Influence of Magnetic Field on Thermomechanical Optical Waves in a Semiconductor Medium with Porosity

Abstract

The manuscript provides a thorough solution for the propagation of waves in a generalized homogeneous and isotropic thermo-photo-electric semiconductor medium. The main focus lies in the analysis of the two-dimensional (2D) issue in the presence of a magnetic field while adding optoelectronic excitation mechanisms. The governing equations exhibit coupling and encompass multiple factors, including relaxation time and porosity (voids) parameters. The governing equations are solved utilizing the normal mode technique, resulting in equations for the quasi-static electric field, heat conduction, carrier density (plasma waves), elastic waves, and the constitutive relationships of the thermo-magneto-photo-electric medium. This investigation considers the boundary conditions of plasma, thermal, and mechanical stress to determine the main physical variables involved. The objective of this research is to examine the effects of several thermo-magneto-photo-electric models, characteristics related to porous structures (voids), as well as time and spatial coordinates on a range of physical quantities. The impacts are examined through the visual analysis of graphical representations.

APPENDIX

The primary variables of Eqs. (20)–(35) can be represented as

$${{q}_{1}} = \frac{{kt{\text{*}}}}{{\rho {{c}_{E}}{{D}_{E}}}},\quad {{q}_{2}} = \frac{k}{{\rho {{c}_{E}}{{D}_{E}}}},\quad {{\varepsilon }_{3}} = \frac{{\kappa K{{\delta }_{n}}t{\text{*}}}}{{\gamma \rho {{c}_{E}}{{D}_{E}}}},\quad {{a}_{1}} = \frac{{\mu + \lambda }}{{\rho c_{T}^{2}}},\quad {{a}_{2}} = \frac{{k + \mu }}{{\rho c_{T}^{2}}},$$

$${{a}_{3}} = \frac{{2\mu + \lambda }}{{\rho c_{T}^{2}}}(1 + {{{v}}_{o}}\omega ),\quad {{a}_{4}} = \frac{{2\mu + \lambda }}{{\rho c_{T}^{2}}},\quad {{a}_{5}} = {{\mu }_{0}}{{\sigma }_{0}}{{H}_{0}},\quad {{a}_{6}} = \frac{{\mu _{0}^{2}{{\sigma }_{0}}{{H}_{0}}kt{\text{*}}}}{{{{\rho }^{2}}{{c}_{E}}}},$$

$${{a}_{7}} = \frac{{{{\mu }_{0}}{{H}_{0}}t{\text{*}}}}{\rho },\quad {{a}_{8}} = \frac{{\rho {{c}_{E}}c_{T}^{2}t{\text{*}}}}{K}({{n}_{1}} + {{\tau }_{o}}\omega ),\quad {{a}_{9}} = \frac{{{{\gamma }^{2}}{{T}_{0}}t{\text{*}}c_{T}^{2}}}{{(2\mu + \lambda )K}}({{n}_{1}} + {{n}_{o}}{{\tau }_{o}}\omega ),$$

$${{a}_{{10}}} = \frac{{{{E}_{g}}\gamma t{{{\text{*}}}^{2}}c_{T}^{2}}}{{\tau {{\delta }_{n}}{{K}^{2}}}},\quad {{a}_{{11}}} = \frac{{2\mu + \lambda }}{\mu },\quad {{a}_{{12}}} = \frac{\lambda }{\mu },\quad {{a}_{{13}}} = \frac{{2\mu + \lambda }}{\mu }(1 + {{{v}}_{o}}\omega ),$$

$${{a}_{{14}}} = \frac{{{{d}_{n}}(2\mu + \lambda )}}{{\mu {{\delta }_{n}}}}(2\mu + 3\lambda ),\quad {{a}_{{15}}} = \frac{{\rho {{c}_{E}}t{\text{*}}c_{T}^{2}}}{{{{\sigma }_{0}}k}},\quad {{a}_{{16}}} = {{\mu }_{0}}t{\text{*}}c_{T}^{2},\quad {{b}_{1}} = {{\mu }_{0}}{{\varepsilon }_{0}}c_{T}^{2},$$

$${{b}_{2}} = \frac{{{{\lambda }_{o}}}}{{\psi \rho t{{{\text{*}}}^{2}}}},\quad {{b}_{4}} = \frac{{{{b}_{2}}}}{{{{a}_{1}} + {{a}_{2}}}},\quad {{b}_{5}} = \frac{{{{\gamma }^{2}}{{T}_{0}}c_{T}^{4}}}{{(2\mu + \lambda )Kt{\text{*}}\psi }},\quad {{b}_{8}} = \frac{{(2\mu + \lambda ){{\lambda }_{o}}\psi t{{{\text{*}}}^{4}}}}{\alpha },$$

