2. Temperature-dependent dispersion relation of terahertz surface magnetoplasmon

Assume that $x=0$ represents the interface between a rippled semiconductor (n-type) ($x\leq 0$) and free space ($x>0$) with the electron density perturbation $n = (n_0/2)(1+\cos q z)$ (Srivastav & Panwar Reference Srivastav and Panwar2023), where $q a\geq 1$, $a$ and $q$ are the amplitude and ripple wavenumber, respectively. Apply the magnetic field $B_{0}\hat {y}$ to the semiconductor (n-type) (cf. figure 1). The effective mass $m_{e}^{\ast }$ of free electrons inside a semiconductor (n-type) is $0.014 m_{e}$. The effective permittivities $\overline {\bar {\epsilon }}$ of the semiconductor (n-type) components are presented by $\varepsilon _{{\rm eff}} = (\epsilon _{x x}^{2} + \epsilon _{x z}^{2})/ (\epsilon _{x x})$, $\epsilon _{x x} = \epsilon _{z z} = \epsilon _r-(\epsilon _r{\omega _{p}^2 (\omega + {\rm i}\nu )})/\omega ((\omega + {\rm i}\nu )^2 - \omega ^2_{{\rm ce}})$, $\epsilon _{x z} =-\epsilon _{ z x} =-{\rm i} {\epsilon _r \omega _{{\rm ce}}\omega _{p}^2}/\omega ((\omega + {\rm i}\nu )^2 - \omega ^2_{{\rm ce}})$, $\omega _{{\rm ce}} = e B_{0}/m_{e}^{\ast }$ is the electron cyclotron frequency, e is electron charge, $\omega _{p}=\sqrt {(n_{0} e^{2})/(m_{e}^{\ast } \epsilon _{0}\epsilon _{r})}$ is the electron plasma frequency, $\nu$ is the collision frequency of electrons and the temperature-dependent electron density $n_{0}= 5.76\times 10^{20}\,\text {T}^{3/2}\exp (-0.26/2\text {k}_{\text {B}}\text {T})$ of n-InSb (Srivastav & Panwar Reference Srivastav and Panwar2023), where $\text {k}_{\text {B}}$ and $\text {T}$ are the Boltzmann constant and temperature, respectively.

The electric field of the SMP wave is given as

(2.1)\begin{equation} \boldsymbol{E} = A \exp({- {\rm i} (\omega t - k_z z)}) \begin{cases} (\hat{z} + \beta_{1} \hat{x}) \exp({- \alpha_{1} x}), & \text{for free space}\ x > 0 \\ (\hat{z} + \beta_{2} \hat{x}) \exp({\alpha_{2} x}), & \text{for n-InSb}\ x \leq 0, \end{cases}\end{equation}

where $A=A_{0}(1+\mu \cos \varOmega (t-z/v_{g}))$, $\beta _{1} = -({\rm i} k_{z}/ \alpha _{1})$, $\beta _{2} = ( {\epsilon _{x z} \alpha _{2}+ \epsilon _{x x} {\rm i} k_{z}}) /( -\epsilon _{x x} \alpha _{2} + \epsilon _{x z} {\rm i} k_{z} )$, $\alpha _{2}^2 = k_z^2 -(\omega ^2/c^2) (({\epsilon ^2_{x z} + \epsilon ^2_{x x}})/{\epsilon _{x x}})$, $\alpha _{1}^2 = k_z^2 -(\omega ^2/c^2)$, $\epsilon _{{\rm eff}} = ({\epsilon ^2_{x z} + \epsilon ^2_{x x}})/{\epsilon _{x x}}$ and $v_{g}$ is the group velocity of the SMP wave.

It adheres to the dispersion relation (Srivastav & Panwar Reference Srivastav and Panwar2023) as

(2.2)\begin{equation} \alpha_{2}=\alpha_{1} \epsilon_{{\rm eff}}+\frac{{\rm i} k_{z} \epsilon_{xz}}{\epsilon_{xx}}.\end{equation}

Srivastav & Panwar (Reference Srivastav and Panwar2022, Reference Srivastav and Panwar2023) reported the dispersion curve of SMPs for different values of the normalized electron cyclotron frequency $\omega _{{\rm ce}}/\omega _{p}$. It shows that the dispersion curve of SMPs shifts away from the dispersion curve of electromagnetic waves in free space for increases in the normalized electron cyclotron frequency.

