The Effects of Variable Thermal Conductivity in Semiconductor Materials Photogenerated by a Focused Thermal Shock

Abstract

In this work, the generalized photo-thermo-elastic model with variable thermal conductivity is presented to estimates the variations of temperature, the carrier density, the stress and the displacement in a semiconductor material. The effects of variable thermal conductivity under photo-thermal transport process is investigated by using the coupled model of thermoelastic and plasma wave. The surface of medium is loaded by uniform unit step temperature. Easily, the analytical solutions in the domain of Laplace are obtained. By using Laplace transforms with the eigenvalue scheme, the fields studied are obtained analytically and presented graphically.

1. Introduction

The theory of bodies explains the properties of the inner structure of some materials when the second law of thermodynamics is used. The conduction of thermal energy in thermo-elastic materials with internal structures is investigated. The laws of the balance of a continuum medium are used from the first moment (heat flux) the heat energy equations with microstructure when are added to the governing equation. With the development of semiconductor integrated circuit technologies and solid-state sensor technologies, MEMS has been widely used in many fields; it has the characteristics of small volume, lightweight and low energy consumption.

To understand the internal structural properties of elastic media—especially in semiconductor media—the electrical properties of these media must be investigated in such a way that considers of their mechanical-thermal properties. The importance of semiconducting materials is due to their recent uses in many beneficial applications—especially in modern technologies based on new energy alternatives. Many of these applications depend on studying the effects of the fall of the sunlight or a beam of laser on the outer surface of the semiconducting material, without taking into account the internal structure of the medium.

Semiconductor materials have only been studied as elastic materials without taking into consideration the influence of light beams on them. However, semiconducting materials are considered nanomaterials in modern technology that have many uses, e.g., industrial photovoltaic solar cells. Electronic and elastic deformations occur when a laser beam strikes the surface of a semiconductor medium. In this case, some surface electrons will be excited and photo-excited free carriers will be generated. In modern technology, the relationship between the uses of photothermal (PT) theory and semiconductors is close. Many scientists have used the PT theory to obtain the values of the temperature. The thermal diffusion of nanocomposite semiconductor materials has also been measured.

Todorovic [1,2] studied thermoelastic and plasma waves in a semiconductor materials. Ailawalia and Kumar [3] investigated the photo-thermo-elastic interactions in semiconducting media caused by the ramp-type heating. Lotfy et al. [4] studied Thomson and electromagnetic effects under photothermal transport process in semiconducting media. Lotfy et al. [5] studied the responses of electromagnetic and Thomson impacts of semiconductor media due to laser pulses during photo-thermal excitations. Lotfy et al. [6] used fractional-order magnetophotothermal models to study the influences of variable thermal conductivity of a semiconductor medium with cavity. Hobiny and Abbas [7] investigated the photo-thermal interactions in a two-dimensional semiconducting half-spaces under the Green–Naghdi model. Othman and Marin [8] studied using the Green–Naghdi model to investigate the effects of thermal load due to laser pulse on thermo-elastic porous media. Alzahrani and Abbas [9] studied the photo-thermoelastic interaction in a two-dimensional semiconductor media without energy dissipations. Jahangir et al. [10] studied photo-thermoelastic interactions in a semiconductor media with two thermal relaxation times. Mondal and Sur [11] investigated the propagations of the photo-thermo-elastic waves in an orthotropic semiconductor media with a spherical hole and memory response. Several authors [12,13,14,15,16,17,18,19,20,21,22,23,24,25] have presented solutions to many problems by using generalized thermoelastic models.

The purpose of this paper is to study the effects of the variability of thermal conductive in a semiconductor material using the eigenvalues approach. By using the eigenvalue scheme and Laplace transforms based on analytic-numeric approaches, the basic equations are derived. The numeric results of the main physical fields in different cases is studied and shown graphically and discussed.

