Plane wave in non-local semiconducting rotating media with Hall effect and three-phase lag fractional order heat transfer

Abstract

This paper deals with the propagation of the plane wave in a nonlocal magneto-thermoelastic semiconductor solid with rotation. The fractional-order three-phase lag theory of thermoelasticity with two temperatures has been applied. When a longitudinal wave is incident on the surface z = 0, four types of reflected coupled longitudinal waves (the coupled longitudinal displacement wave, the coupled thermal wave, coupled carrier density wave, and coupled transverse displacement wave) are identified. The plane wave characteristics such as phase velocities, specific loss, attenuation coefficient, and penetration depth of various reflected waves are computed. The effects of two temperatures, non-local parameter, fractional order parameter, and Hall current on these wave characteristics are illustrated graphically with the use of MATLAB software.

Introduction

The plane wave propagation in a photo-thermo-magneto-elastic solid has gained significant importance due to its applications in the area of semiconductors, magnetometers, solar panels, nuclear fields, geophysics, and other linked topics. Lotfy et al. (2020) discussed Hall current effect in a semiconductor medium exposed to a very strong magnetic field. Lotfy (2017) examined the wave propagation in a semiconductor medium having a spherical cavity using FOT. Ali et al. (2020) examined the reflection of waves over a semiconductor rotating medium using the TPL model with FOT. Tang and Song (2018) studied wave reflection in nonlocal semi-conductor rotating media by using the plasma diffusion equation. Alshaikh (2020) examined the transmission of photo-thermal waves in a semiconductor for diffusion and rotation effects. Kaur et al. (2020a) discussed the propagation of the plane wave in a visco-thermoelastic rotating medium with Hall current. Lata et al. (2021) discussed the propagation of plane harmonic waves thermo-magneto-elastic rotating medium with multi-dual-phase lag heat transfer. Lata and Kaur (2018) discussed the effect of Hall current on a rotating transversely isotropic thermoelastic medium with 2T. Eringen (2004; 1974; 1972) developed the nonlocal continuum mechanics theory to study the micro-scaled/nano-scaled structure problems. These theories exhibit that “consider the state of stress at a point as a function of the states of the strain of all points in the medium. But in classical continuum mechanics, the state of stress at a certain point uniquely depends on the state of strain on that same point”. Also, some other researchers worked on the wave propagation in different media using different theories of thermoelasticity as Lim et al.(1992), Marin (2010; 1996), Abbas and Marin (2018), Kaur et al. (2020b; 2019a), Bhatti et al. (2019; 2020), Marin et al. (2015, 2016, 2020), Zhang et al. (2020a), Bhatti et al. (2021), Lata and Kaur (2019b; 2020; 2019), Pandey et al. (2021), Taye et al. (2021), Zhang et al. (2020b), Bhatti and Abdelsalam (2020), Zhang et al. (2021), and Golewski (2021). Despite the above research, no research has been done for the plane wave propagation with the fractional order three-phase lag two-temperature heat transfer in rotating magneto thermoelastic nonlocal semiconducting medium.

This research investigates the transmission of plane waves in a nonlocal semiconducting rotating medium under the influence of a high magnetic field and Hall current. The governing equations are expressed with TPL-2T FOT of thermoelasticity. For considered 2-D model, when a longitudinal wave is incident on the surface z=0, four types of reflected waves distinguished as coupled longitudinal waves (CLD-wave, CT-wave, CCD-wave, CTD-wave) are identified. The plane wave characteristics of various reflected waves are computed numerically and demonstrated graphically. The effects of two temperatures, non-local parameter, fractional order parameter, and Hall current on wave characteristics illustrated graphically with the use of MATLAB software have been studied.

Basic equations

Following Tang and Song (2018), Othman and Abd-Elaziz (2019), and Mahdy et al. (2020), the equations of motion with Lorentz force is

$$ {\sigma}_{ij,j}+\left(1-{\epsilon}^2{\nabla}^2\right)\ {\mu}_0{\varepsilon}_{ij r}{J}_j{H}_r=\rho \left(1-{\epsilon}^2{\nabla}^2\right){\left\{{\ddot{u}}_i+2{\left(\boldsymbol{\Omega} \times \dot{\boldsymbol{u}}\right)}_i+\left(\boldsymbol{\Omega} \times \Big(\boldsymbol{\Omega} \times \mathbf{u}\right)\right)}_i\Big\}, $$

(1)

where subscript followed by “,” comma denotes partial derivative w.r.t. space variable, and the superimposed dot denotes time derivative. Ω × (Ω × u) represents the centripetal acceleration due to the time-varying motion and $2\boldsymbol{\Omega} \times \dot{\boldsymbol{u}}$ denotes Coriolis acceleration.

For very high magnetic field strength, Hall current term is also introduced, so generalized Ohm’s law (Othman and Abd-Elaziz 2019) is written as

$$ {J}_{\boldsymbol{i}}+{\omega}_e{t}_e{\varepsilon}_{ilk}{J}_l{H}_k={\sigma}_0\left({E}_i+{\mu}_0{\varepsilon}_{ijr}{\dot{u}}_j{H}_r\right), $$

(2)

Equation (2) can also be written as

$$ J=\frac{\sigma_0}{1+{m}^2}\left\{E+{\mu}_0\left(\dot{u}\times H\right)-\frac{\mu_0}{e{n}_e}\left(J\times H\right)\right\} $$

Following Lotfy et al. (2020), the stress-displacement-strain-carrier density function relation is given by

$$ {\sigma}_{ij}=\left(\lambda {u}_{r,r}-\beta T-{\delta}_nN\right){\delta}_{ij}+\mu \left({u}_{i,j}+{u}_{j,i}\right). $$

(3)

where, T = φ − aφ_,ij,

β = (3λ + 2μ)α_T,

δ_n = (3λ + 2μ)d_n,

Here, a > 0 is the two-temperature parameter.