The dispersion curve of SMPs and electromagnetic waves is shown in figure 2 for different semiconductor temperatures $T=310$ K (orange dot dashed line), $T=300$ K (magenta dotted line), $T=290$ K (red dashed line) and $T=280$ K (green double dot dashed line) at the normalized electron cyclotron frequency $\omega _{{\rm ce}}/\omega _{p} = 0.05$. Dispersion curve of SMPs shifts towards a lower normalized THz frequency as the semiconductor (n-InSb) temperature increases. The cutoff frequency of the dispersion curve also shifts to a lower normalized THz frequency with increases in the semiconductor's temperature.

Figure 2.

The $\omega /\omega _{p}$ variation with $k_{z}c/\omega _{p}$ for temperature $T=310$ K (orange dot dashed line), $T=300$ K (magenta dotted line), $T=290$ K (red dashed line) and $T=280$ K (green double dot dashed line) at normalized electron cyclotron frequency $\omega _{{\rm ce}}/\omega _{p} = 0.05$.

3. Linear response of terahertz surface magnetoplasmons

The SMP wave imparts a linear oscillatory velocity to the free electrons of the semiconductors as

(3.1)\begin{equation} v_{\omega}^x = \frac{e}{m_e^{{\ast}}} \tilde{v}_{\omega}^x A \exp({{\alpha }_2 x}) \exp({- {\rm i} (\omega t - k_z z)}) ,\end{equation}

and

(3.2)\begin{equation} v_{\omega}^z = \frac{e}{m_e^{{\ast}}} \tilde{v}_{\omega}^z A \exp({{\alpha }_2 x}) \exp({- {\rm i} (\omega t - k_z z)}) ,\end{equation}

where $\tilde {v}_{\omega }^x = ( \omega _{{\rm ce}} - {\rm i} (\omega + {\rm i} \nu ) {\beta _2} ) / ((\omega + {\rm i} \nu )^2 - \omega _{{\rm ce}}^2)$ and $\tilde {v}_{\omega }^z = - ( \omega _{{\rm ce}} {\beta _2} + {\rm i} (\omega + {\rm i} \nu ) ) / ((\omega + {\rm i} \nu )^2 - \omega _{{\rm ce}}^2)$.

The SMP wave exerts a ponderomotive force $\boldsymbol {F}_p = - (m_e^{\ast }/4) (\boldsymbol {v}_s \boldsymbol {\cdot } \boldsymbol {\nabla } \boldsymbol {v}_s^{\ast } + \boldsymbol {v}_s^{\ast } \boldsymbol {\cdot } \boldsymbol {\nabla } \boldsymbol {v}_s) - (e/4) (\boldsymbol {v}_s \times \boldsymbol {B}_s^{\ast } + \boldsymbol {v}_s^{\ast } \times \boldsymbol {B}_s)$ on the free electrons and it is represented as

(3.3)\begin{equation} \boldsymbol{F}_p = \frac{1}{4} \frac{e^2}{m_e^{{\ast}}} A^2 \exp({(\alpha_2 + \alpha^{{\ast}}_2) x}) (- F_{P x} \hat{x} + F_{P z} \hat{z}),\end{equation}

where

(3.4)\begin{align} F_{P x} & = (\tilde{v}_{\omega}^x \alpha^{{\ast}}_2 \tilde{v}^{{\ast} x}_{\omega} - \tilde{v}_{\omega}^z {\rm i} k_z \tilde{v}^{{\ast} x}_{\omega}) + ( {{\tilde{v}^{{\ast} x}_{\omega}} } \alpha_2 \tilde{v}_{\omega}^x {{+ \tilde{v}^{{\ast} z}_{\omega}} } {\rm i} k_z \tilde{v}_{\omega}^x) \nonumber\\ & \quad + \frac{1}{{\rm i} \omega} (\tilde{v}_{\omega}^{{\ast} z} (\alpha_2 - {\rm i} k_z {\beta_2}) + \tilde{v}_{\omega}^z (\alpha^{{\ast}}_2 + {\rm i} k_z \beta_2^{{\ast}})), \end{align}

and

(3.5)\begin{align} F_{P z} & = \frac{1}{{\rm i} \omega} (\tilde{v}_{\omega}^{{\ast} x} (\alpha_2 - {\rm i} k_z {\beta_2}) + \tilde{v}_{\omega}^x (\alpha^{{\ast}}_2 + {\rm i} k_z \beta_2^{{\ast}})) - ((\tilde{v}_{\omega}^x \alpha^{{\ast}}_2 \tilde{v}^{{\ast} z}_{\omega} \nonumber\\ & \quad - \tilde{v}_{\omega}^z {\rm i} k_z \tilde{v}^{{\ast} z}_{\omega}) + ({{\tilde{v}^{{\ast} x}_{\omega}} } \alpha_2 \tilde{v}_{\omega}^z {{+ \tilde{v}^{{\ast} z}_{\omega}} } {\rm i} k_z \tilde{v}_{\omega}^z)). \end{align}