2. Mathematical Model

The basic equations in the context of photothermal model for a homogeneous and isotropic semiconducting material in the absences of body force and thermal sources are given as in [26,27,28]:

The equations of motion:

$$\mu u_{i,jj}+\left(\lambda +\mu \right)u_{j,ij}-{\gamma }_n{N}_{,i}-{\gamma }_t{T}_{,i}=\rho \frac{{\partial }^2{u}_i}{\partial t^2}$$

(1)

The coupling between plasma and thermoelastic waves can be given by:

$$D_e{N}_{,jj}-\frac{N}{\tau }+\frac{k}{\tau }T-\frac{\partial N}{\partial t}=0.0$$

(2)

The equation of heat conduction:

$${\left(K{T}_{,j}\right)}_j+\frac{E_g}{\tau }N-\rho c_e\frac{\partial T}{\partial t}-{\gamma }_t{T}_o\frac{\partial u_{j,j}}{\partial t}=0.0,$$

(3)

The stress–strain relations:

$${\sigma }_{ij}=\left(\lambda u_{k,k}-{\gamma }_{n}N-{\gamma }_{t}T\right){\delta }_{ij}+\mu \left(u_{i,j}+u_{j,i}\right)$$

(4)

where $\rho$ is the density of material, $N=n-n_o$, $n_o$ is the carrier concentration at equilibrium, $i,j,k=1,2,3$, $\tau$ is the lifetime of photo-generated carrier,$T=T^{*}-T_o$, $T^{*}$ is the variations of temperature, $u_i$ are the displacement components, $T_o$ is the reference temperature, $k=\frac{\partial n_o}{\partial T}$ is the coupling parameter of thermal activation [27],${\sigma }_{ij}$ are the components of stresses, ${\gamma }_t=\left(3\lambda +2\mu \right){\alpha }_t$ and ${\alpha }_t$ is the linear thermal expansion coefficients,$D_e$ is the carrier diffusion coefficient, $t$ is the time, ${\gamma }_n=\left(3\lambda +2\mu \right)d_n$, $d_n$ is the electronic deformation coefficient, $\lambda ,\mu$ are the Lame’s constants,$c_e$ is the heat specific at constant strain and $K$ is the thermal conductivity, which is considered to be temperature-dependent. We will consider the thermal conductivity has the following linear form [29]

$$K\left(T\right)=K_o\left(1+K_{1}T\right)$$

(5)

where $K_1$ is a non-positive small parameter and $K_o$ is the thermal conductivity at the initial temperature $T_o$. The following mapping [29] can be used

$$\varphi =\frac{1}{K_o}{{\displaystyle \int }}_0^{T}K\left(T\right)dT$$

(6)

where the new function $\varphi$ expressing the heat conduction. Substituting from Equation (5) into Equation (6) and integrating, we can obtain [29]

$$\varphi =T+\frac{1}{2}{K}_1{T}^2,$$

(7)

From Equations (6) and (7), we can deduce the following

$$K_o{\varphi }_{,i}=K\left(T\right)T_{,i},,$$

(8)

$$K_o{\varphi }_{,ii}={\left(K\left(T\right)T_{,i}\right)}_{,i},,$$

(9)

$$K_o\frac{\partial \varphi }{\partial t}=K\left(T\right)\frac{\partial T}{\partial t},,$$

(10)

Thus, the basic Equations (1)–(4) can be rewritten in the form:

$$\mu u_{i,jj}+\left(\lambda +\mu \right)u_{j,ij}-{\gamma }_n{N}_{,i}-{\gamma }_t{\varphi }_{,i}=\rho \frac{{\partial }^2{u}_i}{\partial t^2},$$

(11)

$$D_e{N}_{,jj}-\frac{N}{\tau }+\frac{k}{\tau }\varphi =\frac{\partial N}{\partial t},$$

(12)

$$K_o{\varphi }_{,jj}+\frac{E_g}{\tau }N=\rho c_e\frac{\partial \varphi }{\partial t}+{\gamma }_t{T}_o\frac{\partial u_{j,j}}{\partial t},$$

(13)

$${\sigma }_{ij}=\mu \left(u_{i,j}+u_{j,i}\right)+\left(\lambda u_{k,k}-{\gamma }_{n}N-\frac{{\gamma }_t}{K_1}\left(-1+\sqrt{1+2K_1\varphi }\right)\right){\delta }_{ij},$$

(14)

Let us consider a homogeneous, isotropic infinite semiconductor material, whose states can be written in terms of the time $t$ and the space variable $x$, therefore the Equations (11)–(14) are expressed by:

$$\left(\lambda +2\mu \right)\frac{{\partial }^{2}u}{\partial x^2}-{\gamma }_n\frac{\partial N}{\partial x}-{\gamma }_t\frac{\partial \varphi }{\partial x}=\rho \frac{{\partial }^{2}u}{\partial t^2},$$

(15)

$$D_e\frac{{\partial }^{2}N}{\partial x^2}-\frac{N}{\tau }+\frac{k}{\tau }\varphi =\frac{\partial N}{\partial t},$$