For the semiconductors nanostructure medium, for the plasma transportation process, the equation of plasma diffusion is given by

$$ \frac{\partial N\left(x,y,z,t\right)}{\partial t}={D}_E{\nabla}^2N\left(x,y,z,t\right)-\frac{N\left(x,y,z,t\right)}{\tau }+\kappa \frac{T}{\tau } $$

(4)

The fractional-order heat conduction equation with two temperatures (Kaur et al. 2020a, Mahdy et al. 2020) is given by

$$ {K}_{ij}\left(1+\frac{{\left({\tau}_T\right)}^{\alpha }}{\alpha !}\frac{\partial^{\alpha }}{\partial {t}^{\alpha }}\right){\dot{\varphi}}_{, ji}+{K}_{ij}^{\ast}\left(1+\frac{{\left({\tau}_v\right)}^{\alpha }}{\alpha !}\frac{\partial^{\alpha }}{\partial {t}^{\alpha }}\right){\varphi}_{, ji}-\frac{E_g}{\tau}\frac{\partial N\left(r,t\right)}{\partial t}=\left(1+\frac{{\left({\tau}_q\right)}^{\alpha }}{\alpha !}\frac{\partial^{\alpha }}{\partial {t}^{\alpha }}+\frac{{\left({\tau}_q\right)}^{2\alpha }}{2\alpha !}\frac{\partial^{2\alpha }}{\partial {t}^{2\alpha }}\right)\left[{\rho C}_E\ddot{T}+{\beta}_{ij}{T}_0{\ddot{\mathrm{e}}}_{ij}-\rho \dot{Q}\right], $$

(5)

where

$\left\{\begin{array}{c}0<\alpha <1\kern0.5em \mathrm{for}\ \mathrm{weak}\ \mathrm{conductivity},\\ {}\alpha =1\ \mathrm{for}\ \mathrm{normal}\ \mathrm{conductivity},\\ {}1<\alpha \le 2\ \mathrm{for}\ \mathrm{strong}\ \mathrm{conductivity},\end{array}\right.$

${K}_{ij}={K}_i{\delta}_{ij},{K}_{ij}^{\ast }={K}_i^{\ast }{\delta}_{ij},\kern0.5em i$ is not summed.

Method and solution of the problem

Consider a nonlocal semiconducting magneto-thermoelastic homogeneous isotropic medium initially at a constant temperature T₀ and rotating about the y-axis with an angular velocity Ω = (0, Ω, 0). Consider orthogonal Cartesian coordinates (x, y, z) with origin on the surface (z = 0) and the z-axis directing downwards in the semiconductor medium. For the 2-D dynamic problem in xz−plane, we consider displacement vector as

$$ \boldsymbol{u}=\left(u,0,w\right)\left(x,z,t\right). $$

(6)

Consider that a very high-intensity magnetic field H₀ = (0, H₀, 0) is applied in the positive y-direction and also assuming that induced electric field E = 0, therefore from ohms law we have

$$ {J}_y=0. $$

(7)

and J_x and J_z are given as

$$ {J}_x=\frac{\sigma_0{\mu}_0{H}_0}{1+{m}^2}\left(m\frac{\partial u}{\partial t}-\frac{\partial w}{\partial t}\right), $$

(8)

$$ {J}_z=\frac{\sigma_0{\mu}_0{H}_0}{1+\kern0.5em {m}^2}\left(\frac{\partial u}{\partial t}+m\frac{\partial w}{\partial t}\right). $$

(9)

Using Eqs. (6), (7), (8), (9) in Eqs. (1), (4), and (5), the equations for nonlocal 2-D semiconducting medium with 2T in the absence of heat source, i.e., taking Q = 0, are:

$$ \left(\lambda +2\mu \right)\frac{\partial^2u}{\partial {x}^2}+\left(\lambda +\mu \right)\frac{\partial^2w}{\partial x\partial z}+\mu \frac{\partial^2u}{\partial {z}^2}-\beta \frac{\partial }{\partial x}\left\{\varphi -a\left(\frac{\partial^2\varphi }{\partial {x}^2}+\frac{\partial^2\varphi }{\partial {z}^2}\right)\right\}-{\delta}_n\frac{\partial N}{\partial x}-\left(1-{\epsilon}^2\left(\frac{\partial^2}{\partial {x}^2}+\frac{\partial^2}{\partial {z}^2}\right)\right)\frac{\sigma_0{\mu}_0^2{H}_0^2}{1+\kern0.5em {m}^2}\left(\frac{\partial u}{\partial t}+m\frac{\partial w}{\partial t}\right)=\rho \left(1-{\epsilon}^2\left(\frac{\partial^2}{\partial {x}^2}+\frac{\partial^2}{\partial {z}^2}\right)\right)\left(\frac{\partial^2u}{\partial {t}^2}-{\varOmega}^2u+2\varOmega \frac{\partial w}{\partial t}\right), $$

(10)

$$ \left(\lambda +\mu \right)\frac{\partial^2u}{\partial x\partial z}+\mu \frac{\partial^2w}{\partial {x}^2}+\left(\lambda +2\mu \right)\frac{\partial^2w}{\partial {z}^2}-\beta \frac{\partial }{\partial z}\left\{\varphi -a\left(\frac{\partial^2\varphi }{\partial {x}^2}+\frac{\partial^2\varphi }{\partial {z}^2}\right)\right\}-{\delta}_n\frac{\partial N}{\partial z}+\left(1-{\epsilon}^2\left(\frac{\partial^2}{\partial {x}^2}+\frac{\partial^2}{\partial {z}^2}\right)\right)\frac{\sigma_0{\mu}_0^2{H}_0^2}{1+\kern0.5em {m}^2}\left(m\frac{\partial u}{\partial t}-\frac{\partial w}{\partial t}\right)=\rho \left(1-{\epsilon}^2\left(\frac{\partial^2}{\partial {x}^2}+\frac{\partial^2}{\partial {z}^2}\right)\right)\left(\frac{\partial^2w}{\partial {t}^2}-{\varOmega}^2w-2\varOmega \frac{\partial u}{\partial t}\right), $$

(11)

$$ \frac{\partial N}{\partial t}={D}_E\left(\frac{\partial^2N}{\partial {x}^2}+\frac{\partial^2N}{\partial {z}^2}\right)-\frac{N}{\tau }+\kappa \frac{T}{\tau }, $$

