47

48

where .

Boundary conditions

For a poro-semiconductor, the boundary conditions should also account for the fluidic properties of the medium, particularly when porosity or fluid flow is present. The applied force can simultaneously influence carrier density, stress, and displacement, necessitating the consideration of coupled interactions among mechanical, thermal, and electrical effects. This coupling requires solving a system of interconnected equations. When implementing boundary conditions within the framework of normal mode analysis for a hydro-photo-thermoelastic problem, it is crucial to assess how the specified boundary conditions (e.g., at a free surface when ) interact with the governing equations to determine the unknown parameters effectively⁴⁸.

1) The thermal boundary condition here specifies a time-dependent thermal shock at a half-space’s surface, where the thermal shock’s magnitude is not fixed but decays exponentially over time. The surface of the semiconductor is subjected to a decaying thermal shock, represented by:

49

where ,  is the decay parameter (controls the thermal shock rate of decay). When , the thermal shock becomes constant in magnitude, corresponding to a non-decaying thermal shock waveform. This simulates realistic laser pulse heating, wherecontrols how fast the thermal input diminishes over time, impacting the induced thermoelastic waves.

2) The decaying thermal shock affects the time evolution of pore pressure in the hydro-semiconductor medium, leading to a transient response. In this case, in hydro-semiconductor media, the excess water pressure takes the form function of the initial excess water pressure as:

50

3) At the boundary, the carrier density is influenced by the equilibrium carrier concentration, ​, and the effect of photoexcitation or other external perturbations:

51

4) In this case, the surface of the half-space is assumed to be free of any applied mechanical traction. Mathematically, this is expressed as:

52

This condition implies that no external forces are applied to the boundary, allowing the stresses to balance naturally due to internal mechanical and thermal interactions. When applying the normal mode analysis method, the boundary conditions are often expressed in terms of time-dependent functions, particularly when dealing with time-varying fields, like in photo-thermoelastic or wave propagation problems. For a typical boundary condition, the relations are expressed as:

53

54

55

56

The unknown parameters () are typically determined by solving these boundary conditions.

Special cases

Photo-thermoelasticity theories

In photo-thermoelasticity, the classical coupled theory (CD), considering no thermal relaxation time. In contrast, the Lord and Shulman (LS) model introduces a single thermal relaxation time when .

5.2 Multi-Temperature Theories.

• The one-temperature theory is obtained when and .

• The two-temperature theory is obtained when and

• The hyperbolic temperature theory is obtained when and .

Nonlocal parameter effects

When the nonlocal parameter is neglected , the system behaves as if it is governed by local interactions rather than considering the spatially distributed effects that arise from nonlocality. This approximation simplifies the governing equations and eliminates the long-range interactions that nonlocal models account for. In this case, the material properties, such as temperature, stress, or displacement, depend only on the local conditions at a given point in space, without any influence from distant points in the medium.

Thermal conductivity effects

When the variation of thermal conductivity is neglected , the material is assumed to have constant thermal conductivity throughout the medium, regardless of temperature or other factors. Neglecting the variable thermal conductivity simplifies the modeling of heat conduction by treating the material as having uniform thermal properties. While this simplification reduces computational complexity, it may be less accurate in describing the behavior of materials where thermal conductivity varies significantly with temperature.

Numerical results and discussions

In this study, numerical results are presented based on the physical constants of porous silicon (PSi) medium, utilizing a MATLAB program for computational analysis. The focus is on modeling the thermomechanical and optical wave propagation in PSi under specific boundary conditions, incorporating the relevant physical properties such as thermal conductivity, nonlocal, and multi-temperature theories. These properties are used to simulate the behavior of the PSi medium under varying thermal and mechanical influences, with particular emphasis on the impact of porosity on thermal diffusion, and wave dynamics. These numerical simulations allow for a deeper understanding of the interactions between heat, stress, and optical waves in PSi, providing valuable insights into the material’s behavior for advanced semiconductor applications. Table 1 introduces the physical constants of PSi material^39–45:

Table 1.

