==== Front Sci Rep Sci Rep Scientific Reports 2045-2322 Nature Publishing Group UK London 37394610 37801 10.1038/s41598-023-37801-x Article A Floquet engineering approach to optimize Schottky junction-based surface plasmonic waveguides Herath Kosala 1 Gunapala Sarath D. 2 Premaratne Malin malin.premaratne@monash.edu 1 1 grid.1002.3 0000 0004 1936 7857 Advanced Computing and Simulation Laboratory (AχL), Department of Electrical and Computer Systems Engineering, Monash University, Clayton, VIC 3800 Australia 2 grid.20861.3d 0000000107068890 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109 USA 2 7 2023 2 7 2023 2023 13 1069230 1 2023 28 6 2023 © The Author(s) 2023 https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The ability to finely control the surface plasmon polariton (SPP) modes of plasmonic waveguides unveils many potential applications in nanophotonics. This work presents a comprehensive theoretical framework for predicting the propagation characteristics of SPP modes at a Schottky junction exposed to a dressing electromagnetic field. Applying the general linear response theory towards a periodically driven many-body quantum system, we obtain an explicit expression for the dielectric function of the dressed metal. Our study demonstrates that the dressing field can be used to alter and fine-tune the electron damping factor. By doing so, the SPP propagation length could be controlled and enhanced by appropriately selecting the intensity, frequency and polarization type of the external dressing field. Consequently, the developed theory reveals an unexplored mechanism for enhancing the SPP propagation length without altering other SPP characteristics. The proposed improvements are compatible with existing SPP-based waveguiding technologies and could lead to breakthroughs in the design and fabrication of state-of-the-art nanoscale integrated circuits and devices in the near future. Subject terms Electronic devices Electrical and electronic engineering Nanophotonics and plasmonics issue-copyright-statement© Springer Nature Limited 2023 ==== Body pmcIntroduction Meeting the ongoing demand for faster information sharing and data processing through traditional electronic engineering approaches, which are fundamental to all modern electronic devices and components, may be difficult. The thermal and resistor-capacitor time delays pose a challenge in achieving faster and more power-efficient electronic devices1. To overcome this bottleneck, optical waves can be used in place of electronic signals, as optical interconnections can enable high-speed and massive data transmissions. However, the conventional photonic engineering approach has poor integration and miniaturization capabilities due to the diffraction limit, which leads to physical size or dimensional inconsistencies between modern electronic and conventional photonic components2. To bridge the gap between nanoscale electronic components and microscale photonic components, researchers have turned to plasmonic nanostructures, which have the potential to route and actively manipulate light at the nanoscale3. As a result, plasmonics has been investigated in various scientific disciplines, including materials science, chemistry, biology, and communication systems4,5. Surface Plasmon Polaritons (SPPs) are quasiparticles that carry information in plasmonics. They represent a quantum of propagating charge density oscillation at the metal-dielectric interface in the presence of electromagnetic radiation. SPPs are localized at the interface due to their resonant interaction with free electrons in the metal6. SPP-based waveguides are used in modern plasmonic devices to focus and deliver light energy to nanoscale regions. The performance of these waveguides depends on the propagation length and the localization of the field in the SPP mode. Achieving high-performance waveguides based on SPP requires considering the propagation losses imposed by the limitations of signal propagation distance. Various modifications to the metal-dielectric interface have been proposed to improve SPP waveguide performance, including arrayed nanoparticles, sharp metal wedges, nanogrooves, gap waveguides, slot waveguides, and cylindrical waveguides7–12. Recently, active waveguiding solutions have been proposed to achieve better SPP propagation