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
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)