$${{b}_{6}} = \frac{{\rho {{\psi }^{2}}t{{{\text{*}}}^{2}}}}{\alpha }{{\omega }^{2}} + \frac{{{{\omega }_{o}}\psi t{{{\text{*}}}^{3}}}}{\alpha }\omega - \frac{{{{\varsigma }_{1}}\psi t{{{\text{*}}}^{4}}}}{\alpha },\quad {{b}_{7}} = \frac{{{{\lambda }_{o}}\psi t{{{\text{*}}}^{4}}}}{\alpha },\quad {{b}_{9}} = \frac{{{{\lambda }_{o}}c_{T}^{2}}}{{\mu \psi t{{{\text{*}}}^{2}}}},$$

$${{b}_{{10}}} = 1 + \frac{{\mu + k}}{\mu },\quad {{b}_{{11}}} = \frac{{\mu + k}}{\mu },\quad {{b}_{{12}}} = {{b}_{5}}\omega ,\quad {{\alpha }_{{10}}} = {{b}^{2}} + {{b}_{6}},$$

$${{a}_{{17}}} = \frac{1}{{{{a}_{1}} + {{a}_{2}}}},\quad {{a}_{{18}}} = \frac{{{{a}_{3}}}}{{{{a}_{1}} + {{a}_{2}}}},\quad {{a}_{{19}}} = \frac{{{{a}_{4}}}}{{{{a}_{1}} + {{a}_{2}}}},\quad {{a}_{{20}}} = \frac{{{{a}_{5}}{{a}_{7}}}}{{{{a}_{1}} + {{a}_{2}}}},\quad {{a}_{{21}}} = \frac{{{{a}_{5}}{{a}_{6}}}}{{{{a}_{1}} + {{a}_{2}}}},$$

$${{a}_{{22}}} = \frac{\rho }{\mu },\quad {{a}_{{24}}} = {{\sigma }_{0}}({{a}_{{15}}} - {{a}_{{16}}}{{\mu }_{0}}{{H}_{0}}),\quad {{a}_{{25}}} = {{a}_{{17}}}{{\mu }_{0}}{{H}_{0}},\quad {{\alpha }_{1}} = {{b}^{2}} + {{q}_{1}} + {{q}_{2}}\omega ,$$

$${{\alpha }_{2}} = 2{{b}^{2}} + {{a}_{{20}}}\omega + {{a}_{{17}}}{{\omega }^{2}},\quad {{\alpha }_{3}} = {{b}^{4}} + {{b}^{2}}{{a}_{{20}}}\omega + {{b}^{2}}{{a}_{{17}}}{{\omega }^{2}},$$

$${{\alpha }_{4}} = {{a}_{{21}}}\omega ,\quad {{\alpha }_{5}} = {{b}^{2}} + {{a}_{{22}}}{{\omega }^{2}},\quad {{\alpha }_{7}} = {{b}^{2}} + {{a}_{8}}\omega ,\quad {{\alpha }_{8}} = {{a}_{9}}\omega ,\quad {{\alpha }_{9}} = {{a}_{{24}}}\omega - {{a}_{{25}}}{{\omega }^{2}}.$$

The main parameters in Eqs. (64–71) can be expressed as

$$h_{0}^{*} = {{h}_{0}}{{e}^{{ - (\omega t + ibz)}}},\quad \sigma _{0}^{*} = {{\sigma }_{0}}{{e}^{{ - (\omega t + ibz)}}},$$

$${{r}_{n}} = {{a}_{{11}}}k_{n}^{2} - {{a}_{{12}}}{{b}^{2}} - {{a}_{{13}}}{{H}_{{2n}}} + {{a}_{{14}}}{{H}_{{3n}}} + {{b}_{9}}{{H}_{{4n}}};\quad n = 2,3,4,5,6$$

$${{r}_{1}} = - ib{{k}_{1}}({{a}_{{12}}} - {{a}_{{11}}}),$$

$${{S}_{1}} = {{b}_{{11}}}k_{1}^{2} + {{b}^{2}},$$

$${{S}_{n}} = - ib{{k}_{n}}{{b}_{{10}}};\quad n = 2,3,4,5,6$$

$${{C}_{n}} = \left( {\frac{s}{{{{D}_{E}}}} + {{k}_{n}}} \right){{H}_{{3n}}};\quad n = 2,3,4,5,6.$$

