Photo-thermoelastic response of an unbounded semiconductor with a spherical cavity under pulsed heat flux

Abstract

This work explores the propagation of photo-thermoelastic disturbances in an infinite and isotropic semiconductor containing a spherical cavity. The analysis utilizes a fully coupled approach that integrates carrier (plasma) transport, heat conduction, and elastic deformation, effectively portraying the interplay among electronic excitation, increase in temperature, and mechanical deformation. The cavity’s inner boundary excited by a surface heat flux that decays exponentially in time, representing a short pulse. An analytical framework relating to pulsed photo-thermal loading in semiconductors containing internal voids is developed and presented in this research. The governing equations are solved, and boundary conditions are applied to derive closed-form expressions for the physical fields, and the transformed solutions are inverted using eigenvalue decomposition to retrieve the fields in space-time. Numerical illustrations are reported for a silicon-like material to provide quantitative insight into the evolution of temperature, displacement, stress, and carrier density, as well as the associated electrochemical potential energy. The graphical results elucidate how transient behavior depends on radial position, and they highlight the role of coupling among the plasma, thermal, and elastic subsystems in shaping peak magnitudes, decay rates, and spatial attenuation.

Keywords

semiconductor medium spherical cavity laplace transforms eigenvalues approach

Introduction

The interest generated by the laser radiation interactions with semiconductor materials is justified by their applications in microscale heat transport, photothermal diagnostics, and the growing field of optoelectronics. When the energy of a photon, $E$ , is greater than the semiconductor bandgap, $E_{g}$ , an electron may be excited from the valence band to the conduction band, creating a hole. The collection of photoexcited carriers undergoes relaxation to the band edges which is a non-radiative process. Electron-hole pairs exist before the process as a plasma, the spatiotemporal dynamics of which are dominantly described by diffusion in a manner similar to thermal waves. The result of this is that, when the incident laser intensity is varied, the semiconductor material is able to carry thermal waves and a plasma wave similar to the aforementioned. The earliest works recorded focused on simplified models, which, in a way, omitted the individual interactions of the plasma, thermal, and elastic fields.^1–3

Rosencwaig et al.⁴ analyzed localized thermoelastic deformations induced by focused optical beams, while Fournier et al.⁵ combined theoretical and experimental approaches to probe photothermal transport in semiconductors. Song and co-workers^6–9 extended this line of research to generalized thermoelastic vibrations of optically excited semiconducting microcantilevers and wave reflection phenomena under photothermal and thermoelastic frameworks. Nestoros et al.¹⁰ provided quantitative analyses of photothermal reflectance as a function of temperature, and Todorović et al.^11–13 investigated one-dimensional coupled plasma–thermal–elastic interactions through both theoretical modeling and experimental validation.

Lotfy et al.¹⁴ analyzed simultaneously moisture diffusion and photo-magneto-thermoelastic excitation in rotating medium. By using MGT model, El-Sapa and co-authors^15,16 studied the nonlocal thermal diffusions and the effect of moisture. Lotfy¹⁷ examined in detail the thermal and mechanical waves in extended semiconducting structures, while Hosseini and Zhang¹⁸ used strain-gradient models for the case of plasma-initiated wave propagation in nanorods. Awwad et al.¹⁹ have studied how functionally graded semiconductors behave under laser heating. For the first time, Abouelregal et al.²⁰ used a photothermal model with tempered fractional derivatives to describe wave phenomena in free (or unbounded) media. Following that, an expansive assortment of generalized thermoelastic models have been formulated to tackle an array of intricacies in the field. Lotfy²¹ studied the effects of varying thermal conductivity during photothermal diffusion in semiconducting media, while Lotfy et al.²² developed a thermomechanical responses model for reflections photothermal diffusions wave. Abbas²³ applied a dual-phase-lag thermoelastic model to an infinite medium with a circular hole, and Mondal and Sur²⁴ studied photo-thermoelastic waves propagation in an orthotropic semiconducting with spherical cavities and memory-dependent response. Mahdy et al.²⁵ presented an analytical solution for a magneto-photothermal semiconductor model with variable thermal conductivity, whereas Khamis et al.²⁶ examined thermal-piezoelectric interactions under photothermal excitation. Other related studies considered moving-load effects,²⁷ fractional heat conduction in spherical cavities,²⁸ pulse heat flux in semiconducting media with spherical cavities,²⁹ and generalized thermoelastic or micropolar diffusion models.^30–35

