Wave propagation in nonlocal piezo-photo-hygrothermoelastic semiconductors subjected to heat and moisture flux

Nomenclature

T

thermal temperature

c i j k l

elastic stiffness tensor

Q i j T

coupled thermal diffusivity

φ

electric potential

d n T

electronic deformation coefficient

τ i , d i

pyroelectric constants

J

moisture concentration potential

q i j m

coupled moisture diffusivities

α i j T

coupling thermal material coefficients

D i j T

temperature diffusivity

κ

thermal activation coupling parameter

τ

photogenerated carrier lifetime

E g

energy gap of the semiconductor

δ n T

difference in the deformation potential of the conduction and valence bands

κ

thermal activation coupling parameter

e i j

component of strain

β i j T

thermal moduli tensor

D i

electric displacement

η i j k , ε i j

piezothermal moduli tensors

E i

electric field density

C E

specific heat at constant strain

k m

moisture diffusion constant

ρ

density

β i j m

moisture moduli tensor

D i j m

moisture diffusivity

σ i j

component of stress

δ i j

Kroneckar delta

E g N ⁎ τ

photoexcitation effect

1 Introduction

The study of wave propagation in semiconductor materials subjected to coupled thermoelastic, piezoelectric, photothermal, plasma, and moisture diffusion effects has become increasingly important due to its wide applications in microelectronics, optoelectronics, MEMS/NEMS, and advanced sensing technologies. In such materials, the interaction between mechanical, thermal, electric, photothermal, and hygrothermal fields gives rise to a variety of waves whose dynamics directly affect the performance, stability, and lifespan of semiconductor-based devices. These interactions are particularly significant under harsh service conditions involving thermal shocks, moisture ingress, and high-frequency excitations. Recently, the importance of nonlocal elasticity has emerged as a critical factor when analyzing wave propagation in microscale and nanoscale systems, where classical local continuum theories fail to capture the observed dispersive and size-dependent behaviors. The inclusion of the nonlocal parameter accounts for the long-range interatomic interactions that significantly modify the elastic response, especially at a small scales. As semiconductor devices continue to miniaturize, understanding the interplay between piezoelectric, photothermal, moisture, plasma, and nonlocal elastic effects becomes essential for the accurate design and optimization of high-performance components. This coupled and nonlinear multiphysical problem forms the basis of modern research aimed at improving the functionality and reliability of next-generation semiconductor technologies. This has led to the urgent need for more accurate and comprehensive models capturing the simultaneous effects of piezoelectricity, photothermal excitation, plasma waves, and moisture transport in semiconductors. In addition to the classical couplings, the mechanical response of semiconductor materials at small scales is notably affected by nonlocal elastic effects, where the stress at a given point is influenced by strains in its neighborhood rather than just the local strain. The presence of nonlocality becomes critical when wave phenomena are studied in micro- and nanoscale semiconductor structures, as classical local theories often fail to account for observed dispersive and size-dependent behaviors. However, the majority of existing investigations have either neglected the nonlocal influence or restricted it to simple elasticity without considering its interaction with hygrothermoelastic, piezoelectric, and photothermal fields.

Previous research has provided valuable contributions to the understanding of hygrothermal and thermoelastic interactions in solids. The interaction of temperature and moisture diffusion has been explored in polymers, composite structures, and geomaterials, where the coupled diffusion of heat and moisture significantly affects stress development [1,2,3,4,5,6]. The foundation of thermoelasticity itself was laid by Biot [7], and later expanded through the development of generalized thermoelasticity theories [8,9] and their application in modeling thermal, electromagnetic, and elastic wave interactions [10,11,12,13,14]. Similarly, the theory of piezo-thermoelasticity, introduced by Nowacki [15,16] and further developed by Mindlin [17,18], accounts for the coupling between thermal, mechanical, and piezoelectric effects. Chandrasekharaiah [19] refined this theory by introducing finite wave propagation speeds to better describe wave behaviors in realistic materials.