length in plasmonic waveguides by providing loss-compensating energy to the SPP waveguide13–15. One method for enhancing active SPP performance is substituting conventional dielectric material with a doped semiconductor and using electrical pumping to boost performance16,17. There is a great need for a clear theoretical understanding of SPP behavior at the interface between a semiconductor and a metal, known as a Schottky junction. The behavior of SPPs at a Schottky junction is different from conventional SPPs at dielectric-metal interfaces due to the material properties of the semiconductor before and after the Schottky interface formation. The charge carrier density variation in the space charge region formed during the Schottky junction formation impacts the susceptibility and permittivity functions of the interface, offering the ability to manipulate the SPPs propagate along the Schottky junction17. This property has attracted researchers’ attention in modern plasmonics17,18. However, previous studies17,19 on Schottky junction-based SPPs have been restricted to a junction originating with a lossless metal, which is not suitable for real-world plasmonic waveguides with lossy metallic mediums and SPPs with finite propagation lengths. Therefore, it is critical to understand the fundamental mechanism behind Schottky junction-based SPPs that are based on lossy metals to develop next-generation plasmonic waveguides. The purpose of this study is to theoretically analyze the impact of damping effects on Schottky junction-based SPPs and to suggest a novel analytical approach using Floquet engineering to improve the performance of SPPs modes. Recent advancements in laser technology have made Floquet engineering an intriguing topic, which uses high-intensity periodic driving to obtain new characteristics of matter that are not accessible in equilibrium states20. In quantum many-body systems, time-periodic radiation is a powerful tool that can control material properties on an as-needed basis. The Floquet formalism21 is the primary analytical tool used in this area of study, and it can describe the effects of an external field even at high intensities, unlike perturbation techniques. The Floquet formalism allows for better control over material properties, making it an essential tool in the study of quantum many-body systems. Instead of treating quantum particles and electromagnetic fields separately, they should be treated as a single composite quantum system in a non-perturbative approach. This system is commonly referred to as a dressed system, with the high-intensity electromagnetic field acting as the dressing field. While Floquet engineering has found applications in various fields of physics, we will specifically examine its application in plasmonic waveguides. The paper by Herath and Premaratne22 discusses how to improve the performance of SPPs in metal-dielectric plasmonic waveguides using Floquet engineering. However, there has been no evaluation of how external radiation affects Schottky junction-based plasmon waveguides. This study uses the Floquet formalism to identify new phases of dressed Schottky junctions and manipulate SPP propagation lengths. This allows us to change the propagation length of Schottky junction-based SPP modes while maintaining their other properties. Previous studies have shown that the Schottky junction bias voltage can manipulate the propagation length of SPP, but this will also change the space charge region and alter other properties of the SPP modes17,23. The proposed method in this study, using Floquet engineering, can modify the SPP propagation length based on the dressing field’s intensity, frequency, and polarization type. Thus, we believe that Floquet engineering-based method can be applied to nanoplasmonic circuits24 in a very promising manner. For this work, it is essential to assume that the system is in a low-temperature stable state, and the electromagnetic field should be solely dressing to prevent any energy exchange with electrons. The electrons in a metallic system can absorb the field through two mechanisms, namely, electron transitions between different energy levels and transitions between various states within the broadened energy level. To avoid these mechanisms, the frequency of the dressing field should be carefully