Analysis methods have also been refined to accommodate the depth of such issues. In this regard, the eigenvalue approach is especially relevant, as it delivers full solutions in the Laplace domain without any limiting/assumptive conditions pertaining to physical fields. Kuo et al.³⁶ utilized this method to assess the thermal conductivity and interfacial resistance of silicon films via photothermal displacement interferometry. Using the peakforce quantitative nanomechanical mapping mode of an atomic force microscope, Ha, Heebo et al.³⁷ investigated the surface energy characterisation of a single microsphere particle. Lotfy³⁸ examined photothermal interactions within the framework of gravitational fields and internal heat sources. Recent studies have highlighted the importance of advanced modeling in wave propagation, heat-pulse analysis, and coupled transport phenomena. These include investigations of Fano resonance in photonic crystal cavities,³⁹ heat-pulse methods for thermal-property estimation,⁴⁰ and numerical models for nonlinear oscillators, nanofluid heat transfer, and radiative flow problems.^41–43

Several studies have also employed eigenvalue-based methods to solve coupled thermoelastic and photothermal problems. This method was used to study two-temperature generalized thermoelastic interactions in an annular disk by Bera et al.⁴⁴ and fractional-order generalized magneto-thermoelastic media exposed to a moving heat source by Abbas.⁴⁵ Abbas⁴⁶ investigated an unbounded medium with a spherical cavity inside a two-temperature generalized thermoelastic framework, whereas Santra et al.⁴⁷ examined thermoelastic interactions in an infinite elastic solid with voids. Earlier contributions by Das et al.⁴⁸ demonstrated the effectiveness of the eigenvalue approach for three-dimensional and generalized thermoelasticity problems. Further extensions include the effects of relaxation times, moving heat sources, microscale beams, photothermal semiconductor media, while other authors^49–53 have been able, in recent years, to expand the Eigenvalue-based approach to an impressive array of thermal and elastic wave propagation problems.

This study analyzes the photo-thermoelastic interactions in unbounded semiconducting materials with a spherical cavity. An attempt is made to model the fully coupled plasma-thermal-elastic waves to simultaneously capture the carrier density, the temperature field, and the elastic response. An eigenvalue approach is used to obtain regionally the analytical solution in the Laplace domain to guarantee the solution is completely and exactly formulated. The results are analyzed and presented graphically for the interpretation of the temperature, displacement, stress, carrier density, and the electrochemical potential energy of the silicon like-semiconductors and the results are analyzed and presented graphically. This work also accommodates the understanding of the photothermal reaction of semiconductors with internal voids and also provides understanding towards the reliability of optoelectronic devices, the nondestructive evaluation, and the microscale thermal management.

Basic equations

A rigorous description of transport in semiconductors must account for the mutual coupling of carrier, thermal, and elastic fields. The state of the medium is specified by the free-carrier density $n (r, t)$ , the displacement components $u_{i} (r, t)$ and the temperature $T (r, t)$ . Within the photo-thermal framework, these variables obey a set of interdependent balance laws: the equations of motion for the solid, the carrier (plasma) transport equation, and the heat-conduction/energy equation, formulated with constitutive couplings as in Refs.^{11–13,54–56}

σ_{i j, i} = ρ \frac{\partial^{2} u_{i}}{\partial t^{2}}

(1)

D_{m} M_{, i i} = \frac{\partial n}{\partial t} + \frac{n - n_{o}}{τ_{R}}

(2)

k T_{, i i} = (1 + τ_{o} \frac{\partial}{\partial t}) (ρ c_{e} \frac{\partial T}{\partial t} - E_{g} \frac{n - n_{o}}{τ_{R}} + γ_{t} T_{o} \frac{{\partial u}_{i, i}}{\partial t})

(3)

σ_{i j} = μ (u_{i, j} + u_{j, i}) + (λ u_{k, k} - γ_{n} (n - n_{o}) - γ_{t} (T - T_{o})) δ_{i j}

(4)

M = a (T - T_{o}) - γ_{n} u_{k, k} + b (n - n_{o})