The elastic response of semiconductors has also been extensively studied due to their widespread applications in electronics and optoelectronics. Early contributions from Mindlin [20] highlighted the significance of piezoelectricity in semiconductors, and subsequent studies [21,22,23,24,25] examined how photothermal and plasma wave effects influence the dynamic behavior of these materials. Further developments were achieved by Khamis et al. [22] and Lotfy [26], who examined photothermal and thermoelastic wave propagation under dual-phase-lag and memory-dependent models [27]. Hobiny and Abbas [28] further extended the framework by incorporating fractional-order thermoelastic theories in semiconductor materials. While these models provided valuable insights, they largely relied on classical elasticity, neglecting the potential influence of nonlocal elasticity on wave propagation.

Motivated by this limitation, the present work proposes a comprehensive theoretical model that incorporates nonlocal elasticity into the piezo-photo-hygrothermoelastic wave propagation framework for orthotropic semiconductor materials [29,30,31]. The model fully captures the simultaneous interactions between elastic, photothermal, piezoelectric, plasma, and moisture diffusion fields, while accounting for the size-dependent effects inherent in nonlocal elastic theory [32]. The inclusion of the nonlocal parameter in the constitutive relations leads to modified wave characteristics, including dispersion and attenuation patterns, which are critical for accurately predicting wave propagation in micro- and nanoscale semiconductor devices. In practical scenarios, semiconductor devices are often designed with characteristic dimensions at micro- and nanoscales, where nonlocal effects cannot be ignored [33,34,35]. While several models have addressed hygrothermal stresses in polymeric and composite systems, and generalized thermoelastic and piezo-thermoelastic responses in semiconductors, only a few studies have explored nonlocal elasticity in the context of coupled piezo-photo-hygrothermoelastic wave propagation [36,37]. The proposed model fills this gap by integrating nonlocal elasticity with hygrothermoelastic and photothermal effects, enabling a more accurate description of the coupled wave behavior [38].

Wave propagation in semiconductor and elastic media has been extensively investigated in recent years, particularly in the presence of complex coupled effects such as porosity, size dependence, nonlocality, and thermo-piezoelectric interactions. Studies have explored the impact of double and single porosity on SH-wave vibrations in periodic porous lattices, revealing significant alterations in wave behavior due to porosity variations [39]. Additionally, the interaction of surface waves in micropolar thermoelastic media with dual pore connectivity has been analyzed, demonstrating size-dependent effects on wave dispersion and attenuation [40]. Recent work has also addressed moisture and temperature diffusivity in orthotropic hygro-thermo-piezo-elastic materials, providing insights into coupled moisture–temperature interactions [41]. The role of nonlocality in shear wave propagation within fiber-reinforced poroelastic structures has been examined, highlighting its influence on wave speed and stability under interfacial disturbances [42]. Furthermore, vibrational analysis of size-dependent thermo-piezo-photoelectric semiconductor media has been conducted under the memory-dependent Moore–Gibson–Thompson photo-thermoelasticity framework, emphasizing the significance of memory effects in advanced semiconductor applications [43]. Additional studies have explored double poro-magneto-thermoelastic models incorporating microtemperature and initial stress under memory-dependent heat transfer, shedding light on thermal relaxation effects in complex elastic media [44]. The propagation of Rayleigh waves in nonlocal piezo-thermoelastic materials with voids under memory-dependent heat transfer has also been examined, showcasing the intricate interplay between thermal memory and wave characteristics [45]. In a different context, flexoelectric effects and viscoelastic coatings have been shown to significantly alter surface wave dynamics in advanced geomaterial plates [46]. Moreover, fractional and memory effects on wave reflection in pre-stressed microstructured solids with dual porosity have been investigated, offering new insights into wave behavior in engineered materials [47]. The study of wave propagation in complex elastic media has been significantly advanced by incorporating micropolar, microstretch, and diffusion effects, leading to more refined and accurate physical models. While these studies have advanced our understanding of wave propagation in various structured media, the combined effects of nonlocality, temperature-dependent thermal conductivity, and orthotropic piezoelectric micropolar interactions in semiconductor materials remain largely unexplored [48]. The present study aims to bridge this gap by developing a comprehensive model that integrates these factors, providing a more realistic representation of wave dynamics in semiconductor devices at micro- and nanoscales according to microstretch thermoelastic diffusion [49]. Although several studies have been conducted on wave propagation in piezo-photo-thermoelastic and hygrothermoelastic semiconductor media, most of them have ignored the influence of nonlocal effects, which become critical when dealing with micro- and nanoscale semiconductor structures. The neglect of nonlocal elasticity in previous investigations leads to inaccurate predictions of stress, displacement, and temperature distributions, especially under high-frequency or small-scale conditions. Furthermore, the combined interaction of piezoelectric, photothermal, and hygrothermal fields in the presence of nonlocality has not been adequately addressed. Motivated by these shortcomings, the present work aims to develop a comprehensive theoretical model that incorporates nonlocal elasticity within the framework of photo-thermoelasticity for piezo-photo-hygrothermoelastic semiconductors. This formulation provides a more realistic understanding of the coupled wave propagation behavior, capturing the essential physics that was previously overlooked.