The physical constants of PSi medium.

Unit	Symbol	Value	Unit	Symbol	Value
N/m 2	λ μ	3.64×1010 5.46×1010		t
kg/m3
K	T 0	800
S	τ
m 3	d n
m2/S	D E	2.5×10−3	°C−1
eV	E g	1.11
K−1	α s			τ 0	695
	K0	150
kg/m 3	ρ		K
Pa

The other nondimensional quantities which are used are , , , , , and .

The governing equations, after applying the normal mode technique, reduce to a system of algebraic equations in terms of exponential functions. These were solved using MATLAB. The procedure includes:

1. Input Parameters: Physical constants for porous silicon (PSi) were taken from Table 1.

2. Symbolic Solution: The algebraic system (Eqs. 31–37) was solved symbolically to obtain the wave amplitudes and dispersion relationships.

3. Boundary Conditions: Boundary conditions (Eqs. 49–56) were applied to determine unknown coefficients.

4. Field Reconstruction: Using Eqs. (44–48), displacement, temperature, stress, and pore pressure were reconstructed and plotted as functions of space.

5. Plotting and Analysis: Results were plotted using the plot function with clear legends and units.

All simulations assume a 2D half-space geometry with thermal shock applied at the surface, evolving over a constant time. The influence of variable thermal conductivity, nonlocality, and multi-temperature models was studied by varying corresponding parameters , , and relaxation times.

The influence of variable thermal conductivity

To demonstrate the behavior of the proposed model in a realistic setting, we present a dimensionless numerical simulation inspired by the typical physical parameters of porous silicon (PSi). All quantities are normalized using appropriate reference values for length, temperature, displacement, stress, and time. Figure 2 illustrates the propagation of various physical (temperature (T), displacement (u), conductive temperature (), carrier density (N), normal stress (), and excess pore water pressure (P)) as functions of distance (x) under the influence of a nonlocal hydro-semiconductor medium. The simulations are performed using the Lord and Shulman model in the context of hyperbolic temperature theory considering a decay heating parameter  and different values of variable thermal conductivity (​). According to the temperature distribution: for  (constant thermal conductivity), the temperature decays exponentially with distance and stabilizes quickly. As ​ becomes more negative (, ), indicating that thermal conductivity depends on temperature, the decay rate slows, resulting in a broader thermal distribution. This reflects enhanced heat diffusion for temperature-dependent thermal conductivity. The displacement exhibits an initial peak near the origin, which diminishes and stabilizes with increasing x. For variable thermal conductivity (, ), the displacement magnitude decreases compared to constant thermal conductivity (). This suggests a moderating effect of temperature-dependent conductivity on mechanical deformation. The conductive temperature oscillates and decays with distance. The amplitude and spread increase as ​ becomes more negative, implying that variable thermal conductivity enhances the conductive heat transfer, consistent with the behavior observed for T. The carrier density shows a rapid decay near the origin for , stabilizing at larger distances. As becomes more negative, the carrier density diminishes less rapidly, indicating that temperature-dependent thermal conductivity impacts the redistribution of carriers in the semiconductor medium. The normal stress exhibits oscillatory behavior near the origin before decaying with distance. Increased thermal conductivity dependence (negative ​) reduces the amplitude of stress oscillations, indicating a damping effect of temperature-dependent conductivity on stress waves. The pore water pressure displays oscillations that diminish in amplitude and stabilize at more considerable distances. For , , the oscillations are more pronounced and persist longer compared to . This indicates that variable thermal conductivity influences the interaction between thermal and hydrodynamic effects. From this simulation which shows that:

Fig. 2.

Wave propagation of various physical fields against distance for different values of the variable thermal conductivity parameter (​):  (constant thermal conductivity), and (temperature-dependent thermal conductivity) for a nonlocal hydro-semiconductor medium under the Lord and Shulman model with a decay heating parameter.

