Propagation of surface magnetoplasmon polaritons in a symmetric waveguide with two-dimensional electron gas

Mingxiang GAO(高明向),Baojun WANG(王宝军) and Bin GUO(郭斌)

Department of Physics,Wuhan University of Technology,Wuhan 430070,People's Republic of China

Abstract The properties of surface magnetoplasmon polaritons(SMPPs)in a symmetric structure,composed of two semi-infinite regions of high-density two-dimensional electron gas(2DEG) separated by a thin film in Voigt configuration,are investigated.The normal and absorption dispersion relations for the transverse magnetic polarization are derived by correlating Maxwell’s equation and the boundary conditions.It is demonstrated that the features of SMPPs are greatly influenced by the external magnetic field,collision frequency of 2DEG,the dielectric constant,and the thickness of the thin film,suggesting that the locations and propagation lengths of SMPPs can be governed accordingly.It is shown that the symmetry of the physical geometry preserves the symmetry of the dispersion relations of SMPPs.Furthermore,it is discovered that as the external magnetic field increases,the penetration depth of SMPPs decreases,while their energy loss reduces,implying that plasmons can propagate for longer distances.Additionally,it is observed that SMPPs in the symmetric configuration have a longer lifetime than those in the asymmetric configuration.

Keywords: surface plasmon polaritons,magnetized plasma,dispersion relation,propagation length

1.Introduction

It is challenging to reduce the size of photonic devices to the nanoscale level to create genuinely downsized devices since Abbe’s diffraction limit is around half the wavelength of light[1].An unavoidable tendency in the advancement of nanophotonics is the need to figure out how to get over the diffraction limit,regulate and modify the propagation and dispersion of light at the nanoscale,and then create nanointegrated optical devices such as optical modulators and couplers.Fortunately,surface plasmon polaritons(SPPs)emerge.The term ‘SPPs’ always refers to a sub-wavelength scale plasmon collective oscillation produced at the interface between metals and dielectrics,in which the real parts of the metal’s and the dielectric’s dielectric constants have the opposite signs [2-4].By using strong interaction between photons and free electrons to strengthen the local electric field,SPPs can overcome the diffraction limit and localize electromagnetic radiation on the sub-wavelength scale [5,6].SPPs have effectively solved the issues with conventional electrical and photonic devices.SPPs have facilitated the advancement of nanoscience and technology due to their distinct advantage and development prospects.SPPs are currently being used successfully in a variety of applications,including sub-wavelength waveguides [7],metamaterials[8-10],nanofocusing[11,12],sensing and detection[13,14],and photovoltaic devices [15].Additionally,SPPs are still thought to have potential uses in the fields of information[16],energy [17,18],biology and medicine [19].

There has been great interest in the study of SPPs with optical waveguides [20-24].The waveguide is the smallest unit device in integrated optical systems and is mainly a structure used to govern the propagation of electromagnetic waves.It is well known that SPPs are typically excited using either metal-dielectric-metal or dielectric-metal-dielectric waveguides[25-27].Since the waveguide structure’s surface is generally not flat and smooth,it is simpler to excite SPPs using coupling techniques(such as Otto,Kretschmann,and grating configurations) than those with other structures.The waveguide structure also benefits from a straightforward design structure and powerful antielectromagnetic interference properties.Many optoelectronic devices,including ultrafast switches [28,29],nonlinear optical modulators [30],and refractive index sensors [31],have gained from these advantages.

When compared with other conventional materials,plasmas can show superior properties that lead to many intriguing happenings,as well as worth-seeing applications [32-40].There have recently been many investigations in understanding SPPs involved with plasmas [41-46].These studies have demonstrated that introducing plasmas into SPPs can furnish beneficial effects.In our previous work[46],we have explored the features of SPPs generated in an asymmetric waveguide with twodimensional electron gases(2DEG).We have revealed that the external magnetic field,collision frequency of 2DEG,dielectric constant and thickness of the background dielectric,significantly influence the properties of SPPs.We have uncovered that plasmon displays distinguishable behavior for the forward- and backward-propagating modes.In this work,we further examine the characteristics of SPPs yielded by a symmetric waveguide with 2DEG under a static magnetic field.We evaluate the effects of the applied magnetic field,collision frequency of 2DEG,background film dielectric constant and thickness on the properties of surface magnetoplasmon polaritons(SMPPs).We analyze the locations and propagation lengths of SMPPs through the dispersion relations of SMPPs.In addition,we demonstrate how SMPPs behave differently in symmetric and asymmetric waveguides.