(12)

$$ K\left(1+\frac{{\left({\tau}_t\right)}^{\alpha }}{\alpha !}\frac{\partial^{\alpha }}{\partial {t}^{\alpha }}\right)\frac{\partial }{\partial t}\left(\frac{\partial^2\varphi }{\partial {x}^2}+\frac{\partial^2\varphi }{\partial {z}^2}\right)+{K}^{\ast}\left(1+\frac{{\left({\tau}_v\right)}^{\alpha }}{\alpha !}\frac{\partial^{\alpha }}{\partial {t}^{\alpha }}\right)\left(\frac{\partial^2\varphi }{\partial {x}^2}+\frac{\partial^2\varphi }{\partial {z}^2}\right)-\frac{E_g}{\tau}\frac{\partial N}{\partial t}=\left(1+\frac{{\left({\tau}_q\right)}^{\alpha }}{\alpha !}\frac{\partial^{\alpha }}{\partial {t}^{\alpha }}+\frac{{\left({\tau}_q\right)}^{2\alpha }}{2\alpha !}\frac{\partial^{2\alpha }}{\partial {t}^{2\alpha }}\right)\left[{\rho C}_E\frac{\partial^2}{\partial {t}^2}\left[\varphi -a\frac{\partial^2\varphi }{\partial {x}^2}-a\frac{\partial^2\varphi }{\partial {z}^2}\right]+\beta {T}_0\frac{\partial^2}{\partial {t}^2}\left\{\frac{\partial u}{\partial x}+\frac{\partial w}{\partial z}\right\}\right], $$

(13)

and the stress-displacement-carrier density function relation (3) can be written as

$$ {\sigma}_{xx}=\left(\lambda +2\mu \right)\frac{\partial u}{\partial x}+\lambda\ \frac{\partial w}{\partial z}-\beta T-{\delta}_nN, $$

(14)

$$ {\sigma}_{zz}=\lambda \frac{\partial u}{\partial x}+\left(\lambda +2\mu \right)\ \frac{\partial w}{\partial z}-\beta T-{\delta}_nN, $$

(15)

$$ {\sigma}_{xz}=\mu \left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\right). $$

(16)

The dimensionless quantities are assumed as:

$$ \left({x}^{\prime },{z}^{\prime },{u}^{\prime },{w}^{\prime },{\epsilon}^{\prime}\right)=\frac{\omega^{\ast }}{c_1}\left(x,z,u,w,\epsilon \right);{T}^{\prime }=\frac{\beta T}{\lambda +2\mu };{\Omega}^{\prime }=\frac{1}{\omega^{\ast }}\Omega; \left(\ {\sigma}_{xx}^{\prime },{\sigma}_{xz}^{\prime },{\sigma}_{zz}^{\prime}\right)=\frac{1}{\lambda +2\mu}\left({\sigma}_{xx},{\sigma}_{xz},{\sigma}_{zz}\right);a^{\prime }={\left(\frac{\omega^{\ast }}{c_1}\right)}^2,\left({\tau}_T^{\prime },{\tau}_v^{\prime },{\tau}_q^{\prime },{t}^{\prime}\right)={\omega}^{\ast}\left({\tau}_T,{\tau}_v,{\tau}_q,t\right),{\varphi}^{\prime }=\frac{\beta \varphi}{\lambda +2\mu },\left({\phi}^{\prime },{\psi}^{\prime}\right)={\left(\frac{\omega^{\ast }}{c_1}\right)}^2\left(\phi, \psi \right),{N}^{\prime }=\frac{\delta_nN}{\lambda +2\mu },{\omega}^{\ast }=\frac{\rho {C}_E{c}_1^2}{K},{c}_1^2=\frac{\lambda +2\mu }{\rho },{c}_2^2=\frac{\mu }{\rho },{\delta}^2=\frac{c_2^2}{c_1^2},M=\frac{\sigma_0{\mu}_0^2{H}_0^2}{\rho {\omega}^{\ast }}. $$

(17)

Using (17) in Eqs. (10), (11), (12), (13) and after suppressing the primes yields

$$ \frac{\partial^2u}{\partial {x}^2}+\left(1-{\delta}^2\right)\frac{\partial^2w}{\partial x\partial z}+{\delta}^2\frac{\partial^2u}{\partial {z}^2}-\frac{\partial }{\partial x}\left\{\varphi -a\left(\frac{\partial^2\varphi }{\partial {x}^2}+\frac{\partial^2\varphi }{\partial {z}^2}\right)\right\}-\frac{\partial N}{\partial x}=\left(1-{\epsilon}^2\left(\frac{\partial^2}{\partial {x}^2}+\frac{\partial^2}{\partial {z}^2}\right)\right)\left\{\frac{M}{1+{m}^2}\left[\frac{\partial u}{\partial t}+m\frac{\partial w}{\partial t}\right]+\left(\frac{\partial^2u}{\partial {t}^2}-{\Omega}^2u+2\Omega \frac{\partial w}{\partial t}\right)\right\}, $$

(18)

$$ \left(1-{\delta}^2\right)\frac{\partial^2u}{\partial x\partial z}+{\delta}^2\frac{\partial^2w}{\partial {x}^2}+\frac{\partial^2w}{\partial {z}^2}-\frac{\partial }{\partial z}\left\{\varphi -a\left(\frac{\partial^2\varphi }{\partial {x}^2}+\frac{\partial^2\varphi }{\partial {z}^2}\right)\right\}-\frac{\partial N}{\partial z}=\left(1-{\epsilon}^2\left(\frac{\partial^2}{\partial {x}^2}+\frac{\partial^2}{\partial {z}^2}\right)\right)\left\{\frac{-M}{1+{m}^2}\left[m\frac{\partial u}{\partial t}-\frac{\partial w}{\partial t}\right]+\left(\frac{\partial^2w}{\partial {t}^2}-{\Omega}^2w-2\Omega \frac{\partial u}{\partial t}\right)\right\}, $$

(19)

$$ \left[\left(\frac{\partial^2}{\partial {x}^2}+\frac{\partial^2}{\partial {z}^2}\right)-{\delta}_1\left(\frac{\partial }{\partial t}+{\delta}_2\right)\right]N+{\varepsilon}_3\left\{\varphi -a\left(\frac{\partial^2\varphi }{\partial {x}^2}+\frac{\partial^2\varphi }{\partial {z}^2}\right)\right\}=0, $$