(16)

$$K_o\frac{{\partial }^2\varphi }{\partial x^2}+\frac{E_g}{\tau }N=\rho c_e\frac{\partial \varphi }{\partial t}+{\gamma }_t{T}_o\frac{{\partial }^{2}u}{\partial t\partial x},$$

(17)

$${\sigma }_{xx}=\left(\lambda +2\mu \right)\frac{\partial u}{\partial x}-{\gamma }_{n}N-\frac{{\gamma }_t}{K_1}\left(-1+\sqrt{1+2K_1\varphi }\right),$$

(18)

3. Application

We shall also assume that the initial state of the medium is quiescent and are completed by consider the boundary conditions at $x=0$ as

$$u\left(0,t\right)=0,$$

(19)

The surface $x=0.0$ is due to the with thermal shock varying heat:

$$T\left(0,t\right)=T_{1}H\left(t\right),$$

(20)

From Equation (7), then boundary condition (20) is expressed by

$$\varphi \left(0,t\right)=T_{1}H\left(t\right)+\frac{1}{2}{K}_1{\left(T_{1}H\left(t\right)\right)}^2,$$

(21)

where $H\left(t\right)$ is the Heaviside unit function and $T_1$ is a constant temperature. The carrier density boundary condition are difined by

$$D_e{\frac{\partial N\left(x,t\right)}{\partial x}|}_{x=0}=s_{o}N\left(0,t\right) ,$$

(22)

where $s_o$ is the velocity of recombination on the surface. Now, for conveniences, the nondimensional physical variables are given by

$${\varphi }^{*}=\frac{\varphi }{T_o},N^{*}=\frac{N}{n_o},K_1^{*}=T_o{K}_1, {\sigma }_{xx}^{*}=\frac{ {\sigma }_{xx}}{\lambda +2\mu }, \left(x^{*},u^{*}\right)=\eta c\left(x,u\right),\left(t^{*},{\tau }^{*}\right)=\eta c^2\left(t,\tau \right),$$

(23)

where $c^2=\frac{\lambda +2\mu }{\rho }$ and $\eta =\frac{\rho c_e}{K}$.

By using the parameters of dimensionless (23), rewrite the basic equations with neglecting the primes one obtains:

$$\frac{{\partial }^{2}u}{\partial t^2}=\frac{{\partial }^{2}u}{\partial x^2}-x_1\frac{\partial N}{\partial x}-x_2\frac{\partial \varphi }{\partial x},$$

(24)

$$x_3\frac{\partial N}{\partial t}=\frac{{\partial }^{2}N}{\partial x^2}-\frac{x_3}{\tau }N+\frac{\beta }{\tau }\varphi ,$$

(25)

$$\frac{\partial \varphi }{\partial t}=\frac{{\partial }^2\phi }{\partial x^2}+\frac{x_4}{\tau }N-x_5\frac{{\partial }^{2}u}{\partial t\partial x},$$

(26)

$${\sigma }_{xx}=\frac{\partial u}{\partial x}-x_{1}N-x_2\left(-1+\sqrt{1+2K_1\varphi }\right),$$

(27)

$$\left(0,t\right)=0, {\frac{\partial N\left(x,t\right)}{\partial x}|}_{x=0}=x_{6}N\left(0,t\right), \varphi \left(0,t\right)=T_{1}H\left(t\right)+\frac{1}{2}{K}_1{\left(T_{1}H\left(t\right)\right)}^2,$$

(28)

where $x_1=\frac{n_o{\gamma }_n}{\lambda +2\mu }$, $x_2=\frac{T_o{\gamma }_t}{\lambda +2\mu }$, $x_3=\frac{1}{\eta D_e}$, $\beta =\frac{k{T}_o}{n_o{\eta }^2{c}^2{D}_e}$, $x_4=\frac{n_o{E}_g}{\rho c_e{T}_o}$, $x_5=\frac{{\gamma }_t}{\rho c_e}$ and $x_6=\frac{s_o}{\eta c{D}_e}$.