The ponderomotive force at modulation frequency $\varOmega$ can be written as

(3.6)\begin{equation} F_{p \varOmega}^x ={-} \frac{e^2}{m_e^{{\ast}}} A_0^2 \mu F_{P x} \exp({(\alpha_2 + \alpha^{{\ast}}_2) x}) {\exp\left({- {\rm i} \varOmega \left(t - \frac{z}{v_g} \right)}\right)}.\end{equation}

Now, the oscillatory velocity components of the free electrons at modulation frequency $\varOmega$ are given as

(3.7)\begin{equation} v_{\varOmega}^x = \tilde{v}_{\varOmega}^x \frac{e^2}{(m^{{\ast}}_e)^{2}} A_0^2 \mu \exp({(\alpha_2 + \alpha^{{\ast}}_2) x}) {\exp\left({- {\rm i} \varOmega \left( t - \frac{z}{v_g} \right)}\right)},\end{equation}

and

(3.8)\begin{equation} v_{\varOmega}^z = \tilde{v}_{\varOmega}^z \frac{e^2}{(m^{{\ast}}_e)^{2}} A_0^2 \mu \exp({(\alpha_2 + \alpha^{{\ast}}_2) x}) {\exp\left({- {\rm i} \varOmega \left( t - \frac{z}{v_g} \right)}\right)},\end{equation}

where $\tilde {v}_{\varOmega }^x = ({\rm i} (\varOmega + {\rm i} \nu ) F_{P x}) / ((\varOmega + {\rm i} \nu )^2 - \omega _{{\rm ce}}^2)$ and $\tilde {v}_{\varOmega }^z = (\omega _{{\rm ce}} F_{P x}) / ((\varOmega + {\rm i} \nu )^2 - \omega _{{\rm ce}}^2)$.

The oscillatory velocity couples with the modulated electron density to produce current density $\boldsymbol {J}_{\varOmega }$ at modulation frequency $\varOmega$ and it is given as

(3.9)\begin{align} \boldsymbol{J}_{\varOmega} & ={-} \frac{1}{2} n_0 (- \tilde{v}_{\varOmega}^x \hat{x} + \tilde{v}_{\varOmega}^z \hat{z}) \frac{e^3}{(m^{{\ast}}_e)^{2}} A_0^2 \mu h \delta (x) \exp({(\alpha_2 + \alpha^{{\ast}}_2) x}) \nonumber\\ & \quad \times\exp({- {\rm i} \varOmega t}) \exp\left({{\rm i} \left( \frac{\varOmega}{v_g} + q \right) z}\right). \end{align}

4. Excitation process of terahertz surface magnetoplasmon field

By using Maxwell's equation, the $\varOmega$ modulation frequency SMP wave field can be expressed as

(4.1)\begin{equation} \boldsymbol{\nabla} \times \boldsymbol{E}_{s} = {\rm i} \varOmega\boldsymbol{B}_{s},\end{equation}

and

(4.2)\begin{equation} \boldsymbol{\nabla} \times \boldsymbol{B}_{s} = \boldsymbol{J}_{\varOmega} - {\rm i} \varOmega \epsilon_0 \epsilon_{{\rm eff}}^{\varOmega} \boldsymbol{E}_{s},\end{equation}

where $\varepsilon _{{\rm eff}}^{\varOmega } = ((\epsilon _{x x}^{\varOmega })^{2} + (\epsilon _{x z}^{\varOmega })^{2})/ (\epsilon _{x x}^{\varOmega })$, $\epsilon _{x x}^{\varOmega } = \epsilon _{z z}^{\varOmega } = \epsilon _r-(\epsilon _r{\omega _{p}^2 (\varOmega + {\rm i} \nu )})/\varOmega ( (\varOmega + {\rm i}\nu )^2 - \omega ^2_{{\rm ce}} )$, $\epsilon _{ x z}^{\varOmega } =-\epsilon _{ z x}^{\varOmega } =-{\rm i} {\epsilon _r \omega _{{\rm ce}}\omega _{p}^2}/\varOmega ((\varOmega + {\rm i}\nu )^2 - \omega ^2_{{\rm ce}}) \text { for}\ x \leq 0$ and $\varepsilon _{{\rm eff}}^{\varOmega } = 1 \text { for}\ x > 0$, and $s$ represents the SMPs.