selected to be far from the resonant frequency and larger than the scattering-induced energy band broadening. This allows the avoidance of interband and intraband energy absorptions. This work examines the optical properties of dressed Schottky junctions and proposes a method to enhance their performance. The analysis considers more generalized conditions than previous studies. The Floquet states in a dressed metallic system are examined, and their optical characteristics are studied using the Floquet formalism and liner-response theory. An analytical expression for the susceptibility function of the dressed metallic system is derived. The external dressing field can manipulate the damping factor for charge carriers in the metal. The wave-function solution of the Floquet state in a quantum Floquet system depends on the radiation strength, which enables the possibility to tailor the charge transport and SPP characteristics under external radiation. The dispersion relations for possible SPP modes in a dressed Schottky junction are derived, and an analytical technique is proposed to reduce electron-impurity scattering in the dressed metal and manipulate the propagation length of SPP modes. A detailed numerical analysis of the Schottky junction-based dressed SPP properties is performed under differing polarization conditions. This research provides a new possibility for controlling the properties of SPP modes within a Schottky junction, which could benefit next-generation nanoplasmonic components and devices. Theoretical formulation The theoretical basis for Schottky junction-based SPPs and their behavior when exposed to a high-intensity dressing field is described in this section. The optical response of a dressed metal is analyzed, and expressions for its susceptibility and dielectric functions are derived. The Floquet–Fermi golden rule is used to examine the damping effects in dressed metal electrons, demonstrating the possibility of manipulating the metal dielectric function. Ultimately, expressions for the dispersion relations of potential SPP modes at a lossy Schottky junction are derived. Dielectric function of dressed metals This part of the analysis presents the derivation of an expression for the conductivity of a dressed metal film. The equilibrium linear response theory is extended to a driven system out of equilibrium, which is probed by a weak external potential. Moreover, an expression for the current response is derived, which depends not only on the frequency of the probe but also on the frequency of the drive. Finally, an analytical expression for the susceptibility and dielectric functions of a dressed metal is presented. Floquet theory We consider a metal film subjected to a high-intensity periodical dressing field. The wave function of a single electron in the dressed metal can be identified as Floquet states22,25,261 |ψα(t)⟩=exp-iϵαħt|uα(t)⟩. In this case, α represents a discrete set of quantum numbers, |uα(t)⟩ are corresponding Floquet modes, and ϵα are the corresponding quasienergies. The periodicity of the Floquet modes, allow to rewrite them as a Fourier expansion as follows2 |uα(t)⟩=∑nexp(-inΩt)|uαn⟩, where Ω is the angular frequency of the dressing field. The Floquet picture of the Kubo formula In linear response theory, the average current response of an equilibrium system for a probe bias with a vector potential A(k,ω) can be expressed by the general Kubo formula in k-momentum space273 ⟨J^a(k,ω)⟩=limμ→0+-12πħV∑b∫-∞∞∫-∞∞∫-∞∞〈[j^a(k,t),j^b(-k,t′)]〉0Ab(k,t′)exp(iωt)exp[-iω′(t-t′)]ω′+iμdω′dt′dt-e2nmmAa(k,ω). Here, j^ is the single-particle current operator of the system, ω is the frequency of the response current, V is the system volume, a,b∈{x,y,z} present the directional components in three-dimensional coordinate space, and ⟨·⟩0 denotes the statistical thermal average with respect to the system without the probe bias. Due to the presence of high-intensity external dressing in our case, the system will not be in equilibrium. Since the occupation number of Floquet states are independent, we can assume our system to be in a stationary state under the dressing field28. From second quantization formalism, we can expand the current operators j^a,b(k,t) using the Floquet states as the basis for our system as follows294 j^a(k,t)=∑αβj^αβa(k,t)a^α†(t0=0)a^β(t0=0), where