For $f\left(x,t\right)$ function Laplace transforms was written as

$$\overline{f}\left(x,p\right)=L\left[f\left(x,t\right)\right]={\int }_0^{\infty }f\left(x,t\right)e^{-pt}dt, p>0,$$

(29)

where $p$ is the Laplace transforms parameters. Hence, the basic equations can be rewritten by forms as follow

$$\frac{d^2\overline{u}}{d{x}^2}=p^2\overline{u}+x_1\frac{d\overline{N}}{dx}+x_2\frac{d\overline{\varphi }}{dx},$$

(30)

$$\frac{d^2\overline{N}}{d{x}^2}=x_3\left(p+\frac{1}{\tau }\right)\overline{N}-\frac{\beta }{\tau }\overline{\varphi },$$

(31)

$$\frac{d^2\overline{\varphi }}{d{x}^2}=p\varphi -\frac{x_4}{\tau }\overline{N}+x_{5}p\frac{d\overline{u}}{dx},$$

(32)

$${\overline{\sigma }}_{xx}=\frac{d\overline{u}}{dx}-x_1\overline{N}-x_2\left(-1+\sqrt{1+2K_1\overline{\varphi }}\right),$$

(33)

$$\overline{u}\left(0,t\right)=0, {\frac{d\overline{N}\left(x,t\right)}{dx}|}_{x=0}=x_6\overline{N}\left(0,t\right), \overline{\varphi }\left(0,t\right)=\frac{T_1}{p}\left(1+\frac{1}{2p}{T}_1{K}_1\right),$$

(34)

We will now get the solutions of coupled differential system (30)–(32) with the boundary conditions (34) by using the eigenvalues method proposed [30,31,32,33,34,35]. From Equations. (30)–(32), the matrices–vectors can be given by

$$\frac{d\mathrm{V}}{dx}=AV,$$

(35)

where $V={\left[\begin{array}{cccccc}\overline{u}& \overline{N}& \overline{\varphi }& \frac{d\overline{u}}{dx}& \frac{d\overline{N}}{dx}& \frac{d\overline{\varphi }}{dx}\end{array}\right]}^T$ and $A=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ a_{41}& 0& 0& 0& a_{45}& a_{46}\\ 0& a_{52}& a_{53}& 0& 0& 0\\ 0& a_{62}& a_{63}& a_{64}& 0& 0\end{array}\right]$, with

$$a_{41}=p^2,a_{45}=x_1,a_{46}=x_2,a_{52}=x_3\left(p+\frac{1}{\tau }\right), a_{53}=-\frac{\beta }{\tau },a_{62}=-\frac{x_4}{\tau },a_{63}=p,a_{64}=p{x}_5$$

Thus, the characteristic equation of matrix $A$ are presented by:

$${\xi }^6-r_3{\xi }^4+r_2{\xi }^2+r_1=0,$$

(36)

where:

$$r_1=a_{62}{a}_{53}{a}_{41}-a_{41}{a}_{63}{a}_{52}$$

$$r_2=a_{46}{a}_{52}{a}_{64}+a_{41}{a}_{52}-a_{45}{a}_{53}{a}_{64}+a_{41}{a}_{63}-a_{53}{a}_{62}+a_{52}{a}_{63}$$

$$r_3=a_{46}{a}_{64}+a_{52}+a_{41}+a_{63}$$

The matrix eigenvalue of $A$ are the six roots of Equation (36) which define by the forms $\pm {\xi }_1, \pm {\xi }_2, \pm {\xi }_3$. Thus, the eigenvectors $Y$ are computed as:

$$\begin{gathered} Y_1=a_{46}\left(a_{52}-{\xi }^2\right)\xi -\xi a_{53}{a}_{45},Y_2=\left(a_{41}-{\xi }^2\right)a_{53} \\ {Y}_3=-\left(a_{41}-{\xi }^2\right)\left(a_{52}-{\xi }^2\right), Y_4=\xi Y_1 , Y_5=\xi Y_{2 }, Y_6=\xi Y_3 \end{gathered}$$

(37)

The solutions of Equation (35) have the following from:

$$\mathrm{V}\left(x,p\right)={\sum }_{i=1}^3\left(B_i{Y}_i{e}^{-{\xi }_{i}x}+B_{i+1}{Y}_{i+1}{e}^{{\xi }_{i}x}\right),$$

(38)

Due to the regularity condition of the solution, the increasing exponential nature with the variable $x$ has been eliminated to infinity, so the general solutions (34) can be presented as:

$$\mathrm{V}\left(x,p\right)={\sum }_{i=1}^3{B}_i{Y}_i{e}^{-{\xi }_{i}x},$$

(39)

where $B_1,B_2$ and $B_3$ are constants which can be determined by using the boundary conditions of the problem. Thus, the general solutions of physical quantities can be taken the forms:

$$\overline{u}\left(x,p\right)={\sum }_{i=1}^3{B}_i{U}_i{e}^{-{\xi }_{i}x},$$

(40)

$$\overline{N}\left(x,p\right)={\sum }_{i=1}^3{B}_i{N}_i{e}^{-{\xi }_{i}x},$$

(41)

$$\overline{\phi }\left(x,p\right)={\sum }_{i=1}^3{B}_i{T}_i{e}^{-{\xi }_{i}x},$$