(5)

where

D_{m} =

\frac{σ_{o}}{q_{o}^{2}}

b = \frac{q_{o}^{2} D_{e}}{σ_{o}}

a = \frac{E_{g}}{T_{o}}

γ_{n} = (3 λ + 2 μ) α_{n}

γ_{t} = (3 λ + 2 μ) α_{t}

and

i, j, k = 1, 2, 3

Consider an infinite, homogeneous, elastic semiconductor that contains a spherical cavity, so the material domain is $r \geq b$ . Owing to the spherical symmetry, each field variable depends solely on the radial coordinate $r$ and time $t$ . In the spherical coordinate system $(r, ψ, ϕ)$ , the components of the displacement vector can therefore be expressed by:

u_{r} = u (r, t), u_{ψ} = u_{ϕ} = 0

(6)

So that the equations (1)-(5) can be rewritten as:

\frac{\partial σ_{r r}}{\partial r} + \frac{2 σ_{r r} - σ_{ψ ψ} - σ_{ϕ ϕ}}{r} = ρ \frac{\partial^{2} u}{\partial t^{2}}

(7)

D_{m} (\frac{\partial^{2} M}{\partial r^{2}} + \frac{2}{r} \frac{\partial M}{\partial r}) = \frac{\partial n}{\partial t} + \frac{n - n_{o}}{τ}

(8)

k (\frac{\partial^{2} T}{\partial r^{2}} + \frac{2}{r} \frac{\partial T}{\partial r}) = (1 + τ_{o} \frac{\partial}{\partial t}) (ρ c_{e} \frac{\partial T}{\partial t} - E_{g} \frac{n - n_{o}}{τ} + γ_{t} T_{o} \frac{\partial}{\partial t} (\frac{\partial u}{\partial r} + \frac{2 u}{r}))

(9)

σ_{r r} = (λ + 2 μ) \frac{\partial u}{\partial r} + λ \frac{2 u}{r} - γ_{n} (n - n_{o}) - γ_{t} (T - T_{o})

(10)

σ_{ψ ψ} = σ_{ϕ ϕ} = (λ + 2 μ) \frac{u}{r} + λ (\frac{\partial u}{\partial r} + \frac{u}{r}) - γ_{n} (n - n_{o}) - γ_{t} (T - T_{o})

(11)

M = a (T - T_{o}) - γ_{n} (\frac{\partial u}{\partial r} + \frac{2 u}{r}) + b (n - n_{o})

(12)

Initial and boundary conditions

Before proceeding to the solution, the problem’s boundary and initial conditions must be specified. The initial state is prescribed as follows:

u (r, 0) = 0, \frac{\partial u (r, 0)}{\partial t} = 0, T (r, 0) = T_{o}, \frac{\partial T (r, 0)}{\partial t} = 0, n (r, 0) = n_{o}, \frac{\partial n (r, 0)}{\partial t} = 0

(13)

The boundary conditions are defined under the assumption that the cavity inner surface is as.

u (b, t) = 0

(14)

An exponentially decaying heat pulse is applied to the sphere’s surface at $r = b$ ⁵⁷.

k \frac{\partial T (b, t)}{\partial r} = - q_{o} \frac{t^{2} e^{- \frac{t}{t_{p}}}}{16 t_{p}^{2}}

(15)

For the processes of transport and recombination of photocarriers, the boundary condition governing the carrier density can be stated as:

D_{e} {\frac{\partial N (r, t)}{\partial r} |}_{r = b} = S_{s} N (a, t)

(16)

For convenience, the foregoing relations are nondimensionalized. The dimensionless groups are defined as follows:

\begin{array}{l} (r^{'}, u^{'}) = ω c (r, u), N^{'} = \frac{n - n_{o}}{n_{o}}, θ^{'} = \frac{T - T_{o}}{T_{o}}, (σ_{r r}^{'}, σ_{ψ ψ}^{'}, σ_{ϕ ϕ}^{'}) = \frac{(σ_{r r}, σ_{ψ ψ}, σ_{ϕ ϕ})}{λ + 2 μ}, \\ M^{'} = \frac{M}{b n_{o}}, (t^{'}, τ_{o}^{'}, τ^{'}, t_{p}^{'}) = ω c^{2} (t, τ_{o}, τ, t_{p}), Q_{o}^{'} = \frac{Q_{o}}{ω c k T_{o}} \end{array}

(17)

where

c^{2} = \frac{λ + 2 μ}{ρ}

ω = \frac{ρ c_{e}}{K}

. By omitting the prime notation and rewriting equations (7)–(16), we obtain:

\frac{\partial^{2} u}{\partial r^{2}} + \frac{2}{r} \frac{\partial u}{\partial r} - \frac{2}{r^{2}} u - e_{2} \frac{\partial N}{\partial r} - e_{3} \frac{\partial θ}{\partial r} = \frac{\partial^{2} u}{\partial t^{2}}

(18)

\frac{\partial^{2} M}{\partial r^{2}} + \frac{2}{r} \frac{\partial M}{\partial r} = α (\frac{\partial N}{\partial t} + \frac{N}{τ})

(19)

\frac{\partial^{2} θ}{\partial r^{2}} + \frac{2}{r} \frac{\partial θ}{\partial r} = (1 + τ_{o} \frac{\partial}{\partial t}) (\frac{\partial θ}{\partial t} - e_{6} \frac{N}{τ} + e_{7} \frac{\partial}{\partial t} (\frac{\partial u}{\partial r} + \frac{2 u}{r}))

(20)

σ_{r r} = \frac{\partial u}{\partial r} + e_{1} \frac{2 u}{r} - e_{2} N - e_{3} θ

(21)

σ_{ψ ψ} = σ_{ϕ ϕ} = \frac{u}{r} + e_{1} (\frac{\partial u}{\partial r} + \frac{u}{r}) - e_{2} N - e_{3} θ

(22)

M = e_{4} θ - e_{5} (\frac{\partial u}{\partial r} + \frac{2 u}{r}) + N

(23)

u (b, t) = 0

(24)

\frac{\partial T (b, t)}{\partial r} = - q_{o} \frac{t^{2} e^{- \frac{t}{t_{p}}}}{16 t_{p}^{2}}

(25)

{\frac{\partial N (r, t)}{\partial r} |}_{r = b} = β N (a, t)

(26)

where

α = \frac{1}{b ω D_{m}}, e_{1} = \frac{λ}{(λ + 2 μ)}, e_{2} = \frac{γ_{n} n_{o}}{(λ + 2 μ)}, e_{3} = \frac{γ_{t} T_{o}}{(λ + 2 μ)}, e_{4} = \frac{T_{o} a}{b n_{o}}, e_{5} = \frac{γ_{n}}{b n_{o}}, e_{6} = \frac{n_{o} E_{g}}{T_{o} ρ c_{e}}, e_{7} = \frac{γ_{t}}{ρ c_{e}}, β = \frac{S_{s}}{{ω c D}_{e}}

Solution in the laplace transform domain

For $f (r, t)$ function Laplace transforms were written as

\bar{f} (r, s) = L [f (r, t)] = \int_{0}^{\infty} f (r, t) e^{- s t} d t, s > 0

(27)

\bar{f} (r, s)

where

p

is the Laplace transform parameter. Hence, the above equations can be rewritten as

\frac{d^{2} \bar{u}}{d r^{2}} + \frac{2}{r} \frac{d \bar{u}}{d r} - \frac{2}{r^{2}} \bar{u} - e_{2} \frac{d \bar{N}}{d r} - e_{3} \frac{d \bar{θ}}{d r} = s^{2} \bar{u}

(28)

\frac{d^{2} \bar{M}}{d r^{2}} + \frac{2}{r} \frac{d \bar{M}}{d r} = α (s + \frac{1}{τ}) \bar{N}

(29)

\frac{d^{2} \bar{θ}}{d r^{2}} + \frac{2}{r} \frac{d \bar{θ}}{d r} = (1 + s τ_{o}) (s \bar{θ} - e_{6} \frac{\bar{N}}{τ} + e_{7} s (\frac{d \bar{u}}{d r} + \frac{2 \bar{u}}{r}))

(30)

{\bar{σ}}_{r r} = \frac{d \bar{u}}{d r} + e_{1} \frac{2 \bar{u}}{r} - e_{2} \bar{N} - e_{3} \bar{θ}

(31)

{\bar{σ}}_{ψ ψ} = {\bar{σ}}_{ϕ ϕ} = \frac{\bar{u}}{r} + e_{1} (\frac{d \bar{u}}{d r} + \frac{\bar{u}}{r}) - e_{2} \bar{N} - e_{3} \bar{θ}

(32)

\bar{M} = e_{4} \bar{θ} - e_{5} (\frac{d \bar{u}}{d r} + \frac{2 \bar{u}}{r}) + \bar{N}