(20)

$$ \left(1+\frac{{\left({\tau}_t\right)}^{\alpha }}{\alpha !}\frac{\partial^{\alpha }}{\partial {t}^{\alpha }}\right)\frac{\partial }{\partial t}\left(\frac{\partial^2\varphi }{\partial {x}^2}+\frac{\partial^2\varphi }{\partial {z}^2}\right)+\frac{K^{\ast }}{K{\omega}^{\ast }}\left(1+\frac{{\left({\tau}_v\right)}^{\alpha }}{\alpha !}\frac{\partial^{\alpha }}{\partial {t}^{\alpha }}\right)\left(\frac{\partial^2\varphi }{\partial {x}^2}+\frac{\partial^2\varphi }{\partial {z}^2}\right)+{\varepsilon}_2\frac{\partial N}{\partial t}=\left(1+\frac{{\left({\tau}_q\right)}^{\alpha }}{\alpha !}\frac{\partial^{\alpha }}{\partial {t}^{\alpha }}+\frac{{\left({\tau}_q\right)}^{2\alpha }}{2\alpha !}\frac{\partial^{2\alpha }}{\partial {t}^{2\alpha }}\right)\left[\frac{\partial^2}{\partial {t}^2}\left\{\varphi -a\left(\frac{\partial^2\varphi }{\partial {x}^2}+\frac{\partial^2\varphi }{\partial {z}^2}\right)\right\}+{\varepsilon}_1\left\{\frac{\partial \ddot{u}}{\partial x}+\frac{\partial \ddot{w}}{\partial z}\right\}\right], $$

(21)

where

$$ {\delta}_1=\frac{c_1^2}{D_E{\omega}^{\ast }},{\varepsilon}_3=\frac{\kappa K{d}_n}{\alpha_T\rho {C}_E{D}_E{\omega}^{\ast}\tau^{\prime }},{\varepsilon}_2=\frac{E_g{\alpha}_T}{d_n\rho {C}_E{\left({\omega}^{\ast}\right)}^2\tau^{\prime }},{\varepsilon}_1=\frac{\beta^2{T}_0}{\rho {C}_E\left(\lambda +2\mu \right)},{\delta}_2=\frac{1}{\tau }, $$

The parameter ε₁ is thermoelastic coupling parameter as it depends on thermoelastic properties (i.e., specific heat, Lame’s elastic constants, and temperature T₀). The parameter ε₃ is thermoelectric coupling parameter as it depends on thermoelectrical properties (i.e., coefficient of electronic deformation d_n).

By using Eq. (17) in Eqs. (14), (15), (16) and after suppressing the primes, it yields

$$ {\sigma}_{\mathrm{xx}}\left(\mathrm{x},\mathrm{z},\mathrm{t}\right)=\frac{\partial u}{\partial x}+\left(1-2{\delta}^2\right)\frac{\partial w}{\partial z}-\left\{\varphi -\mathrm{a}\left(\frac{\partial^2\varphi }{{\partial x}^2}+\frac{\partial^2\varphi }{{\partial z}^2}\right)\right\}-N, $$

(22)

$$ {\sigma}_{\mathrm{zz}}\left(\mathrm{x},\mathrm{z},\mathrm{t}\right)=\left(1-2{\delta}^2\right)\frac{\partial u}{\partial x}+\frac{\partial w}{\partial z}-\left\{\varphi -\mathrm{a}\left(\frac{\partial^2\varphi }{{\partial x}^2}+\frac{\partial^2\varphi }{{\partial z}^2}\right)\right\}-N, $$

(23)

$$ {\sigma}_{\mathrm{xz}}\left(\mathrm{x},\mathrm{z},\mathrm{t}\right)={\delta}^2\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\right), $$

(24)

We now present the potential functions ϕ and ψ as

$$ u=\frac{\partial \phi }{\partial x}-\frac{\partial \psi }{\partial z},w=\frac{\partial \phi }{\partial z}+\frac{\partial \psi }{\partial x},e={\nabla}^2\phi, \frac{\partial w}{\partial x}-\frac{\partial u}{\partial z}={\nabla}^2\psi, $$

(25)

Using (25) in Eqs. (18), (19), (20), (21) yields

$$ {\delta}^2{\nabla}^2\psi +\left\{\varphi -a{\nabla}^2\varphi \right\}+N=\left(1-{\epsilon}^2{\nabla}^2\right)\left\{\frac{M}{1+{m}^2}\left[\frac{\partial \psi }{\partial t}-m\frac{\partial \phi }{\partial t}\right]+\left(\frac{\partial^2\psi }{\partial {t}^2}-{\Omega}^2\psi -2\Omega \frac{\partial \phi }{\partial t}\right)\right\}, $$

(26)

$$ {\nabla}^2\phi =\left(1-{\epsilon}^2{\nabla}^2\right)\left\{\frac{M}{1+{m}^2}\left[m\frac{\partial \psi }{\partial t}+\frac{\partial \phi }{\partial t}\right]+\left(\frac{\partial^2\phi }{\partial {t}^2}-{\Omega}^2\phi +2\Omega \frac{\partial \psi }{\partial t}\right)\right\}, $$

(27)

$$ \left[{\nabla}^2-{\delta}_1\left(\frac{\partial }{\partial t}+{\delta}_2\right)\right]N+{\varepsilon}_3\left\{\varphi -a{\nabla}^2\varphi \right\}=0 $$

(28)

$$ \left\{\left(1+\frac{{\left({\tau}_t\right)}^{\alpha }}{\alpha !}\frac{\partial^{\alpha }}{\partial {t}^{\alpha }}\right)\frac{\partial }{\partial t}+\frac{K^{\ast }}{K{\omega}^{\ast }}\left(1+\frac{{\left({\tau}_v\right)}^{\alpha }}{\alpha !}\frac{\partial^{\alpha }}{\partial {t}^{\alpha }}\right)\right\}{\nabla}^2\varphi +{\varepsilon}_2\frac{\partial N}{\partial t}=\left(1+\frac{{\left({\tau}_q\right)}^{\alpha }}{\alpha !}\frac{\partial^{\alpha }}{\partial {t}^{\alpha }}+\frac{{\left({\tau}_q\right)}^{2\alpha }}{2\alpha !}\frac{\partial^{2\alpha }}{\partial {t}^{2\alpha }}\right)\left[\frac{\partial^2}{\partial {t}^2}\left\{\varphi -a{\nabla}^2\varphi \right\}+{\varepsilon}_1\frac{\partial^2}{{\partial t}^2}{\nabla}^2\phi \right], $$