a^α†(t) and a^α(t) are creation and annihilation operators for α-th Floquet state. These operators should satisfy the following relationships5 a^α†(t)|0⟩=|ψα(t)⟩,a^α(t)|0⟩=0,[a^α(t),a^β†(t)]±=δαβ,and[a^α(t),a^β(t)]±=[a^α†(t),a^β†(t)]±=0. Here, |0⟩ is the vacuum state containing no particle, and positive (negative) subscripts refer to the fermionic anticommutators (commutators) for the Floquet state particles. Furthermore, j^αβa(k,t) is the single particle matrix element given in Schrödinger’s picture by6 j^αβa(k,t)=⟨ψα(t)|j^a(k)|ψβ(t)⟩=⟨uα(t)|expiϵαtħj^a(k)exp-iϵβtħ|uβ(t)⟩, where j^a(k)=j^a(k,t=0). Considering the Fourier series expand of the time-periodic Floquet modes, we can identify that7 j^αβa(k,t)=∑n1,n2=-∞∞expiħ[(ϵα-ϵβ)+(n1-n2)ħΩ]t〈uαn1|j^a(k)|uβn2〉, where n1,n2 are integers. Next, we can evaluate the commutator operator in equation (3) using the above derived second quantization expansion8 [j^a(k,t),j^b(-k,t′)]=∑αβj^αβa(k,t)j^βαa(-k,t′)(a^α†a^α-a^β†a^β). Here, a^α†a^α represent the number operator for the α-th Floquet state. Thus, we introduce the distribution functions for the Floquet state particles in the system as follows9 Fα:=⟨a^α†a^α⟩,andFβ:=〈a^β†a^β〉. It is important to note that these distribution functions not necessarily to be equilibrium distribution functions, however it is assumed that these are time-independent. Thus, the statistical thermal expectation value of the above commutator becomes10 〈[j^a(k,t),j^b(-k,t′)]〉0=∑αβ∑n1,…,n4=-∞∞expiħ[(ϵα-ϵβ)+(n1-n2)ħΩ]texpiħ[(ϵβ-ϵα)+(n3-n4)ħΩ]t′×〈uαn1|j^a(k)|uβn2〉〈uβn3|j^b(-k)|uαn4〉(Fα-Fβ). Then, we can represent the vector potential corresponding to the probe bias in the frequency domain as follows11 Ab(k,t′)=12π∫-∞∞Ab(k,ω′′)exp(-iω′′t′)dω′′, and submitting these back into the Kubo formula and evaluating the time integrals and ω′ integral, we can obtain12 ⟨J^a(k,ω)⟩=limμ→0+-1ħV∑b∫-∞∞∑αβ∑n1,…,n4=-∞∞δω-ω′′+(n1-n2+n3-n4)Ωω+1ħ(ϵα-ϵβ)+(n1-n2)Ω+iμ×〈uαn1|j^a(k)|uβn2〉〈uβn3|j^b(-k)|uαn4〉Ab(k,ω′′)Fα-Fβdω′′-e2nmmAa(k,ω). We can specify the probe electric field relation with vector potential using the electromagnetic theory13 Ab(k,ω)=-i(ω+iγ)Eb(k,ω). Here, the factor γ is a phenomenological way to include electron scattering-caused damping effects in the quantum calculations. Then, we can rewrite the Kubo formula in a compact form14 ⟨J^a(k,ω)⟩=∑b∫-∞∞σab(k,ω,ω′′)Eb(k,ω′′)dω′′, by initiating the conductivity tensor for the dressed metallic system15 σab(k,ω,ω′′)=limμ→0+iħV∑αβ∑n1,…,n4=-∞∞δω-ω′′+(n1-n2+n3-n4)Ωω+1ħ(ϵα-ϵβ)+(n1-n2)Ω+iμ×〈uαn1|j^a(k)|uβn2〉〈uβn3|j^b(-k)|uαn4〉Fα-Fβω+iγ+ie2nmm(ω+iγ)δ(ω-ω′′)δab, Based on this expression, the current is no longer a simple product of conductivity and perturbation electric field as it is convoluted over the bias frequency ω′′, as opposed to the un-driven case. There is significance in the fact that the conductivity tensor depends on both the response frequency ω and the bias frequency ω′′. With the above-derived general expression for conductivity, we can determine a new expression for our Floquet system by specifying the conditions which affect the parameters. First, we assume that the response and bias frequency ω and ω′′ are in the central Floquet zone16 |ω|,|ω′′|<Ω/2⇒|ω-ω′′|<Ω. Under this condition, we can identify that the delta distribution in Eq. (15) only can be non-zero with the condition17 n1-n2+n3-n4=0. Focusing on more special case of conductivity by analyzing only the longitudinal conductivity where a=b=x, and assuming that the current response is spatially homogeneous (k→0), we can obtain18 σxx(ω)=limμ→0+iħV(ω+iγ)e2m2∑αβ∑n1,…,n4=-∞∞Fα-Fβω+1ħ(ϵα-ϵβ)+(n1-n2)Ω+i0+〈uαn1|p^x|uβn2〉〈uβn3|p^x|uαn4〉+ie2nmm(ω+iγ), where p^x is the x-directional momentum operator. Polarization and current responses are physically indistinguishable effects of the conductivity tensor. Accordingly, susceptibility function of the system can be expressed in general as a function of linear conductivity3019 χxx(ω)=σxx(ω)-iω. Subsequently, we can identify the susceptibility function of the dressed quantum system as20 χxx(ω)=limμ→0+1ħω(ω+iγ)e2m2V∑αβ∑n1,…,n4=-∞∞Fα-Fβω+1ħ(ϵα-ϵβ)+(n1-n2)Ω+iμ〈uαn1|p^x|uβn2〉〈uβn3|p^x|uαn4〉+e2nmmω(ω+iγ). The study involves a dressed metallic system that utilizes the free electron model to explain its transport properties. The particle distribution functions can be