The wave equation may be written as

(4.3)\begin{equation} \nabla^2 \boldsymbol{E}_{s} - \boldsymbol{\nabla} (\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{E}_{s}) - \frac{\omega^2}{c^2} \left(\overline{\bar{\epsilon}} \boldsymbol{E}_{s} \right) ={-}\mu_{0} {\rm i} \varOmega \boldsymbol J_{\varOmega}.\end{equation}

Here, $\overline {\bar {\epsilon }}$ is an n-type semiconductor effective permittivity at frequency $\varOmega$. When the right-hand side term of (4.2) is zero, that is $\boldsymbol {J}_{\varOmega }$ is absent, then the SMP eigenmode structure with

(4.4)\begin{equation} \boldsymbol{E}_{s} = \boldsymbol{E}_{s,0}, \boldsymbol{B}_{s} = \boldsymbol{B}_{s,0},\end{equation}

where $\boldsymbol {E}_{s,0}$ and $\boldsymbol {B}_{s,0}$ is given as

(4.5)\begin{equation} \boldsymbol{\nabla} \times \boldsymbol{E}_{s,0} = {\rm i} \varOmega\boldsymbol{B}_{s,0},\end{equation}

and

(4.6)\begin{equation} \boldsymbol{\nabla} \times \boldsymbol{B}_{s,0} ={-} {\rm i}\varOmega \epsilon_0 \epsilon_{{\rm eff}}^{\varOmega} \boldsymbol{E}_{s,0}.\end{equation}

The SMP electric field can be represented as

(4.7)\begin{equation} \boldsymbol{E}_{s,0} = \exp({- {\rm i} (\varOmega t - k_z^{\varOmega} z)}) \begin{cases} (\hat{z} + \beta_{1}^{\varOmega} \hat{x}) \exp({- \alpha_{1}^{\varOmega} x}), & \text{for free space}\ x > 0 \\ (\hat{z} + \beta_{2}^{\varOmega} \hat{x}) \exp({\alpha_{2}^{\varOmega} x}), & \text{for n-InSb}\ x \leq 0 \end{cases},\end{equation}

where $\beta _{1}^{\varOmega } \!=\! -({\rm i} k_{z}^{\varOmega }/ \alpha _{1}^{\varOmega })$, $\beta _{2}^{\varOmega } \!=\! ( {\epsilon _{x z}^{\varOmega } \alpha _{2}^{\varOmega }+ \epsilon _{x x}^{\varOmega } {\rm i} k_{z}^{\varOmega }})/({ -\epsilon _{x x}^{\varOmega } \alpha _{2}^{\varOmega } + \epsilon _{x z}^{\varOmega } {\rm i} k_{z}^{\varOmega } })$, $(\alpha _{2}k_z^{\varOmega })^2 = (k_z^{\varOmega })^2 -(\varOmega ^2/c^2) \epsilon _{{\rm eff}}^{\varOmega }$, $(\alpha _{1}^{\varOmega })^2 = (k_z^{\varOmega })^2 -(\varOmega ^2/c^2)$, $k_{z}^{\varOmega } = q + \varOmega /v_{g}$ and $q$ provides the extra wave-number for the phase matching condition.

Figure 3 represents the variation of normalized ripple wavenumber $qc/\omega _{p}$ with normalized electron cyclotron frequency $\omega _{{\rm ce}}/\omega _{p}$ for the three values of THz frequency $\omega = 0.5$ THz (green line), $\omega = 1.0$ THz (blue dashed line) and $\omega = 1.5$ THz (red dot dashed line). Normalized ripple wavenumber $qc/\omega _{p}$ increases with an increase in the normalized electron cyclotron frequency $\omega _{{\rm ce}}/\omega _{p}$ at $\omega _{p} = 2.7$ THz. Also, it increases with a decrease in the THz frequency.

Figure 3.

Variation of normalized ripple wavenumber $q$ with normalized electron cyclotron frequency $\omega _{{\rm ce}}/\omega _{p}$ for THz frequency $\omega = 0.5$ THz (green line), $\omega = 1.0$ THz (blue dashed line) and $\omega = 1.5$ THz (red dot dashed line).