(29)

where

$$ {\nabla}^2\equiv \frac{\partial^2}{{\partial x}^2}+\frac{\partial^2}{{\partial z}^2}. $$

The stress displacement carrier density relations becomes

$$ {\sigma}_{\mathrm{xx}}\left(\mathrm{x},\mathrm{z},\mathrm{t}\right)=\left(\frac{\partial^2\phi }{\partial {x}^2}-\frac{\partial^2\psi }{\partial x\partial z}\right)+\left(1-2{\delta}^2\right)\left(\frac{\partial^2\phi }{\partial {z}^2}+\frac{\partial^2\psi }{\partial x\partial z}\right)-\left\{\varphi -\mathrm{a}\left(\frac{\partial^2\varphi }{{\partial x}^2}+\frac{\partial^2\varphi }{{\partial z}^2}\right)\right\}-N, $$

(30)

$$ {\sigma}_{\mathrm{zz}}\left(\mathrm{x},\mathrm{z},\mathrm{t}\right)=\left(1-2{\delta}^2\right)\left(\frac{\partial^2\phi }{\partial {x}^2}-\frac{\partial^2\psi }{\partial x\partial z}\right)+\left(\frac{\partial^2\phi }{\partial {z}^2}+\frac{\partial^2\psi }{\partial x\partial z}\right)-\left\{\varphi -\mathrm{a}\left(\frac{\partial^2\varphi }{{\partial x}^2}+\frac{\partial^2\varphi }{{\partial z}^2}\right)\right\}-N, $$

(31)

$$ {\sigma}_{\mathrm{xz}}\left(\mathrm{x},\mathrm{z},\mathrm{t}\right)={\delta}^2\left(2\frac{\partial^2\phi }{\partial x\partial z}-\frac{\partial^2\psi }{\partial {z}^2}+\frac{\partial^2\psi }{\partial {x}^2}\right), $$

(32)

Plane-wave propagation

Consider the plane wave solution of the Eqs. (26), (27), (28), (29) of the form

$$ \left(\begin{array}{c}\phi \\ {}\psi \\ {}\varphi \\ {}N\end{array}\right)=\left(\begin{array}{c}\overline{\phi}\\ {}\overline{\psi}\\ {}\overline{\varphi}\\ {}\overline{N}\end{array}\right){e}^{\left( i\xi \Big(x\ sin\theta +z\ cos\theta \right)- i\omega t\Big)}, $$

(33)

where sinθ, cosθ indicates the projection of wave normal to the x − z plane, ω represents angular frequency and ξ denotes the wavenumber of a plane wave propagating in x − z plane and $\overline{\phi},\overline{\psi},\overline{\varphi},\overline{N}$ are the constants to be determined .

Using Eq. (33) in Eqs. (26), (27), (28), (29) yields

$$ \left[{\zeta}_2+{\zeta}_4{\xi}^2\right]\overline{\phi}+\left[{\zeta}_1+{\zeta}_3{\xi}^2\right]\overline{\psi}+\left[1+a{\xi}^2\right]\overline{\varphi}+\overline{N}=0, $$

(34)

$$ \left[{\zeta}_6{\xi}^2+{\zeta}_5\right]\overline{\phi}+\left[{\zeta}_8{\xi}^2+{\zeta}_7\right]\overline{\psi}-=0, $$

(35)

$$ {\varepsilon}_3\left[1+a{\xi}^2\right]\overline{\varphi}+\left[{\zeta}_9-{\xi}^2\right]\overline{N}=0, $$

(36)

$$ {\zeta}_{13}{\xi}^2\overline{\phi}+\left[{\zeta}_{12}{\xi}^2+{\zeta}_{10}\right]\overline{\varphi}+{\varepsilon}_2 i\omega \overline{N}=0. $$

(37)

And the stress-strain relations can be written as

$$ {\sigma}_{xx}=\left[-{\xi}^2\left({\mathit{\sin}}^2\theta +\left(1-2{\delta}^2\right){\mathit{\cos}}^2\theta \right)\overline{\phi}+{\xi}^2\left(-2{\delta}^2\right)\frac{\sin 2\theta }{2}\overline{\psi}-\left[1+a{\xi}^2\right]\overline{\varphi}-\overline{N}\right]{e}^{\left( i\xi \Big(x\ sin\theta +z\ cos\theta \right)- i\omega t\Big)} $$

(38)

$$ {\sigma}_{\mathrm{zz}}=\left[-{\xi}^2\left(\left(1-2{\delta}^2\right){\mathit{\sin}}^2\theta +{\mathit{\cos}}^2\theta \right)\overline{\phi}+{\xi}^2\left(-2{\delta}^2\right)\frac{\sin 2\theta }{2}\overline{\psi}-\left[1+a{\xi}^2\right]\overline{\varphi}-\overline{N}\right]{e}^{\left( i\xi \Big(x\ sin\theta +z\ cos\theta \right)- i\omega t\Big)}, $$

(39)

$$ {\sigma}_{\mathrm{xz}}\left(\mathrm{x},\mathrm{z},\mathrm{t}\right)=-{\xi}^2{\delta}^2\left(\sin 2\theta \overline{\phi}+\cos 2\theta \overline{\psi}\right){e}^{\left( i\xi \Big(x\ sin\theta +z\ cos\theta \right)- i\omega t\Big)}, $$

(40)

Where, ${\zeta}_1=\left(\frac{i\omega M}{1+{m}^2}+{\omega}^2+{\Omega}^2\right),$

$$ {\zeta}_2=-\left(\frac{Mmi\omega}{1+{m}^2}+2\omega \Omega i\right), $$

$$ {\zeta}_3={\zeta}_1{\epsilon}^2-{\delta}^2, $$

$$ {\zeta}_4={\zeta}_2{\epsilon}^2, $$

$$ {\zeta}_5=\frac{Mi\omega}{1+{m}^2}+{\omega}^2+{\Omega}^2, $$

$$ {\zeta}_6={\zeta}_5{\epsilon}^2-1, $$

$$ {\zeta}_7=\frac{Mmi\omega}{1+{m}^2}+2\omega \Omega i, $$

$$ {\zeta}_8={\zeta}_7{\epsilon}^2, $$

$$ {\zeta}_9={\delta}_1\left( i\omega -{\delta}_2\right), $$

$$ {\zeta}_{10}=\left[1+\frac{\tau_q^{\alpha }}{\alpha !}{\left(- i\omega \right)}^{\alpha }+\frac{\tau_q^{2\alpha }}{2\alpha !}{\left(- i\omega \right)}^{2\alpha}\right]{\omega}^2, $$

$$ {\zeta}_{11}=\left[1+\frac{\tau_T^{\alpha }}{\alpha !}{\left(- i\omega \right)}^{\alpha}\right] i\omega -\frac{K^{\ast }}{K{\omega}^{\ast }}\left[1+\frac{\tau_v^{\alpha }}{\alpha !}{\left(- i\omega \right)}^{\alpha}\right], $$