Calculating the values of the fundamental coefficients in equation (36), which can be done with software like Mathematica and Matlab, is as simple as doing the following

$${{A}_{1}} = {{a}_{{18}}}{{\alpha }_{8}} - {{b}_{3}}{{b}_{7}} + {{\alpha }_{1}} + {{\alpha }_{2}} + {{\alpha }_{7}} + {{\alpha }_{{10}}},$$

$$\begin{gathered} {{A}_{2}} = 2{{b}^{2}}{{a}_{{18}}}{{\alpha }_{8}} - 2{{b}^{2}}{{b}_{3}}{{b}_{7}} + {{a}_{{18}}}{{\alpha }_{1}}{{\alpha }_{8}} + {{a}_{{18}}}{{\alpha }_{{10}}}{{\alpha }_{8}} - {{a}_{{18}}}{{b}_{{12}}}{{b}_{7}} + {{a}_{{19}}}{{\alpha }_{8}}{{\varepsilon }_{3}} - {{\alpha }_{1}}{{b}_{3}}{{b}_{7}} \\ \, - {{\alpha }_{7}}{{b}_{3}}{{b}_{7}} - {{\alpha }_{8}}{{b}_{3}}{{b}_{8}} - {{a}_{{10}}}{{\varepsilon }_{3}} + {{\alpha }_{1}}{{\alpha }_{2}} + {{\alpha }_{1}}{{\alpha }_{7}} + {{\alpha }_{1}}{{\alpha }_{{10}}} + {{\alpha }_{2}}{{\alpha }_{7}} + {{\alpha }_{2}}{{\alpha }_{{10}}} - {{\alpha }_{4}}{{\alpha }_{9}} \\ + \,{{\alpha }_{7}}{{\alpha }_{{10}}} + {{b}_{{12}}}{{b}_{8}} - {{\alpha }_{3}}, \\ \end{gathered} $$

$$\begin{gathered} {{A}_{3}} = {{b}^{4}}{{a}_{{18}}}{{\alpha }_{8}} - {{b}^{4}}{{b}_{3}}{{b}_{7}} + 2{{b}^{2}}{{a}_{{18}}}{{\alpha }_{1}}{{\alpha }_{8}} + 2{{b}^{2}}{{a}_{{18}}}{{\alpha }_{{10}}}{{\alpha }_{8}} - 2{{b}^{2}}{{a}_{{18}}}{{b}_{{12}}}{{b}_{7}} + 2{{b}^{2}}{{a}_{{19}}}{{\alpha }_{8}}{{\varepsilon }_{3}} - 2{{b}^{2}}{{\alpha }_{1}}{{b}_{3}}{{b}_{7}} \\ - \,2{{b}^{2}}{{\alpha }_{7}}{{b}_{3}}{{b}_{7}} - 2{{b}^{2}}{{\alpha }_{8}}{{b}_{3}}{{b}_{8}} + {{a}_{{10}}}{{b}_{3}}{{b}_{7}}{{\varepsilon }_{3}} - {{b}_{{12}}}{{a}_{{18}}}{{\alpha }_{1}}{{\alpha }_{7}} + {{a}_{{18}}}{{\alpha }_{1}}{{\alpha }_{8}}{{\alpha }_{{10}}} + {{a}_{{19}}}{{\alpha }_{8}}{{\alpha }_{{10}}}{{\varepsilon }_{3}} - {{a}_{{19}}}{{b}_{{12}}}{{b}_{7}}{{\varepsilon }_{3}} \\ - \,{{b}_{3}}{{b}_{7}}{{\alpha }_{1}}{{\alpha }_{7}} - {{\alpha }_{1}}{{\alpha }_{8}}{{b}_{3}}{{b}_{8}} - {{a}_{{10}}}{{\alpha }_{2}}{{\varepsilon }_{3}} - {{a}_{{10}}}{{\alpha }_{{10}}}{{\varepsilon }_{3}} + {{\alpha }_{1}}{{\alpha }_{2}}{{\alpha }_{7}} + {{\alpha }_{1}}{{\alpha }_{2}}{{\alpha }_{{10}}} - {{\alpha }_{1}}{{\alpha }_{4}}{{\alpha }_{9}} + {{\alpha }_{1}}{{\alpha }_{7}}{{\alpha }_{{10}}} + {{\alpha }_{1}}{{b}_{{12}}}{{b}_{8}} \\ - \,{{\alpha }_{2}}{{\alpha }_{7}}{{\alpha }_{{10}}} + {{\alpha }_{2}}{{b}_{{12}}}{{b}_{8}} - {{\alpha }_{7}}{{\alpha }_{4}}{{\alpha }_{9}} - {{\alpha }_{{10}}}{{\alpha }_{4}}{{\alpha }_{9}} - {{\alpha }_{1}}{{\alpha }_{3}} - {{\alpha }_{3}}{{\alpha }_{7}} - {{\alpha }_{3}}{{\alpha }_{{10}}}, \\ \end{gathered} $$