If the current density $\boldsymbol {J}_{\varOmega }$ is present, then the electric and magnetic fields of the THz SMP wave can be expressed as

(4.8a,b)\begin{equation} \boldsymbol{E}_{s} = a(t) \boldsymbol{E}_{s,0},\quad \boldsymbol{B}_{s} = b(t) \boldsymbol{B}_{s,0}.\end{equation}

Solving (4.1) and (4.2) with (4.8a,b), and using (4.4), we get

(4.9)\begin{equation} (a - b) {\rm i} \varOmega \epsilon_0 \epsilon_{{\rm eff}}^{\varOmega} {\boldsymbol{E}_{s,0}} = \boldsymbol{J}_{\varOmega} + {\boldsymbol{E}_{s,0}} \frac{\partial a }{\partial t} \epsilon_0 \epsilon_{{\rm eff}},\end{equation}

and

(4.10)\begin{equation} b - a = \frac{1}{{\rm i} \varOmega} \frac{\partial b (t)}{\partial t}.\end{equation}

Using (4.10) in (4.9) and taking $\partial a /\partial t \approx \partial b /\partial t$, we get

(4.11)\begin{equation} \frac{\partial a}{\partial t} {\boldsymbol{E}_{s,0}} ={-} \frac{1}{2 \epsilon_0 \epsilon_{{\rm eff}}^{\varOmega}} \boldsymbol{J}_{\varOmega}.\end{equation}

Multiplying (4.11) by $\boldsymbol {E}_{s,0}^{\ast }\,{{\rm d}x}$ and integrating $- \infty$ to $\infty$, we obtain

(4.12)\begin{equation} \frac{\partial a }{\partial t} ={-} \frac{1}{2 \epsilon_0 \epsilon_{{\rm eff}}^{\varOmega}} \frac{I_2}{I_1},\end{equation}

where $I_{1} = \int ^{\infty }_{- \infty } \boldsymbol {E}_{s,0} \boldsymbol {\cdot } \boldsymbol {E}_{s,0}^{\ast }\,{{\rm d}x}$ and $I_{2} = \int ^{\infty }_{- \infty } \boldsymbol {J}_{\varOmega } \boldsymbol {\cdot } \boldsymbol {E}_{s,0}^{\ast }\,{{\rm d}x}$.

Solving the (4.12), one may obtain

(4.13)\begin{equation} \frac{\partial a }{\partial t} = \frac{A_0^{2} \mu h}{2 \epsilon_{{\rm eff}}^{\varOmega}} \frac{n_0 e^3}{\epsilon_0 (m_e^{{\ast}})^{2}} \frac{( - \tilde{v}_{\varOmega}^x ({\beta^{\varOmega}_2})^{{\ast}} + \tilde{v}_{\varOmega}^z )}{\left( \dfrac{1 + \beta^2_2}{\alpha^{\varOmega}_2} + \dfrac{1 + \beta^2_1}{\alpha^{\varOmega}_1} \right)}.\end{equation}

Letting $\partial /\partial t = - {\rm i} \varOmega$, we obtain

(4.14)\begin{equation} \left|\frac{a}{A_{0}}\right| = \left|{\rm i} \frac{A_0 e h \mu \omega^2_{p}}{2 \varOmega \epsilon_{{\rm eff}}^{\varOmega} m_e^{{\ast}}} \frac{(- \tilde{v}_{\varOmega}^x ({\beta^{\varOmega}_2})^{{\ast}} + \tilde{v}_{\varOmega}^z )}{\left( \dfrac{1 + \beta^2_2}{\alpha^{\varOmega}_2} + \dfrac{1 + \beta^2_1}{\alpha^{\varOmega}_1} \right)}\right|.\end{equation}

The amplitude ratio of THz SMPs to the SMP pump ($a/A_{0}$) is represented by (4.14). It shows that $a/A_{0}$ is directly proportional to the amplitude $A_{0}$ and modulation index $\mu$ of the SMP wave. The amplitude ratio of THz SMPs ($a/A_{0}$) also depends upon the electron cyclotron frequency $\omega _{{\rm ce}}$ (external magnetic field $B_{0} \hat {y}$) and n-type semiconductor temperature. The following parameter modulation index $\mu = 0.3$, ripple height ${h} = 50\,\mathrm {\mu }$m, effective mass of free electrons $m_{e}^{\ast } = 0.014 m_{e}$, $\epsilon _{r} = 15.68$ are used for the numerical study of the THz SMP wave.