• As becomes more negative, thermal waves propagate further, and temperature decay slows, indicating enhanced thermal diffusion with temperature-dependent conductivity.

• The mechanical displacement exhibits oscillatory behavior with attenuation, and the amplitude decreases with stronger conductivity variation.

• Stress waves demonstrate similar oscillatory attenuation, with broader waveforms observed for higher thermal diffusivity.

These results reflect the interplay between thermal diffusion and mechanical wave propagation in a porous semiconductor framework and support the applicability of the model to nanoscale photoacoustic and optothermal behavior.

The influence of nonlocal parameters

Figure 3 compares the wave propagation of various physical fields against distance (x) for two cases: a local medium and a nonlocal medium. Based on the Lord and Shulman (LS) model with variable thermal conductivity and a decay heating parameter, the computational results demonstrate significant differences between the two cases. For the nonlocal medium, the wave profiles exhibit slower decay, broader distributions, and smoother oscillatory behavior compared to the local medium, where sharper decay and higher oscillation amplitudes are observed. These differences are particularly evident in the temperature (T), conductive temperature (), and excess pore water pressure (P), where the nonlocal medium exhibits enhanced thermal diffusion and prolonged wave effects. The displacement (u) and carrier density (N) also show noticeable damping in the nonlocal case. The results indicate that nonlocal effects, which account for spatial interactions and microstructural dependencies, significantly influence the thermal, mechanical, and hydrodynamic behavior of the medium, highlighting the importance of nonlocality in accurately modeling wave propagation in hydro-semiconductor materials. This comparison underscores the importance of incorporating nonlocal effects to accurately capture the dynamic responses of hydro-semiconductor materials, particularly in applications involving high-speed thermal processes, variable conductivity, and microstructural interactions.

Fig. 3.

Wave propagation of various physical fields against distance for two cases: local medium (dotted lines) and nonlocal medium (dashed lines). The results are computed under the effect of variable thermal conductivity and a decay heating parameter using the Lord and Shulman (LS) model within the hyperbolic temperature framework.

The multi-temperature theories

Figure 4 illustrates the wave propagation of the primary physical fields in a nonlocal hydro-semiconductor medium under the effects of variable thermal conductivity, decay heating parameter, and different multi-temperature theories (one-temperature, two-temperature, and hyperbolic temperature) based on the LS model. Temperature (T) and Conductive Temperature (): The hyperbolic temperature theory predicts a slower decay and broader wave distribution due to its finite thermal signal speed, while the one-temperature theory exhibits sharper attenuation. The two-temperature theory lies between the two, reflecting intermediate behavior. Displacement (u): The hyperbolic model leads to smoother oscillations compared to the one-temperature and two-temperature cases, highlighting its effectiveness in capturing finite speed thermal effects. Carrier Density (N) and Normal Stress (​): Both parameters show notable differences between the three models. The hyperbolic temperature theory induces higher amplitude and extended oscillations, whereas the one-temperature model reflects steeper gradients and more localized effects. The two-temperature theory again provides a middle ground. Excess Pore Water Pressure (P): Similar trends are observed, with the hyperbolic model showing sustained oscillatory behavior, reflecting its enhanced ability to model nonlocal and transient thermal effects.

Fig. 4.

Wave propagation profiles of key physical fields as functions of distance in a nonlocal hydro-semiconductor medium were analyzed under variable thermal conductivity and a decay heating parameter using the Lord and Shulman (LS) model. The figure compares results across three multi-temperature theories: one-temperature (solid lines), two-temperature (dashed lines), and hyperbolic temperature (dotted lines).

To validate the model under realistic conditions, we conducted simulations using physical dimensions and properties of porous silicon (PSi). The half-space domain was set to 100 μm in depth. A thermal shock of s duration and decay rate was applied at the surface. The carrier diffusion coefficient was taken as 3.5 × 10 − 33.5 \times 10^{−3}3.5 × 10–3 m²/s, and the thermal conductivity varied with . Figure 4 presents the main physical ddistributions over the dimensionaless distance x∈[0,8], showing a rapid but finite thermal wave propagation. The displacement amplitude reached 0.5 nm, consistent with optoacoustic scale phenomena.