described using the Fermi–Dirac distribution. Additionally, very low-temperature conditions (T→0) are considered, leading to a re-writing of the Fermi–Dirac distribution21 F(ϵ)=limT→01exp(ϵ-ϵFkBT)+1≈Θ(ϵF-ϵ), where, ϵF is the Fermi energy, kB is the Boltzmann constant, T is the absolute temperature, and Θ(·) is the Heaviside function. Now, we can restructure the general susceptibility function according to the dressed metallic system22 χxx(ω)=limμ→0+1ħω(ω+iγ)e2m2V∑αβ∑n1,…,n4=-∞∞Θ(ϵF-ϵα)-Θ(ϵF-ϵβ)ω+1ħ(ϵα-ϵβ)+(n1-n2)Ω+iμ〈uαn1|p^x|uβn2〉〈uβn3|p^x|uαn4〉+e2nmmω(ω+iγ). Considering the transport properties of metallic electrons, it’s only necessary to analyze the behavior of the conduction electrons. This limits the analysis to electrons with energy similar to the Fermi energy. This leads to assume that ϵα=ϵ′, and ϵβ=ϵ′′, where ϵ′,ϵ′′→ϵF. Next, we can apply these changes back into the susceptibility function and obtain23 χxx(ω)=limμ→0+limϵ′,ϵ′′→ϵF1ħω(ω+iγ)e2m2V∑αβ∑n1,…,n4=-∞∞Θ(ϵF-ϵ′)-Θ(ϵF-ϵ′′)ω+1ħ(ϵ′-ϵ′′)+(n1-n2)Ω+iμ〈uαn1|p^x|uβn2〉〈uβn3|p^x|uαn4〉+e2nmmω(ω+iγ). In our study, we assume that ω≠0 and ω<|Ω/2|. It follows that the denominator of the above expression cannot be very small. Therefore, we can see that the first term of the above expression goes to zero. Finally, now we can assume that there is no contribution from the first term in the susceptibility function. Moreover, we can derive final expression for the susceptibility function for a dressed metallic quantum system as24 χxx(ω)=ϵ0ωpm2ω(ω+iγ),whereωpm=e2nmϵ0m. Here, γ0 is the un-driven damping factor. Finally, we can identify the dressed metal dielectric function25 εm(ω)=εhm-χxx(ω)ϵ0=εhm-ωpm2ω(ω+iγ), where εhm is the high-frequency permittivity of the metal. Although the derived expression is the same as the general Drude-Sommerfeld model description, we need to consider the effects on the damping factor γ induced by the dressing field. As the previous literature22,31 describes, the dressing field modifies the electron wave functions. As long as the electron scattering rate and the electron transport damping factor depend on the wave function, we can control the damping factor as well as the optical properties of the system by applying an external dressing field. These effects can be analytically evaluated by applying the Floquet–Fermi golden rule for a dressed metallic system22. This enables us to achieve high-performance SPP modes in Schottky junction-based waveguides. Next, we can rewrite the x-directional metal dielectric function by introducing the normalized damping factor γ~ as follows26 εm(ω)=εhm-ωpm2ω(ω+iγ~γ0),whereγ~=γγ0. Here, γ0 is the un-driven damping factor. Moreover, γ~ depends on the intensity and the frequency of the applied dressing field22. For details on the derivation of these results, see Section 1 and 2 of the Supplementary Information. Under the results section, we investigate numerically how metal dielectric function can be manipulated under various driving fields. SPP modes at the Schottky junction Our primary interest is to analyze the optical properties of Schottky junctions and achieve improved SPP modes at the interface. The interface between a semi-infinite metal and a semi-infinite n-type semiconductor film in the xy plane of the Cartesian coordinate space is considered as shown in Fig. 1a top section. The contact between metals and semiconductors leads to the flow of free charge carriers until their Fermi levels align according to thermodynamic principles, causing changes in charge density and creating a space charge region near the interface of the two materials. Notably, the width of the space charge region d can be manipulated by applying an external voltage across the interface. Due to this, the Schottky junction plays a crucial role in modern electronics. In general, a complex charge density profile can be approximated by a piecewise-linear fluctuation. Therefore, to model the charge profile of the semiconductor region, a piecewise-linear variation is used as shown in Fig. 1a bottom section. This model has been used in earlier literature17,19, and is also employed in this study for comparison purposes. Additionally, the frequencies of the electromagnetic fields are carefully selected to be in an off-resonant regime to minimize photon absorption by the system.Figure 1 (a) Formation of a