The primary goal of this work is to investigate how nonlocal elasticity affects the transient and steady-state responses of the semiconductor medium when exposed to heat, moisture, and photothermal excitations. Using the normal mode technique, the governing equations are analytically solved, and the obtained expressions describe the distributions of displacement, temperature, carrier density, moisture concentration, stress, and electric fields. Numerical simulations are conducted for the cadmium selenide (CdSe) semiconductor material to demonstrate the impact of the nonlocal parameter on wave profiles. The results show that the presence of nonlocal elasticity leads to substantial changes in wave amplitudes, attenuation rates, and coupling characteristics between the various fields, providing deeper insights into the behavior of semiconductor devices operating under coupled thermoelectrical–mechanical–moisture environments, particularly at small scales.

2 Basic equation

This section contains the fundamental governing equations describing the propagation of coupled piezo-photo-hygrothermoelastic waves in an orthotropic nonlocal semiconductor with temperature-dependent thermal conductivity. In this study, we consider an orthotropic semiconductor medium subjected to simultaneous mechanical, thermal, electric, and moisture effects, where the photothermal excitation and moisture flux are applied on the surface. The material is assumed to be homogeneous, electrically and thermally conductive, and exhibits piezoelectric characteristics. The coordinate system is chosen such that the x-axis is normal to the surface of the semiconductor, while the z-axis lies along the surface, describing a semi-infinite domain (x ≥ 0) (Figure 1). The physical quantities involved in the formulation include the displacement vector u ( r → , t ) , temperature change T ( r → , t ) , moisture concentration m ( r → , t ) , and carrier density N ( r → , t ) . These variables collectively describe the piezo-photo-hygrothermoelastic behavior of the nonlocal semiconductor, and their interactions will be mathematically modeled in the following sections [21,22].

Figure 1

Geometry of the problem.

The stress–strain–temperature–moisture relation for an orthotropic piezo-hygrothermoelastic nonlocal semiconductor is given by

The nonlocal parameter ε characterizes the influence of long-range interatomic interactions within the material, which plays a crucial role in capturing size-dependent mechanical behavior, especially when dealing with materials at the micro- and nanoscale. In the experimental findings reported by Yadav et al. [50,51,52], it has been identified that there exists a mutual interaction between the traditional Fourier heat flux and Fick moisture flux, leading to the emergence of additional cross-coupling fluxes commonly referred to as the Dufour and Soret effects. Specifically, the heat conduction process is influenced by moisture gradients, giving rise to the Dufour flux, while the moisture diffusion process is affected by temperature gradients, resulting in the Soret flux. Consequently, the expressions for both the total heat flux q i and the total moisture flux f i are modified to include these coupling terms, as presented by Yadav et al. [50,51,52]:

In this formulation, q F represents the conventional Fourier heat flux, which is governed by the thermal diffusivity ( D T ) and is expressed as q i F = − D i j T T , i . Similarly, f F denotes the classical Fick moisture flux potential, which depends on the moisture diffusivity ( D m ) and is defined as f i F = − D i j m m , i . The term q D refers to the additional heat flux generated by the Dufour effect, while f s corresponds to the additional moisture flux arising from the Soret effect. These cross-effects introduce new coupling parameters, namely the cross-diffusivities q i j d and Q i j T , which describe the interaction between heat and moisture transport mechanisms. Specifically, the Dufour-related contribution is expressed as q i D = − q i j d m , i , and the Soret-induced moisture flux is given by f i s = − Q i j T T , i [50].

Considering 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 yields

From Eqs. (5) and (6),

Taking into account the Thomson effect within Fourier’s heat conduction law, and considering the additional influence of plasma on heat transport while disregarding the Peltier contribution, the modified heat conduction behavior for semiconductors can be formulated [51,52]:

By differentiating Eq. (8) for x i , the following relation is obtained [53]:

Substituting Eq. (9) into Eq. (10) yields the governing equation for temperature and plasma diffusion within the piezoelectric hygrothermal semiconductor medium [54]:

Similarly, the moisture-plasma diffusion equation governing the behavior in a piezoelectric hygrothermal semiconductor medium can be expressed as [31]

The equation of motion for a nonlocal semiconductor medium [32,33] is

3 Problem formulation

By substituting Eqs. (1) and (2) into Eq. (12), the governing equations for the two-dimensional (2D) case are obtained under the assumption u ⇀ ( x , y , z , t ) = ( u , 0 , w ) that [55]

where ∇ 2 = ∂ 2 ∂ x 2 + ∂ 2 ∂ z 2 .

The corresponding 2D plasma-thermal diffusion equation governing the interaction between the carrier density and temperature fields is expressed as follows [56]:

The 2D heat-plasma diffusion equation for the piezoelectric hygrothermal semiconductor medium, derived from Eq. (10), can be expressed as

The moisture-plasma diffusion equation describing the coupled behavior in the piezoelectric hygrothermal semiconductor medium, as obtained from Eq. (11), takes the following form:

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 , a 33 T represent the linear thermal expansion constants, α 11 m , α 33 m refer to the moisture expansion constants, and d 11 N , d 33 N express the electronic deformation coefficient.

4 Solution to the problem

To analyze the propagation of harmonic waves in the considered medium, we focus on waves traveling, with the wave normal lying in the xz-plane. To obtain analytical solutions for Eqs. (13)–(18) governing the coupled physical fields, the normal mode method is employed by assuming the following form for the physical variables. The normal mode technique is particularly advantageous in solving multi-physics wave propagation problems, as it effectively reduces the system of partial differential equations to an algebraic system in the transformed domain. This method allows for the direct analysis of wave dispersion, attenuation, and coupling characteristics between the thermoelastic, piezoelectric, photothermal, and plasma fields, making it highly suitable for complex semiconductor models involving nonlocal effects and hygrothermal interactions [57,58,59]:

where ω expresses the angular frequency, i = − 1 , b is a wave number along the z -direction, T ˜ ( x ) , N ˜ ∗ ( x ) , u ˜ ( x ) , w ˜ ( x ) , m ˜ ( x ) , and , φ ˜ ( x ) are the amplitude functions depending on the normal direction x . By applying the normal mode representation introduced in Eq. (19) to Eqs. (13)–(18), the system is reduced to a set of six coupled homogeneous equations governing the field variables:

where D = 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 + 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 + ω , a 43 = q 1 m , a 44 = − b 2 q 3 m , a 45 = E g τ , a 46 = − T 0 ρ C E β 1 T ω , a 47 = − i b ω T 0 ρ C E β 3 T , a 48 = i b ω T 0 τ 3 , a 51 = D 1 m , a 52 = − ( b 2 D 3 m + ω ) , a 53 = Q 1 T , a 54 = − b 2 Q 3 T , a 55 = a 45 , a 56 = − m 0 k m β 1 m D 1 m ω , a 57 = − i b ω m 0 k m β 3 m D 3 m , 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 .