ζ₁₃ =  − ζ₁₀ε₁,

$$ {\zeta}_{12}=\left({\zeta}_{11}-{\zeta}_{10}a\right). $$

Eliminating $\overline{\phi},\overline{\psi},\overline{\varphi}$ and $\overline{N}$ from the Eqs. (34), (35), (36), (37) yields the characteristic equation as

$$ A{\xi}^8+B{\xi}^6+C{\xi}^4+D{\xi}^2+E=0, $$

(41)

where

A =  − ζ₁₃ζ₉ζ₈a + ζ₁₂ζ₄ζ₈ − ζ₃ζ₆ζ₁₂,

B =  − aε₃ζ₁₃ζ₈ + ε₂iωε₃ζ₄ζ₈a − aε₂iωε₃ζ₆ζ₃ − ζ₁₃ζ₉ζ₇a + ζ₁₃ζ₉ζ₈a − ζ₁₃ζ₉ζ₈a + ζ₁₂ζ₂ζ₈ − ζ₁ζ₆ζ₁₂ − ζ₁₂ζ₉ζ₄ζ₈ + ζ₁₀ζ₄ζ₈ + ζ₁₂ζ₉ζ₆ζ₃ − ζ₃ζ₆ζ₁₀ + ζ₁₂ζ₄ζ₇ − ζ₁₂ζ₅ζ₃,

C = ζ₁₃ζ₉ζ₈ + ζ₁₃ζ₉ζ₇a − ζ₁₃ζ₉ζ₇ − ε₃ζ₁₃ζ₈ − ε₃ζ₁₃ζ₇ + (ε₂ε₃ζ₄ζ₈ − ε₂ε₃ζ₆ζ₃)iω + (ε₂ε₃ζ₂ζ₈ + ε₂ε₃ζ₄ζ₇ − ε₂ε₃ζ₆ζ₁ − ε₂ε₃ζ₅ζ₃)aiω − ζ₁₂ζ₉ζ₂ζ₈ + ζ₁₀ζ₂ζ₈ + ζ₁₂ζ₆ζ₁ζ₉ − ζ₁₀ζ₁ζ₆ − ζ₁₀ζ₉ζ₄ζ₈ − ζ₁₂ζ₅ζ₁ + ζ₁₂ζ₂ζ₇ − ζ₁₂ζ₄ζ₇ζ₉ + ζ₁₀ζ₄ζ₇ + ζ₁₂ζ₉ζ₅ζ₃ − ζ₁₀ζ₅ζ₃,

$$ D={\zeta}_{13}{\zeta}_9{\zeta}_7-{\varepsilon}_3{\zeta}_{13}{\zeta}_7+ i\omega \left({\varepsilon}_2{\varepsilon}_3{\zeta}_2{\zeta}_8+{\varepsilon}_2{\varepsilon}_3{\zeta}_4{\zeta}_7-{\varepsilon}_2{\varepsilon}_3{\zeta}_6{\zeta}_1-{\varepsilon}_2{\varepsilon}_3{\zeta}_5{\zeta}_3+a{\varepsilon}_2{\varepsilon}_3{\zeta}_2{\zeta}_7-a{\varepsilon}_2{\varepsilon}_3{\zeta}_5{\zeta}_1\right)-{\zeta}_{10}{\zeta}_9{\zeta}_2{\zeta}_8+{\zeta}_{10}{\zeta}_9{\zeta}_1{\zeta}_6+{\zeta}_{12}{\zeta}_9{\zeta}_5{\zeta}_1-{\zeta}_{10}{\zeta}_1{\zeta}_5+{\zeta}_{10}{\zeta}_7{\zeta}_2-{\zeta}_{10}{\zeta}_9{\zeta}_4{\zeta}_7+{\zeta}_{10}{\zeta}_9{\zeta}_3{\zeta}_5 $$

E = iω(ε₂ε₃ζ₂ζ₇ − ε₂ε₃ζ₅ζ₁) + ζ₁₀ζ₉ζ₅ζ₁ − ζ₁₂ζ₉ζ₂ζ₇ − ζ₁₀ζ₉ζ₂ζ₇ + ζ₁₀ζ₉ζ₃ζ₆.

The solution of Eq. (41) give eight roots in ξ that is, ±ξ₁, ± ξ₂, ± ξ₃, ± ξ₄, and we are concerned with the positive imaginary parts of the roots. When a coupled longitudinal wave falls on the boundary z = 0, four reflected waves are generated. It exhibits that the generated waves are coupled in nature. Corresponding to positive four roots and descending order of their velocities, four coupled waves are transmitted, specifically CLD wave related with ϕ transmitting with the maximum speed V₁, CT-wave linked with the φ having speed V₂ and CCD-wave related with N having speed V₃ and CTD-wave linked with the vector potential ψ transmitting with the lowest speed V₄. Following Lata et al. (2021), the characteristics properties of these waves are obtained by the following expressions

• (i) Phase velocity

The phase velocities of the plane wave is represented as

$$ {V}_j=\frac{\omega }{\mathit{\operatorname{Re}}\left({\xi}_j\right)},\mathrm{j}=1,2,3,4 $$

• (ii) Attenuation coefficient

The attenuation coefficient of the plane wave is represented as

$$ {Q}_j= Img\left({\xi}_j\right),\mathrm{j}=1,2,3,4. $$

• (iii) Specific loss

The specific loss of the plane wave is represented as:

$$ {W}_j={\left(\frac{\Delta W}{W}\right)}_j=4\pi \left|\frac{Img\left({\xi}_j\right)}{\mathit{\operatorname{Re}}\left({\xi}_j\right)}\right|,\mathrm{j}=1,2,3,4. $$

ΔW is the energy dissipated and W is the energy stored.