Schottky junction (top) and piecewise-linear variation of charge density in the Schottky junction (bottom) under the full depletion approximation. Here, ns is the bulk charge density in the semiconductor, nm is the charge density in the metal, and d represents the width of the space charge region. (b) A schematic illustration of the Schottky junction, including divided of the junction regions A, B, C and D are based on the carrier density distribution. Furthermore, carrier density n(z) and plasma frequency ωP(z) for each region are depicted. The plots presented in this figure may not adhere to a linear scale on the vertical axis. The complex dielectric functions for n-type semiconductor εs can be described using the following form32:27 εs(ω)=εintra(ω)+εinter(ω) The proposed approach explicitly distinguishes between the intraband effects εintra and the interband effects εinter. In this context, the intraband effects are generated by the free electrons, whereas the interband effects are generated by the bound electrons. The intraband component of the dielectric function is can be characterized by the Drude model3328 εintra(ω)=ε∞-ωp2ω2+iωςs. Here, ε∞ is the high-frequency permittivity of the semiconductor, ωp is the plasma frequency of the semiconductor, and ςs is the damping factor of the semiconductor induced by the electron scattering in the material. The interband component of the dielectric function can be represented by both a real part and an imaginary part:εinter(ω)=εinter′(ω)+iεinter′′(ω). Under the absence of interband absorptions we can neglect the imaginary part of the interband component. This leads to29 εs(ω)=εhs-ωp2ω2+iωςs, where εhs=ε∞+εinter′(ω). Generally, εhs exhibits dependence on the angular frequency. However, for certain spectral ranges, it can be approximately treated as constant34. Since the semiconductor plasma frequency is contingent upon the electron density, the dielectric function is also influenced by variations in electron density. In our system, the electron density of the semiconductor varies with the coordinates along the z-direction. Hence, the dielectric function of the semiconductor relies on both the angular frequency of the SPP excitation radiation and the spatial coordinate in the z-direction. Therefore, we can determine the dielectric function of the semiconductor in our Schottky junction as follows3530 εs(z,ω)=εhs1-ωps2(z)ω2+iωςs,whereωps(z)=n(z)e2εhsε0m. Here, n(z) is position-dependent free charge carrier density, ε0 is the vacuum permittivity, and m is the effective mass of the free charge carriers. In addition, we assume that the damping factor is not dependent on the free charge carrier density. Moreover, we define the plasma frequency of the bulk semiconductor region as31 ωps0=nse2εhsε0m, and it is independent of the position. Since the charge density inside the semiconductor of the Schottky junction varies, we can identify four main regions inside a Schottky junction, as depicted in Fig. 1b. For the metal region, we use the general dielectric function expression derived in Eq. (26). With the dressing field removed, one is left with the well-known Drude model-based metal dielectric function expression.Then, we summarize these characteristics of each region under the Table 1.Table 1 Characteristics of each region in the Schottky junction. Here, r=1/(d2-d1), and h=-rd1. Region Range Charge carrier density n(z) Plasma frequency ωp Dielectric function ε(z,ω) A z≤0 nm ωpm 1-ωpm2ω2+iωγ~γ0 B 00 solutions (with z<0 values) as we need only the decaying solution on the outside of the interface, and EA is an unknown real constant. Region B In this region, ε(z,ω)=εhs is a real-valued constant. Thus,37 ∂2Ez(z)∂z2-κB2Ez(z)=0,whereκB2=kx2-ω2εhsc2. For κB2>0, we can express the solutions as38 Ez(z)=EB1eκBz+EB2e-κBzandEx(z)=iκBkxEB1eκBz-EB2e-κBz, where EB1 and EB2 are unknown constants. However, under the scenario κB2<0, we need to introduce different type of solutions such as39 Ez(z)=EB1cos-κB2z+EB2sin-κB2z,andEx(z)=i-κB2kx-EB1sin-κB2z+EB2cos-κB2z. Region C In this region, charge carrier density varies linearly. Thus, the permittivity function of this region is also dependent on the z-directional coordinates. Before further analysis, we should consider the practical aspect of these parameters. Our analysis uses an excitation field frequency ω