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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 − ,ij,

β = (3λ + 2μ)αT,

δn = (3λ + 2μ)dn,

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 Jx and Jz 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 dn).

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 =  − 3ζ13ζ8 + ε2iωε3ζ4ζ8a − 2iωε3ζ6ζ3 − ζ13ζ9ζ7a + ζ13ζ9ζ8a − ζ13ζ9ζ8a + ζ12ζ2ζ8 − ζ1ζ6ζ12 − ζ12ζ9ζ4ζ8 + ζ10ζ4ζ8 + ζ12ζ9ζ6ζ3 − ζ3ζ6ζ10 + ζ12ζ4ζ7 − ζ12ζ5ζ3,

C = ζ13ζ9ζ8 + ζ13ζ9ζ7a − ζ13ζ9ζ7 − ε3ζ13ζ8 − ε3ζ13ζ7 + (ε2ε3ζ4ζ8 − ε2ε3ζ6ζ3) + (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 = (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

  1. (i)

    Phase velocity

The phase velocities of the plane wave is represented as

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

    Attenuation coefficient

The attenuation coefficient of the plane wave is represented as

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

  1. (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. 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

    1. i.

      Three-phase lag FOT (TPL-FOT)

      If τq > τT > τv ≥ 0.

    2. ii.

      Dual-phase lag FOT (DPL-FOT)

      If \( {\tau}_v=0,{K}_{ij}^{\ast }=0 \) τq > τT ≥ 0.

    3. iii.

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

      If τT = 0, τv = 0, τq = τ0 > 0 and \( {K}_{ij}^{\ast }=0 \), and ignoring \( {\tau}_q^2 \).

    4. iv.

      Three-phase lag (TPL)

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

    5. v.

      Dual-phase lag (DPL)

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

    6. vi.

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

      If τT = 0, τv = 0, τq = τ0 > 0 α = 1,and \( {K}_{ij}^{\ast }=0 \), and ignoring \( {\tau}_q^2 \).\( {G}_{\tau_q}= i\omega \)

    7. vii.

      GN theory of Type-III

      If τT = 0, τv = 0, τq = 0 α = 1,and \( {K}_{ij}^{\ast}\ne 0,{K}_{ij}\ne 0 \)

    8. viii.

      GN theory of Type-II

      If τT = 0, τv = 0, τq = 0 α = 1, and Kij = 0

    9. ix.

      GN theory of Type-I

      If τT = 0, τv = 0, τq = 0 \( {K}_{ij}^{\ast }=0,\alpha =1. \)

  2. 2.

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

  3. 3.

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

  4. 4.

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

  5. 5.

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

  6. 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
figure 1

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

Fig. 2
figure 2

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

Fig. 3
figure 3

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

Fig. 4
figure 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
figure 5

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

Fig. 6
figure 6

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

Fig. 7
figure 7

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

Fig. 8
figure 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
figure 9

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

Fig. 10
figure 10

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

Fig. 11
figure 11

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

Fig. 12
figure 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
figure 13

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

Fig. 14
figure 14

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

Fig. 15
figure 15

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

Fig. 16
figure 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

δij Kronecker delta

t0 the pulse rise time

w lateral deflection of the beam

Kij thermal conductivity

T0 reference temperature

tij stress tensors

eij strain tensors

ui displacement components

βij thermal elastic coupling tensor

ρ medium density

CE specific heat

T temperature change

I moment of inertia of cross-section

t time

Ei intensity tensor of the electric field

me mass of the electron

te electron collision time

cijkl elastic parameters

MT thermal moment of inertia

β1MT thermal moment of the beam

φ conductive temperature

μ0 magnetic permeability

αij linear thermal expansion coefficient

aij two-temperature parameter

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

λi pyromagnetic coefficient

τq phase lags of the heat flux

τT phase lags of the temperature gradient

Jj conduction current density tensor

εilr permutation symbol

Hr magnetic strength

e charge of the electron

ne electron number density

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

τv phase lags of the thermal displacement

\( {K}_{ij}^{\ast } \) materialistic constant

tij(x) non-local stress tensor

σij(x) local stress tensor

ϵ nonlocal parameter

a internal characteristic length

e0 constant characterizes the nonlocal effect of material

dn coefficient of electronic deformation

αT coefficient of linear thermal expansion

λ, μ Lame’s elastic constants

N carrier density

DE carrier diffusion coefficients

τ photo-generated carrier lifetime

Eg 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

Availability of data and materials

For the numerical results, cobalt material has been taken for thermoelastic material from Mahdy et al. (2020).

Abbreviations

TPL:

Three-phase lag

2T:

Two temperatures

FOT:

Fractional order theory

CLD:

Coupled-longitudinal displacement

CT:

Coupled thermal

CCD:

Coupled carrier density

CTD:

Coupled transverse displacement

TIT:

Transversely isotropic thermoelastic

SPL-FOT:

Single-phase lag FOT

GN:

Green-Naghdi

TPL-FOT:

Three-phase lag FOT

DPL-FOT:

Dual-phase lag FOT

2-D:

Two-dimensional

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Authors and Affiliations

Authors

Contributions

Iqbal Kaur: idea formulation, conceptualization, formulated strategies for mathematical modeling, methodology refinement, formal analysis, validation, writing—review and editing. Kulvinder Singh: conceptualization, effective literature review, experiments, simulation, investigation, methodology, software, supervision, validation, visualization, writing—original draft. Both authors read and approved the final manuscript.

Corresponding author

Correspondence to Iqbal Kaur.

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Kaur, I., Singh, K. Plane wave in non-local semiconducting rotating media with Hall effect and three-phase lag fractional order heat transfer. Int J Mech Mater Eng 16, 14 (2021). https://doi.org/10.1186/s40712-021-00137-3

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