• (iv) Penetration depth

The penetration depth is given by

$$ {S}_j=\frac{1}{Img\left({\xi}_j\right)},\mathrm{j}=1,2,3,4. $$

Particular cases

• 1. For nonlocal semiconductor medium with rotation, Hall current, and two temperatures ( m, a, α, Ω, ξ) > 0, from the above relations, the following cases can also be obtained

  • i. Three-phase lag FOT (TPL-FOT)

    If τ_q > τ_T > τ_v ≥ 0.

  • ii. Dual-phase lag FOT (DPL-FOT)

    If ${\tau}_v=0,{K}_{ij}^{\ast }=0$ τ_q > τ_T ≥ 0.

  • iii. Single-phase lag FOT (SPL-FOT) or Lord–Shulman MDD

    If τ_T = 0, τ_v = 0, τ_q = τ₀ > 0 and ${K}_{ij}^{\ast }=0$, and ignoring ${\tau}_q^2$.

  • iv. Three-phase lag (TPL)

    If ${\tau}_q>{\tau}_T>{\tau}_v\ge 0,\alpha =1,{G}_{\tau_q}={G}_{\tau_v}={G}_{\tau_T}= i\omega$

  • v. Dual-phase lag (DPL)

    If ${\tau}_v=0,{K}_{ij}^{\ast }=0$ α = 1,and${\tau}_q>{\tau}_T\ge 0,{G}_{\tau_q}={G}_{\tau_T}= i\omega$

  • vi. Single-phase lag (SPL) or Lord–Shulman model

    If τ_T = 0, τ_v = 0, τ_q = τ₀ > 0 α = 1,and ${K}_{ij}^{\ast }=0$, and ignoring ${\tau}_q^2$.${G}_{\tau_q}= i\omega$

  • vii. GN theory of Type-III

    If τ_T = 0, τ_v = 0, τ_q = 0 α = 1,and ${K}_{ij}^{\ast}\ne 0,{K}_{ij}\ne 0$

  • viii. GN theory of Type-II

    If τ_T = 0, τ_v = 0, τ_q = 0 α = 1, and K_ij = 0

  • ix. GN theory of Type-I

    If τ_T = 0, τ_v = 0, τ_q = 0 ${K}_{ij}^{\ast }=0,\alpha =1.$

• 2. For local semiconductor medium ϵ = 0, for all the above a − j cases

• 3. For semiconductor medium without rotation, Ω = 0, for all the above a − j cases

• 4. For semiconductor medium without Hall current m=0, for all the above a − j cases

• 5. For semiconductor medium without two temperatures a = 0, for all the above a − j cases

• 6. For semiconductor medium without FOT α = 0, for all the above a − j cases

Numerical results and discussion

To demonstrate the theoretical results and effect of Hall current, fractional order parameter, two temperatures, and non-local parameter, the physical data for semiconducting medium taken from Mahdy et al. (2020) is given as

$$ \lambda =3.64\times {10}^{10}\ N{m}^{-2},\kern0.75em \mu =5.46\times {10}^{10}\ N{m}^{-2},\kern0.75em \beta =7.04\times {10}^6N{m}^{-2}{\mathit{\deg}}^{-1},\kern0.75em {d}_n=-9\times {10}^{-31}\ {m}^{-3},\kern1em \rho =2.33\times {10}^3K{gm}^{-3},\kern0.5em {C}_E=695\ JK{g}^{-1}{K}^{-1},K=150\ W{m}^{-1}{K}^{-1},{K}^{\ast }=1.54\times {10}^2 Ws,\kern0.5em {\mathrm{T}}_0=800\ \mathrm{K},{\tau}_T=1\times {10}^{-7}{\tau}_v=2\times {10}^{-8}s,{\tau}_q=2\times {10}^{-7}\mathrm{s},{D}_E=2.5\times {10}^{-3}\ {m}^2{s}^{-1},{\mathrm{H}}_0=1\ \mathrm{J}{\mathrm{m}}^{-1}\mathrm{n}{\mathrm{b}}^{-1},\uptau =5\times {10}^{-5}\ \mathrm{s},\kern0.75em {N}_0={10}^{20}{m}^{-3},{s}_0=2\ m{s}^{-1},{\upvarepsilon}_0=8.838\times {10}^{-12}\mathrm{F}{\mathrm{m}}^{-1},{E}_g=1.11 eV,{\alpha}_T=3\times {10}^{-6}{K}^{-1}. $$

Figures 1, 2, 3, and 4 indicate the change of phase velocities w.r.t. frequency ω respectively. Figure 1 illustrates the change in phase velocity with the change in fractional order heat transfer parameter α. Figure 2 illustrates the change in phase velocity with the change in Hall current parameter m. As the Hall current increases, phase velocity decreases. Figure 3 illustrates the change in phase velocity with the change in two-temperature parameter a. The higher the value of two temperatures, the lower is the phase velocity of the plane wave.

Fig. 1

Phase velocity w.r.t ω and fractional order heat transfer parameter α

Fig. 2

Phase velocity w.r.t ω and Hall current parameter m

Fig. 3

Phase velocity w.r.t ω and two-temperature parameter a

Fig. 4

Phase velocity w.r.t ω and non-local parameter ϵ

Figure 4 illustrates the change in phase velocity with the change in non-local parameter ϵ. The higher the value of ϵ, the lower is the phase velocity of plane wave.

Figures 5, 6, 7, and 8 indicate the change of attenuation coefficients w.r.t. frequency ω respectively. Figure 5 illustrates the change in attenuation coefficients with the change in fractional order heat transfer parameter α. For the initial value of the frequency, attenuation coefficients decrease sharply. The higher the value of α, the higher is the attenuation coefficients. Figure 6 illustrates the change in attenuation coefficients with the change in Hall current parameter m. For the initial value of the frequency, attenuation coefficients decrease sharply. However, As the Hall current increases, attenuation coefficients decrease. Figure 7 illustrates the change in attenuation coefficients with the change in two-temperature parameter a. The higher the value of two temperature, the lower is the attenuation coefficients of a plane wave. Figure 8 illustrates the change in attenuation coefficients with the change in non-local parameter ϵ . The higher the value of parameter ϵ, the lower is the attenuation coefficients of a plane wave.