in ∼1015s-1 order. In addition, we model the semiconductor using the parameters of n-type doped gallium arsenide material, and we can identify that its damping factor is in ∼1012s-1 order. Under these practical conditions, we can assume that ςs≪ω, and neglect the effects of damping in the semiconductor region. It is important to notice that for d10 solutions (with z>d2 values) as we need only the decaying solution on the outside of the interface, and ED is an unknown real constant. Dispersion relation of SPP modes at the Schottky junction The dispersion relation for possible SPP modes relates the excitation light’s angular frequency to the SPP mode’s in-plane wavenumber magnitude. The dispersion relation can be found by applying the self-consistent boundary conditions required by Maxwell’s equations3645 εR1Ez,R1(z)=εR2Ez,R2(z),andEx,R1(z)=Ex,R2(z), where R1,R2∈{A, B, C, D} at z=0, z=d1, and z=d2 to ensure the continuity of the SPP electromagnetic field. Using the previously identified wave modes for each region in the Schottky junction, we can derive six linear simultaneous equations for these boundary conditions, and represent them in a matrix equation46 M6×6(ω,kx)E6×1=0. Here, M6×6(ω,kx) is the matrix of coefficients, and it can be represented as47 M6×6(ω,kx)=εm(ω)-εhs-εhs000κA-κBκB0000eκBd1e-κBd1-Ai′(ζd1)ζd1-Bi′(ζd1)ζd100κBeκBd1-κBe-κBd1-Ai(ζd1)ϑ-Bi(ζd1)ϑ0000Ai′(ζd2)ζd2Bi′(ζd2)ζd2-e-κDd2000Ai(ζd2)ϑBi(ζd2)ϑκDe-κDd2, for κB2>0. However, under the κB2<0 condition, we can find that48 M6×6(ω,kx)=εm(ω)-εhs0000κA0-κB′0000cos(κB′d1)sin(κB′d1)-Ai′(ζd1)ζd1-Bi′(ζd1)ζd100-κB′sin(κB′d1)κB′cos(κB′d1)-Ai(ζd1)ϑ-Bi(ζd1)ϑ0000Ai′(ζd2)ζd2Bi′(ζd2)ζd2-e-κDd2000Ai(ζd2)ϑBi(ζd2)ϑκDe-κDd2, where κB′=-κB2, and E6×1=[EA,EB1,EB2,EC1,EC2,ED]T. A nontrivial solution to E6×1 can be obtained by ensuring that the determinant of the coefficient matrix M6×6(ω,kx) is equal to zero49 det(M6×6(ω,kx))=0, and this leads to a secular equation relating ω and kx. This is the dispersion relation for the possible SPP modes in a Schottky junction. Results and discussion This section presents the numerical results we received for the characteristics of SPP modes in dressed Schottky junction-based waveguides. In this analysis, we consider silver (Ag) metallic material and n-doped gallium arsenide (GaAs) semiconductor material. To ensure the validity of the free electron model, a plasmonic metal with minimal natural damping effects needs to be chosen. Ag was selected as it has the lowest natural damping factor among popular plasmonic materials34. Additionally, Ag is a better candidate for dressed SPP modes with long propagation lengths, while other properties remain unaffected22. In prior studies on Schottky junctions, n-doped silicon (Si) or n-doped GaAs were used as the semiconductor material16,17. The selection of the semiconductor material for the Schottky junction does not impact the research outcomes since the main focus is on reducing propagation losses in the metallic region. However, selecting a semiconductor with a higher plasma frequency can provide a higher frequency range for operating SPP modes. GaAs was selected over doped Si as it has a lower effective electron mass, resulting in a higher plasma frequency. Unless specified otherwise, the following empirical material parameters are used in the numerical calculations37,38: the metal plasma frequency ωpm=1.352×1016rads-1, natural damping factor γ0=0.2eV, the high-frequency permittivity εhm=5, and energy ϵF=5.5eV for Ag. The effective mass m is 1.0me, and 0.067me for Ag and GaAs respectively. Here, me is the mass of an electron. Furthermore, we assume static dielectric constant εhs=12.3, and bulk carrier density ns=2×1024m-3 for GaAs. According to Newman’s research39, this bulk carrier density is achievable for n-doped GaAs. With these empirical data, we can evaluate the bulk plasma frequency for the semiconductor region and obtain ωps0=8.789×1013rads-1. In addition, we assume that the dressing field frequency Ω=1×1014rads-1, and the reference dressing field intensity I0=3kWcm-2. The full mathematica code for the numerical calculations is available under the Supplementary files. Dispersion relation with lossy metal The previous literature17,19 assumed that the dielectric function of the metal is a real-valued function for all frequency ranges. Under the ω~