Surface Magnetoplasmon Polaritons (SMPs) are the electromagnetic (EM) waves that propagate along the interface of two different media. This is known to be the interaction between surface plasmons and external magnetic fields that can enable effective manipulation of light. The field exhibits a wave-like behavior as one moves parallel to the interface but exponentially decays when moving away from the interface into either of the two media. The dispersion properties of SMPs have been extensively studied by various researchers [1], [2], [3], [4]. Comprehensive experimental reviews regarding the collective electronic excitations at material surfaces can also be found in the literature, including works by Ritchie [5], Feibelman [6], Liebsch [7] and Raether [8], [9]. In this context, the Drude or Drude-Sommerfeld models were employed to elucidate the optical and electronic characteristics of SMPs. Models such as the Boltzmann transport equations may also be used for a more accurate description of electronic behavior in materials. The model equations provide a more general framework for examining collective dynamics. Employing the kinetic theory, one can gain insights into the impact of temperature on different characteristics of the material. The understanding may play an important role for designing and optimizing semiconductor devices that operate under diverse temperature conditions, as encountered in practical applications. In semiconducting mediums, finite-temperature effects become noticeable at temperatures exceeding T>102K, particularly when the electron number density (ne) is on the order of ne≥ 1016 to 1018cm−3 (see e.g., [10]). Medium like plasma in semiconductors are often considered to be partially degenerate, which implies that the medium’s temperature “T” approaches the Fermi temperature “TF” and the equilibrium distribution function transitions from the Maxwell–Boltzmann (MB) distribution to the Fermi–Dirac (FD) distribution [11], [12], [13], [14], [15], [16]. The study of ordinary plasma waves with arbitrary temperature degeneracy can be found in the literature [17], [18], [19], [20], [21], [22], [23], [24] . Dispersion characteristics of electromagnetic plasma waves with arbitrary electron degeneracy have also been analyzed by few of us [25], [26], [27].

Numerous studies have focused on modeling and experimenting with SPPs and SMPs at finite temperatures considering different geometrical conditions. Ahmad et al. [28] derived the dispersion of the SPP mode with finite temperature and nonzero chemical potential for the complex-frequency analysis for the Transverse Electric mode in graphene. Finite temperature effects are also crucial in the context of low-frequency magneto-electronic excitations in armchair graphene nanoribbons (AGNRs) [29]. These effects lead to an increase in electrical conductivity (σ) in InSb, resulting in a larger number of electrons compared to holes within the material. Thermal excitation shifts the Fermi level towards the conduction band’s edge at higher temperatures, leading to an enhanced power factor due to Fermi level optimization [30]. In femtosecond time-resolved pump-probe experiments on metallic nanoparticles, the collective modes of surface plasmons are excited by electrons through thermal effects. The absorption spectrum of the hot electron gas in the nanoparticle exhibits dependencies on temperature, surface plasmon linewidth, and resonance frequency [31]. These effects have also been observed experimentally [32], [33]. Liang et al. [34] examined the impact of variable finite temperatures on SMPs in a Voigt reflection geometry. This was accomplished through time-domain THz spectroscopy, allowing for the observation of transverse-field magneto-optical effects in InSb. Their study revealed that at elevated temperatures, changes in thermal carrier density modify the dispersions of magneto-plasmon bands, and these changes can be accurately tuned by temperature.

Our current research objective is to determine the dispersion relation of SMPs propagating between the semiconductor-vacuum interface in the Voigt geometry using Vlasov-Maxwell’s model approach and applicable to terahertz frequency domain at finite temperatures. The investigation represents a more generalized approach compared to previous studies [35], which examined SMP dispersion relations in dilute (Maxwell–Boltzmann distributed) and dense (Fermi-distributed at T=0) media. We consider the dielectric tensor components given in Ref. [35] and express them in terms of Plasma Dispersion Functions for equilibrium Fermi–Dirac distribution function for nonzero temperature. Incorporating finite temperatures into the analysis of SMPs in Voigt geometry is essential for a comprehensive understanding of their behavior in the terahertz domain, ensuring realistic modeling for both theoretical investigations and practical applications. We analyze the high-frequency dispersion relations of SMPs while considering thermal corrections which is crucial to account for the influence of thermal electron motion within a magnetized medium. The inclusion of thermal corrections may be essential for comprehending the behavior of high-frequency SMPs, as they have a significant impact on resonance width and wave propagation. The present investigation into graphene plasmonics on structured metasurfaces offers insights that extend to the study of Surface Magnetoplasmon Polaritons (SMPs) in Voigt reflection geometry. Extensive examination has been conducted on the impact of loss on Surface Plasmon Polaritons (SPPs) within a quantum regime, with particular attention paid to the significance of the loss coefficient as a crucial parameter in characterizing these effects [36]. Graphene is also considered to be an alternative material to support surface plasmon polaritons (SPPs) with its excellent properties such as the tight electromagnetic field localization, low dissipative loss, and versatile tunability. A theoretical and simulated studies about the excitation of SPPs by an injected electron beam with periodic graphene ribbon arrays deposited on a dielectric medium has also been studied by Liu et al. [37]. They obtained analytical expression of the dispersion relation, propagation loss, and field pattern in the THz band. Graphene provides an excellent possibility to control the properties of the electromagnetic mode from the optical band to the terahertz (THz) band. Understanding the dispersion theory and propagation characteristics of SMPs along structured interfaces can provide a powerful framework for designing innovative photonic, plasmonic, and optoelectronic devices across a wideband spectrum. By leveraging the unique properties of materials like graphene and exploring their interaction with structured metasurfaces, researchers can unlock novel functionalities and enhance device performance. Additionally, this approach can be extended to other materials, such as transition metal dichalcogenides (TMDs) and two-dimensional materials like black phosphorus, opening up new avenues for advancing photonics and optoelectronics [38].

The paper’s structure is as follows: Section 2 briefly introduce the SMPs dispersion relation at the Vacuum-matter interface in the Voigt reflection geometry. The characteristics of FD distribution at finite temperatures and the gyrodielectric tensor components obtained from the Vlasov-Maxwell model equations are presented. In particular, the results are presented in terms of the Plasma Dispersion Functions for FD distribution function. The derived results are discussed numerically and graphically considering Vacuum-InSb interface within a specific temperature range. Section 3 summarizes the paper’s findings.