(33)

\bar{u} (b, s) = 0

(34)

\frac{d \bar{θ} (b, s)}{d r} = - \frac{{- q}_{o} t_{p}}{8 {(s t_{p} + 1)}^{3}}

(35)

{\frac{d \bar{N} (r, s)}{d r} |}_{r = b} = β \bar{N} (b, s)

(36)

Differentiating equation (29) and (30) with respect to $r$ and using equations (28) and (33), which can be written as

(\frac{d^{2}}{d r^{2}} + \frac{2}{r} \frac{d}{d r} - \frac{2}{r^{2}}) \bar{u} = e_{8} \bar{u} + e_{2} \frac{d \bar{N}}{d r} + e_{3} \frac{d \bar{θ}}{d r}

(37)

(\frac{d^{2}}{d r^{2}} + \frac{2}{r} \frac{d}{d r} - \frac{2}{r^{2}}) \frac{d \bar{N}}{d r} = e_{16} \bar{u} + e_{17} \frac{d \bar{N}}{d r} + e_{18} \frac{d \bar{θ}}{d r}

(38)

(\frac{d^{2}}{d r^{2}} + \frac{2}{r} \frac{d}{d r} - \frac{2}{r^{2}}) \frac{d \bar{θ}}{d r} = e_{13} \bar{u} + e_{14} \frac{d \bar{N}}{d r} + e_{15} \frac{d \bar{θ}}{d r}

(39)

where

e_{8} = s^{2}

e_{9} = \frac{α}{τ} (1 + s τ)

e_{10} = s (1 + s τ_{o})

e_{11} = (1 + s τ_{o}) \frac{e_{6}}{τ}

e_{12} = s (1 + s τ_{o}) e_{7}

e_{13} = e_{8} e_{12}

e_{14} = (e_{2} e_{12} - e_{11})

e_{15} = (e_{10} + e_{3} e_{12})

e_{16} = \frac{(e_{13} (e_{5} e_{3} - e_{4}) + e_{5} e_{8} e_{8})}{(1 - e_{5} e_{2})}, e_{17} = \frac{(e_{9} + e_{5} e_{8} e_{2} - (e_{4} - e_{5} e_{3}) e_{14})}{(1 - e_{5} e_{2})}

e_{18} = \frac{(e_{5} e_{8} e_{3} - (e_{4} - e_{5} e_{3}) e_{15})}{(1 - e_{5} e_{2})}

The coupled differential equations (32)-(34) are addressed using an eigenvalue-based method following Refs.^58,59 For compactness, the system is rewritten in state-space form as a first-order linear matrix–vector ordinary differential equation.

(\frac{d^{2}}{d r^{2}} + \frac{2}{r} \frac{d}{d r} - \frac{2}{r^{2}}) V = A V

(40)

where

V = {[\begin{array}{l} \bar{u} & \frac{d \bar{N}}{d r} & \frac{d \bar{T}}{d r} \end{array}]}^{T}

and

A = [\begin{array}{l} e_{8} & e_{2} & e_{3} \\ e_{16} & e_{17} & e_{18} \\ e_{13} & e_{14} & e_{15} \end{array}]

After that, matrix $A$ 's characteristic equation looks like

\begin{array}{l} ω^{3} - ω^{2} (e_{8} + e_{17} + e_{15}) - ω (- e_{8} e_{17} + e_{3} e_{13} + e_{18} e_{14} - e_{8} e_{15} - e_{17} e_{15}) \\ + e_{3} e_{17} e_{13} - e_{2} e_{18} e_{13} + e_{8} e_{18} e_{14} - e_{8} e_{17} e_{15} = 0 \end{array}

(41)

Evaluating the characteristic polynomial in (37) produces the eigenvalue triple $ω_{1}, ω_{2}, ω_{3}$ . The solution to Eq. (40) that remains finite in the limit $r \to \infty$ takes the form

V (r, s) = \sum_{i = 1}^{3} A_{i} X_{i} r^{- 1 / 2} K_{3 / 2} (n_{i} r)

(42)

Here

n_{i} = \sqrt{ω_{i}}

K_{3 / 2}

denotes the modified Bessel function of order

3 / 2

, and

A_{1}, A_{2}, A_{3}

are constants fixed by the boundary conditions. Consequently, the field variables for

r

and

s

take the forms:

\bar{u} (r, s) = \sum_{i = 1}^{3} A_{i} U_{i} r^{- 1 / 2} K_{3 / 2} (n_{i} r)

(43)

\bar{N} (r, s) = - \sum_{i = 1}^{3} A_{i} N_{i} \frac{r^{- 1 / 2}}{n_{i}} K_{1 / 2} (n_{i} r)

(44)

\bar{θ} (r, s) = - \sum_{i = 1}^{3} A_{i} T_{i} \frac{r^{- 1 / 2}}{n_{i}} K_{1 / 2} (n_{i} r)

(45)

The distributions of carriers, displacement, temperature, and stress were recovered in time using the Stehfest numerical inverse Laplace procedure,⁶⁰ which approximates $f (r, t)$ from $\overset{`}{f} (r, p)$ through the relation:

f (r, t) = \frac{\ln 2}{t} \sum_{j = 1}^{H} V_{j} f (r, j \frac{\ln 2}{t})

(46)

where

V_{j} = {(- 1)}^{\frac{H}{2} + 1} \sum_{k = ⌊ \frac{i + 1}{2} ⌋}^{\min (i, \frac{H}{2})} \frac{k^{\frac{H}{2} + 1} (2 k)!}{(\frac{H}{2} - k)! k! (i - k)! (2 k - 1)! (k - 1)!}

Numerical results and discussion

To demonstrate the theoretical results derived above, numerical parameter values are employed. For the numerical illustration, a silicon-like semiconductor is considered using the physical constants reported in Ref.⁵⁵ Silicon is chosen as a representative material due to its wide use in microelectronic, optoelectronic, and photothermal applications. These constants provide a realistic basis for examining the coupled plasma–thermal–elastic response. The adopted values are listed as follows:

λ = 3.64 \times 10^{10} (N) (m^{- 2}), μ = 5.46 \times 10^{10} (N) (m^{- 2}), T_{o} = 300 (k), α_{t} = 3 \times 10^{- 6} (k^{- 1}), ρ = 2330 (k g) (m^{- 3}), τ = 5 \times 10^{- 5} (s), c_{e} = 695 (J) ({k g}^{- 1}) (k^{- 1}), α_{n} = - 9 \times 10^{- 31} (m^{3}), E_{g} = 1.12 (e V), D_{e} = 2.5 \times 10^{- 3} (m^{2}) (s^{- 1}), n_{o} = 10^{20} (m^{- 3}), q_{o} = 1.6 \times 10^{- 19} (C), t = 0.2, t_{p} = 0.215, τ_{o} = 0.05, Q_{o} = 100, σ_{o} = 1.56 \times 10^{- 3} (S) (m^{- 1}) .

The distributions of carrier density, temperature, displacement, electrochemical potential energy, radial and hoop stresses in the r-direction in fully coupled photo-thermal theory were calculated using the numerical approach described above. All variables are measured in their dimensional forms, and their visual representations are shown in Figures 1–12. The calculations were run at a time value of $t = 0.2$ .

The influence of the thermal relaxation time $τ_{o}$ on the coupled field variables is illustrated in Figures 1–6. As shown in Figure 1, the temperature decreases monotonically with the radial distance from the cavity surface for all considered values of $τ_{o}$ . Increasing the thermal relaxation time reduces the temperature magnitude and makes the thermal disturbance more localized near the cavity surface. This behavior reflects the non-Fourier heat-conduction effect, where the finite speed of thermal-wave propagation weakens the immediate thermal response compared with the classical Fourier case.

The displacement distribution corresponding to the above is shown in Figure 2. The radial displacement is initially zero at the boundary of the cavity, then gradually increases to a small maximum near the inner surface and then decreases with increasing (r). A higher value of the parameter $τ_{o}$ will lead to a smaller peak displacement and a faster decay rate. This decrease is a direct result of the lessening of the thermal expansion due to the lesser temperature field.

From the above carrier-density variation, as illustrated in Figure 3, it is evident that the plasma field is also influenced by the thermal relaxation. The closer to the cavity surface, the greater the response of the carrier density with increasing $τ_{o}$ , whereas the farther away from the cavity, the faster the decay of the carrier-density responses and the crossing of the profiles. This suggests that the relaxation time affects both the amount and the distribution of the photo-generated carriers by the coupled thermal–plasma interaction.