By employing the elimination method to solve the system of Eqs. (20)–(25), we obtain the following results:

The explicit expressions for the coefficients Δ i , i = 1 , .... . , 7 are provided in Appendix A. The solution of Eq. (26) in factorization form is

where k i 2 ( i = 1 → 7 ) are the roots of Eq. (27), chosen such that their real parts are positive to ensure the boundedness of the solution as x → ∞ . Owing to the linearity of the problem, the solution of Eq. (27) can be expressed as

Similarly, the solutions corresponding to the other physical variables take the following form:

such that

where 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 dependent on the parameters b and ω , which can be determined by applying the boundary conditions.

5 Boundary conditions

The solution of the governing equations requires the specification of appropriate boundary conditions that accurately reflect the physical situation under consideration. In the present problem, the piezoelectric hygrothermal semiconductor medium is subjected to coupled mechanical, thermal, plasma, moisture, and electric fields at its surface. Therefore, boundary conditions are formulated to account for external actions such as thermal and plasma excitations, moisture variation, and mechanical stresses. These conditions play a crucial role in capturing the interaction between temperature, moisture, carrier density, electric potential, and stress fields, which are vital for understanding the behavior of the medium under practical engineering applications, including thermal management, optoelectronic devices, and moisture-sensitive semiconductor components. The boundary conditions are chosen to ensure the physical realism of the model and the proper decay of the disturbances away from the excitation region. To obtain the constants M i , we will assume the following:

(I) Since the surface is subjected to a decaying thermal shock, the temperature at the boundary is not constant but follows an exponentially decaying function with time at x = 0 :

where ζ is the decay constant (controls how fast the thermal shock decays). This boundary condition describes the application of an initial thermal disturbance at the free surface that dissipates exponentially with time, modeling situations where the surface is exposed to a rapid thermal shock, such as laser pulses or transient heating, commonly encountered in semiconductor manufacturing and thermal testing. In this case, we have

(II) The normal component of the electric field at the free surface x = 0 is assumed to vanish, which is a common assumption for electrically insulated or free surfaces in piezoelectric and semiconductor media. This condition is expressed as

Therefore,

This boundary condition ensures that no electric flux crosses the boundary, reflecting practical situations where the surface is either free of charge carriers or insulated, which is highly relevant in microelectronic and piezoelectric device applications.

(III) At the boundary x = 0 , the plasma diffusion function is prescribed as

Hence,

where n 0 is a known plasma distribution or excitation function at the surface, which could result from external plasma injection, laser excitation, or boundary-controlled processes. This boundary condition plays a significant role in describing the behavior of charge carriers at the surface, which is essential for analyzing semiconductor devices, sensors, and plasma-affected systems.

(Ⅳ) Since the surface is assumed to be traction-free, the boundary conditions for mechanical stresses are given by

Furthermore,

These conditions represent a mechanically free surface, common in thin films, coatings, and layered semiconductor structures, where the outer surface is not subjected to external mechanical loads.

(V) The tangent mechanical conditions at x = 0 are

Hence,

(VI) At the boundary x = 0 , the moisture concentration is specified by

So,

where m 0 represents the imposed moisture concentration on the surface. This condition models the direct influence of environmental humidity or controlled hygrothermal processes on the medium. It is crucial in the analysis of hygro-thermoelastic and piezoelectric materials used in advanced sensing, coatings, and smart structures.

The simultaneous solution of the established boundary conditions yields the unknown parameters M i ( b , ω ) , following the methodologies outlined in previous studies [60,61].

6 Numerical results and discussions

In this section, numerical simulations are carried out based on the physical constants of the CdSe semiconductor material. The obtained computational results are valuable for illustrating the influence of the proposed model and may contribute to advancements in physics-based technologies, with potential applications across various sectors of the modern mechanical and semiconductor industries. The physical properties of CdSe used in the simulation are listed in Table 1, where all constants are expressed in SI units [36,37].