Conclusions

This study developed a novel model for multi-temperature photoacoustic wave propagation in nonlocal hydro-semiconductor media with temperature-dependent thermal conductivity. The following key conclusions were drawn based on analytical and numerical results:

• Variable thermal conductivity () leads to slower temperature decay and broader thermal distributions, impacting mechanical deformation, stress oscillations, and carrier redistribution.

• Nonlocal effects () smoothen wavefronts, extend the spatial reach of wave propagation, and reduce peak amplitudes, consistent with microstructural interaction phenomena at the nanoscale.

• Simulations under different temperature theories (one-temperature, two-temperature, hyperbolic) show:

  • The hyperbolic temperature theory exhibits smoother waveforms and delayed thermal diffusion due to finite signal speed, aligning with physical expectations in high-frequency or short-pulse thermal processes.
  • The two-temperature model shows intermediate behavior, reflecting separate but coupled evolution of lattice and carrier temperatures.
  • The one-temperature model predicts steeper gradients and faster attenuation but may oversimplify wave behavior under rapid transients.

While qualitative comparisons were demonstrated in Fig. 4, a full quantitative analysis of model accuracy across different thermal theories requires benchmark experimental validation and is proposed for future work.

This model offers practical insight for the design of photoexcited MEMS switches, laser-irradiated detectors, and porous thermal barrier structures in semiconductor-based electronics and optoelectronics. Its modular framework allows extension to fractional-order or magneto-thermoelastic interactions in future studies.

Applications

The proposed model has potential applications in several advanced semiconductor technologies. For example, in silicon-based microprocessors used in high-speed computing, accurate prediction of temperature-dependent wave propagation is essential for designing thermal management strategies and avoiding thermal hotspots. In laser-activated photodetectors and MEMS switches, the interaction of thermal and optical waves with nonlocal effects plays a critical role in sensor sensitivity and switching speed. Furthermore, GaN-based power electronics used in electric vehicles and high-power amplifiers rely on porous substrates for thermal regulation; our model can inform the design of such substrates under variable thermal conditions. Finally, in porous thermoelectric generators, where efficiency is highly sensitive to thermal gradients, understanding the coupling between thermal, acoustic, and electronic fields is crucial for performance optimization. Thus, the current study offers a foundational tool for improving design and reliability across a range of nanoelectronic and photonic systems.

Abbreviations

λ,  μ 

Lame’s parameters

 

Deformation potential coefficient

d_n

The electronic deformation coefficient

T

Thermodynamical temperature

T₀

Reference temperature

 

The thermal expansion of volume

α_s

The thermal expansion coefficient of semiconductor grains

ρ_s

Density of semiconductor grains

ρ_w

Density of pore water

ρ

The density of the medium

e

Cubical dilatation

C_e

Specific heat of the medium

K

The thermal conductivity

D_E

The carrier diffusion coefficient

τ

Lifetime

t

Time variable

E_g

The energy gaps

u_i

Displacement vector

N

Carrier concentration (density)

m

Volumetric heat capacity of the medium

n_o

Porosity

τ₀

Thermal memory (thermal relaxation time)

P

Excess pore water pressure

α_w

Thermal expansion coefficient of pore water

g

Gravity

σ_ij

The stress tensor

θ

The conductive temperature

C_w

Heat capacity of pore water

C_s

Heat capacity of semiconductor grains

K_d

Coefficient of permeability

a, b₀, c₀, c₁

Chosen constants

Author contributions

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

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author upon reasonable request.

Declarations

Conflict of interest

The corresponding author states that there is no conflict of interest.

Use of AI tools declaration

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

Ethical approval

Not applicable.