(42)

The numeric inversions scheme adopted the general solutions of the carrier density, the displacement, the temperature and the stress distribution. The Stehfest method [36] was taken as a numeric inversions scheme. In this method, the Laplace transforms inverses for $\overline{f}\left(x,p\right)$ are approximated as:

$$f\left(x,t\right)=\frac{ln\left(2\right)}{t}{\sum }_{n=1}^N{V}_n\overline{f}\left(x,n\frac{ln\left(2\right)}{t}\right),$$

(43)

where:

$$V_n={\left(-1\right)}^{\left(\frac{N}{2}+1\right)}{\displaystyle \sum }_{p=\frac{n+1}{2}}^{min\left(n,\frac{N}{2}\right)}\frac{\left(2p\right)!p^{\left(\frac{N}{2}+1\right)}}{p!\left(n-p\right)!\left(\frac{N}{2}-p\right)!\left(2n-1\right)!}$$

where $N$ is the term numbers.

4. Numeric Results and Discussion

For numeric computations, silicon medium was selection for purposes of numeric estimation. The parameters values for silicon (Si) medium are given by [37]:

$$d_n=-9\times {10}^{-31}\left({\mathrm{m}}^3\right), \lambda =3.64\times {10}^{10}\left(\mathrm{N}\right)\left({\mathrm{m}}^{-2}\right), \mu =5.46\times {10}^{10}\left(\mathrm{N}\right)\left({\mathrm{m}}^{-2}\right),T_o=300\left(\mathrm{k}\right)$$

$$c_e=695\left(\mathrm{J}\right)\left(\mathrm{k}{\mathrm{g}}^{-1}\right)\left({\mathrm{k}}^{-1}\right),\rho =2330\left(\mathrm{kg}\right)\left({\mathrm{m}}^{-3}\right), E_g=1.11 \left(\mathrm{eV}\right), {\alpha }_t=3\times {10}^{-6}\left({\mathrm{k}}^{-1}\right)$$

$$s_o=2 \left(\mathrm{m}\right)\left({\mathrm{s}}^{-1}\right),n_o={10}^{20}\left({\mathrm{m}}^{-3}\right),\tau =5\times {10}^{-5}\left(\mathrm{s}\right), D_e=2.5\times {10}^{-3}\left({\mathrm{m}}^2\right)\left({\mathrm{s}}^{-1}\right), T_1=1$$

The numeric computations are carried out for the time $t=1.5$. Based on the above values of parameters, the values of physical quantities (numeral) with respect to the distances $x$ with the coupled theory of plasma and thermoelastic wave are presented in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5. The variations of carrier density, the variations of temperature, the variations of displacement and the variation of stress along the distances $x$ in the context of the coupled photo-thermo-elastic model are computed numerically. Figure 1 shows the variations of temperature versus $x$. It is observed that the temperature has maximum values $\left(T_1=1\right)$ which satisfy the problem boundary conditions when $x=0$ after that it gradually reduces with the rising of the distance $x$ until it reached to zero. Otherwise, Figure 2 exhibits the variations of the heat conduction along the distance $x$. Figure 3 displays the variation of carrier density with respect to the distance $x$. It is observed that it begins with its ultimate value at the surface $x=0$ then it gradually reduces with the rising of the distances $x$ until it close to zeros. Figure 4 depicts the variation of displacement along the distance $x$. It is clear that the displacement begin from zero which satisfy boundary conditions of the problem when $x=0$ then it is rising gradually up to peak values then reduces to zero. Figure 5 demonstrates the variations of stress versus the distance $x$. It is observed that it attains maxmum negative value after that it increases gradually to zeros.

Finally, in the compressions between the solutions, it can be concluded that considering the coupling photo-thermoelastic models with variable thermal conductivity is significant phenomenon and have important effects on the distributions of the studding fields.

5. Conclusions

In the present work, I study the effects of variable thermal conductivity in semiconductor materials photogenerated by a focused thermal shock. The resulting nondimensional equations were solved employing the Laplace transforms techniques and were solved using the eigenvalues approach. The significant effects of the variable thermal conductivity are discussed for all physical quantities. The results carried out in this work can be used to design various semiconductor elements to meet special engineering requirements.