The radial and hoop stress distributions are displayed in Figures 4 and 5, respectively. Both stress components are compressive near the cavity boundary and tend toward zero as the radial distance increases. Increasing $τ_{o}$ reduces the compressive stress magnitudes and promotes faster stress relaxation. This confirms that thermal relaxation mitigates the mechanical intensity generated by the pulsed heat flux around the cavity.

The electrochemical potential energy is shown in Figure 6. It decreases continuously with radial distance and becomes smaller as $τ_{o}$ increases. Since the electrochemical potential depends on temperature variation, carrier-density perturbation, and elastic deformation, its reduction reflects the overall weakening of the coupled plasma–thermal–elastic response under stronger non-Fourier effects.

The effect of the characteristic pulse-duration parameter ( $t_{p}$ ) is presented in Figures 7–12. As observed in Figure 7, increasing ( $t_{p}$ ) raises the temperature near the cavity surface and broadens the thermal penetration region. A longer pulse supplies thermal energy over a greater time interval, which strengthens the temperature field and slows its spatial decay.

The displacement response in Figure 8 follows the same trend. Larger pulse duration produces a higher displacement peak and extends the region affected by deformation. This behavior is consistent with the enhanced thermal expansion caused by the stronger and more sustained heat input.

The carrier-density distribution in Figure 9 also increases near the cavity surface when (t_o) becomes larger. The longer heating duration intensifies the photo-thermal disturbance and strengthens the interaction between the thermal and plasma fields. Farther from the cavity, the curves gradually approach zero as the carrier disturbance attenuates with distance.

The radial and hoop stresses shown in Figures 10 and 11 indicate that increasing ( $t_{p}$ ) intensifies the compressive stress near the cavity surface and delays its relaxation. This is because a longer thermal pulse produces stronger local heating and, consequently, greater thermoelastic deformation and stress concentration.

Finally, Figure 12 shows that the electrochemical potential energy increases with pulse duration near the cavity boundary and decays gradually with radial distance. This confirms that the pulse duration controls not only the thermal field but also the associated carrier transport and mechanical response. Overall, the results show that thermal relaxation time tends to suppress and localize the coupled response, whereas pulse duration enhances the magnitude and spatial extent of the plasma–thermal–elastic fields.

Conclusion

Using numerical inversion of the Gaver-Stehfest method, we developed the first eigenvalue-based exact Laplace-domain solution of the fully coupled photo-thermoelastic responses of an infinite semiconductor with a spherical cavity subject to an exponentially decaying surface heat pulse for all of the relevant coupled fields, which allows for the first time quantification of the coupled response of the plasma, thermal, and elastic fields, and how they shape the response of the coupled fields and the response to the coupled fields with respect to the radial distance from the center of the cavity. From this study, two primary control parameters can be identified. First, the thermal relaxation time, ( $τ_{o}$ ), (non-Fourier effect) reduces the severity of thermal pulse response and radial displacements at the cavity wall, reduces the severity of compressive radial and hoop stress, and accelerates the rate of radial decay of all coupled fields, including the fields, and can reorder the profiles of the carrier density, increasing the density of the carriers at the cavity and decreasing the density of the carriers beyond that location. The second control parameter, pulse duration ( $t_{p}$ ), has a response scaling effect that is exactly the opposite of that of ( $τ_{o}$ ). Collectively, these results reveal how important control parameters can be used to optically manipulate the semiconductor component with embedded voids for specified thermal management and/or non-destructive evaluation.

In photothermal diagnostics and non-destructive evaluation, the predicted variations of temperature, stress, and carrier density can assist in interpreting measured photothermal signals and detecting the influence of internal voids or defects. In microscale thermal management, the results provide insight into how pulse duration and non-Fourier thermal effects can be controlled to regulate heat localization and spatial attenuation in semiconductor materials. Future work may extend the present model by examining the effects of additional semiconductor parameters, such as carrier lifetime, recombination velocity, and diffusion coefficient. Other boundary conditions, including a traction-free cavity surface, and more complex material models or external fields, may also be considered to provide a broader understanding of coupled photo-thermoelastic behavior in semiconductor devices and related engineering applications.

Footnotes

ORCID iD

Ibrahim Abbas

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 authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work was supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R742), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Declaration of conflicting interests

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

Appendix

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