$$\begin{gathered} {{A}_{4}} = {{b}^{4}}{{a}_{{18}}}{{\alpha }_{1}}{{\alpha }_{8}} + {{b}^{4}}{{a}_{{18}}}{{\alpha }_{8}}{{\alpha }_{{10}}} - {{b}^{4}}{{a}_{{18}}}{{b}_{7}}{{b}_{{12}}} + {{b}^{4}}{{a}_{{19}}}{{\alpha }_{8}}{{\varepsilon }_{3}} - {{b}^{4}}{{a}_{1}}{{b}_{3}}{{b}_{7}} - {{b}^{4}}{{a}_{7}}{{b}_{7}}{{b}_{3}} \\ - \,{{b}^{4}}{{a}_{8}}{{b}_{3}}{{b}_{8}} + 2{{b}^{2}}{{a}_{{10}}}{{b}_{3}}{{b}_{7}}{{\varepsilon }_{3}} - 2{{b}^{2}}{{b}_{{12}}}{{a}_{{18}}}{{\alpha }_{1}}{{\alpha }_{7}} + 2{{b}^{2}}{{a}_{{18}}}{{\alpha }_{1}}{{\alpha }_{8}}{{\alpha }_{{10}}} + 2{{b}^{2}}{{a}_{{19}}}{{\alpha }_{8}}{{\alpha }_{{10}}}{{\varepsilon }_{3}} - 2{{b}^{2}}{{a}_{{19}}}{{b}_{{12}}}{{b}_{7}}{{\varepsilon }_{3}} \\ - \,2{{b}^{2}}{{b}_{3}}{{b}_{7}}{{\alpha }_{1}}{{\alpha }_{7}} - 2{{b}^{2}}{{\alpha }_{1}}{{\alpha }_{8}}{{b}_{3}}{{b}_{8}} - {{a}_{{10}}}{{\alpha }_{4}}{{\alpha }_{9}}{{\varepsilon }_{3}} + {{\alpha }_{{10}}}{{\alpha }_{1}}{{\alpha }_{2}}{{\alpha }_{7}} + {{\alpha }_{1}}{{\alpha }_{2}}{{b}_{8}}{{b}_{{12}}} - {{\alpha }_{1}}{{\alpha }_{4}}{{\alpha }_{7}}{{\alpha }_{9}} \\ - \,{{\alpha }_{1}}{{\alpha }_{4}}{{\alpha }_{9}}{{\alpha }_{{10}}} - {{\alpha }_{4}}{{\alpha }_{7}}{{\alpha }_{9}}{{\alpha }_{{10}}} - {{\alpha }_{4}}{{\alpha }_{9}}{{b}_{8}}{{b}_{{12}}} + {{a}_{{10}}}{{\alpha }_{3}}{{\varepsilon }_{3}} - {{\alpha }_{1}}{{\alpha }_{3}}{{\alpha }_{7}} - {{\alpha }_{1}}{{\alpha }_{3}}{{\alpha }_{{10}}} - {{\alpha }_{3}}{{\alpha }_{7}}{{\alpha }_{{10}}} - {{\alpha }_{3}}{{b}_{8}}{{b}_{{12}}}, \\ \end{gathered} $$

$$\begin{gathered} {{A}_{5}} = {{b}^{4}}{{a}_{{10}}}{{b}_{7}}{{b}_{3}}{{\varepsilon }_{3}} + {{b}^{4}}{{a}_{{18}}}{{\alpha }_{8}}{{\alpha }_{{10}}} - {{b}^{4}}{{a}_{{18}}}{{\alpha }_{1}}{{\alpha }_{8}}{{\alpha }_{{10}}}{{\varepsilon }_{3}} - {{b}^{4}}{{a}_{{19}}}{{b}_{{12}}}{{b}_{7}}{{\varepsilon }_{3}} - {{b}^{4}}{{a}_{1}}{{b}_{3}}{{b}_{7}}{{\alpha }_{7}} \\ - \,{{b}^{4}}{{\alpha }_{1}}{{\alpha }_{8}}{{b}_{3}}{{b}_{8}} + {{a}_{{10}}}{{\alpha }_{4}}{{\alpha }_{9}}{{\alpha }_{{10}}}{{\varepsilon }_{3}} - {{\alpha }_{1}}{{\alpha }_{4}}{{\alpha }_{7}}{{\alpha }_{9}}{{\alpha }_{{10}}} - {{\alpha }_{1}}{{\alpha }_{4}}{{\alpha }_{9}}{{b}_{8}}{{b}_{{12}}} + {{a}_{{10}}}{{\alpha }_{3}}{{\alpha }_{{10}}}{{\varepsilon }_{3}} - {{\alpha }_{1}}{{\alpha }_{3}}{{\alpha }_{7}}{{\alpha }_{{10}}} - {{\alpha }_{1}}{{\alpha }_{3}}{{b}_{8}}{{b}_{{12}}}. \\ \end{gathered} $$