Fig. 5

Attenuation coefficient w.r.t ω and fractional order heat transfer parameter α

Fig. 6

Attenuation coefficient w.r.t ω and Hall current parameter m

Fig. 7

Attenuation coefficient w.r.t ω and two temperature parameter a

Fig. 8

Attenuation coefficient w.r.t ω and non-local parameter ϵ

Figures 9, 10, 11, and 12 indicate the change of specific loss w.r.t. frequency ω respectively. Figure 9 illustrates the change in specific loss with the change in fractional order heat transfer parameter α. The higher the value of α, the higher is the specific loss. Figure 10 illustrates the change in specific loss with the change in Hall current parameter m. As the Hall current increases, specific loss decreases. Figure 11 illustrates the change in specific loss with the change in two-temperature parameter a. The higher the value of two temperatures, the lower is the specific loss of plane wave. Figure 12 illustrates the change in specific loss with the change in non-local parameter ϵ. The higher the value of ϵ, the lower is the specific loss of plane wave.

Fig. 9

Specific loss w.r.t ω and fractional order heat transfer parameter α

Fig. 10

Specific loss w.r.t ω and Hall current parameter m

Fig. 11

Specific loss w.r.t ω and two temperature parameter a

Fig. 12

Specific loss w.r.t ω and non-local parameter ϵ

Figures 13, 14, 15, and 16 indicate the change of penetration depth w.r.t. frequency ω respectively. Figure 13 illustrates the change in penetration depth with the change in fractional order heat transfer parameter α The higher the value of α, the higher is the penetration depth. Figure 14 illustrates the change in penetration depth with the change in Hall current parameter m. For the initial value of the frequency, penetration depth increases sharply, and after half range of frequency, it decreases. However, the higher the value of Hall current increases, the higher is the penetration depth. Figure 15 illustrates the change in penetration depth with the change in two temperature parameter a. The higher the value of two temperature, the lower is the penetration depth of plane wave. Figure 16 illustrates the change in penetration depth with the change in non-local parameter ϵ. The higher the value of non-local parameter ϵ, the lower is the penetration depth of plane wave.

Fig. 13

Penetration depth w.r.t ω and fractional order heat transfer parameter α

Fig. 14

Penetration depth w.r.t ω and Hall current parameter m

Fig. 15

Penetration depth w.r.t ω and two temperature parameter a

Fig. 16

Penetration depth w.r.t ω and non-local parameter ϵ

Conclusions

• In this study, the propagation of plane harmonic waves in magneto-thermoelastic rotating semiconducting medium has been studied.

• The semiconducting medium is rotating with angular frequency Ω and is under the influence of high magnetic field. The governing equations are modeled using the Hall current effect and fractional order three phase lag heat transfer with two temperature.

• The non-dimensional expressions for penetration depth, phase velocities, specific loss, and attenuation coefficients of various reflected waves are calculated and drawn graphically with the help of MATLAB software.

• Effect of fractional order heat transfer, Hall current, two-temperature, and non-local parameter ϵ on the penetration depth, phase velocities, specific loss, and attenuation coefficients of various reflected waves are represented graphically. The results exhibit that as the value of fractional order heat transfer parameter α increases, variations in the penetration depth, phase velocities, specific loss, and attenuation coefficients also increases. The higher the value of Hall current, the lower will be the penetration depth, phase velocities, specific loss, and attenuation coefficients of the plane wave. However, two-temperature parameters show different behavior with different characteristics of a plane wave.

• The non-local parameter ϵ has a significant effect on the penetration depth, phase velocities, specific loss, and attenuation coefficients of various reflected waves. The deviation in penetration depth, phase velocities, specific loss, and attenuation coefficients of various reflected waves is higher when ϵ = 0, as the value of ϵ increases, the variations in penetration depth, phase velocities, specific loss, and attenuation coefficients of various reflected waves decrease.

• The study may help in the design of semiconductor nano-devices, Hall effect sensors, magnetic switches, applications in the automotive world, geology, and seismology as well as semiconductor nanostructure devices such as MEMS/NEMS.

Nomenclature

δ_ij Kronecker delta

t₀ the pulse rise time

w lateral deflection of the beam

K_ij thermal conductivity

T₀ reference temperature

t_ij stress tensors

e_ij strain tensors

u_i displacement components

β_ij thermal elastic coupling tensor

ρ medium density

C_E specific heat

T temperature change

I moment of inertia of cross-section

t time

E_i intensity tensor of the electric field

m_e mass of the electron

t_e electron collision time

c_ijkl elastic parameters

M_T thermal moment of inertia

β₁M_T thermal moment of the beam

φ conductive temperature

μ₀ magnetic permeability

α_ij linear thermal expansion coefficient

a_ij two-temperature parameter

m Hall effect parameter$m={\omega}_e{t}_e=\frac{\sigma_0{\mu}_0{H}_0}{e{n}_e}$

λ_i pyromagnetic coefficient

τ_q phase lags of the heat flux

τ_T phase lags of the temperature gradient

J_j conduction current density tensor

ε_ilr permutation symbol

H_r magnetic strength

e charge of the electron

n_e electron number density

σ ₀ $\mathrm{electrical}\ \mathrm{conductivity}\ \mathrm{and}\kern0.5em ={\sigma}_0=\frac{n_e{e}^2{t}_e}{m_e}$

τ_v phase lags of the thermal displacement

${K}_{ij}^{\ast }$ materialistic constant

t_ij(x) non-local stress tensor

σ_ij(x) local stress tensor

ϵ nonlocal parameter

a internal characteristic length

e₀ constant characterizes the nonlocal effect of material

d_n coefficient of electronic deformation

α_T coefficient of linear thermal expansion

λ, μ Lame’s elastic constants

N carrier density

D_E carrier diffusion coefficients

τ photo-generated carrier lifetime

E_g energy gap of the semiconductor parameter

$\kappa =\frac{\partial {N}_0}{\partial T}$ coupling parameter for thermal activation

N₀ carrier concentration at equilibrium position

s₀ velocity of recombination on the surface

M Hartmann number or magnetic parameter for semiconductor elastic medium