The paper is organized as follows.In section 2,we start by introducing our model,and then we give the corresponding analytical formulas to calculate the SMPPs dispersion relations.Numerical results are presented and discussed in section 3.Finally,we summarize our main results in section 4.

2.Theoretical model and basic equations

The symmetric structure being referred to is a design consisting of two semi-infinite high-density 2DEG that are separated by a thin film,as shown in figure 1.The corresponding dielectric constant of the film and the 2DEG layer are indicated byε2and,respectively.It is worth noting that in our analysis,we consider the 2DEG as a type of plasma.By using plasma properties to describe the behavior of the 2DEG,we can gain a better understanding of its characteristics and behavior.It has been reported that plasma cannot be affected by the external magnetic field along inxdirection orzdirection under transverse-magnetic(TM) polarization [45].We here consider that an applied static magnetic fieldis along theydirection that is parallel to the surface in both regions ofz＜-d/2 andz＞d/2.Thus under the TM polarizations,as shown in figure 1,the SMPPs are excited atz=-d/2 andz=d/2 interfaces,correspondingly.We assume that the film is a non-magnetic material.Under the Voigt configuration,the permittivity of 2DEG can be expressed by the following tensor

Following our previous work in [46],we can obtain the magnetic field

and thex-components of electric fields

whereA1,F,G,andA3are the undetermined coefficients that indicate the amplitudes of the electromagnetic fields,respectively.qis thex-component of the wavevector.represent thez-component of the wavevector in 2DEG and the background change of thez-component of the electromagnetic field,which is caused by the collisions of the particles in the 2DEG.Therefore,equation(4) can be simplified as

dielectric film,respectively.Herein,cis the velocity in the free space,andis the effective dielectric function of 2DEG,which can be obtained from the wave equation ∇×∇×E-(ω2/c2)D=0.

Employing the boundary conditionsE1x=E2x=E3xandH1y=H2y=H3yin equations(2) and(3),one can have

It is of note that we take the collisions into account in the present study.Thus,the wavevectorqand the longitude attenuation coefficientsκi(i=1,2,3) must be complex.For the sake of simplicity,we assume thatεν=εr+iεi,q=qr+iqi,andα/ε1=η+iγ.Due to the nonretarded effect,the two solutions corresponding to the two different attenuation coefficients must be overlapped to satisfy the boundary conditions such that the electric vector field is outside to the wavevector plane which is composed of the SMPPs wave vector direction and the interface normal where

herein,q′ depictsqrorqi.Moreover,the product ofqrandqishould meet the requirement ofqrqi＞0.

We can further directly solve equation(5) by separating its real and imaginary parts [45,46].Then,we can get the normal dispersion relation of SMPPs(corresponding to the real part of the equation)

and the absorption dispersion relation of SMPPs(corresponding to the imaginary part of the equation)

whereA,B,C,andDare expressed as follows

direction.Therefore,ideally,the studies of SMPPs are always carried out under the nonretarded effect [47,48],that isq≫ω/c,which is,mathematically,equivalent to takingc→∞,then we can easily obtainκ1≈κ2≈κ3≈|qr|+i|qi|.Herein,|qr| is the attenuation coefficients of the electromagnetic field along thezdirection,and|qi|is the small phase The propagation lengthLof SPPs then can be given by[2]

Figure 2.Dispersion relation of SMPPs for different external magnetic fields with ωc/ωp=0(green solid line),ωc/ωp=1(red solid line),and ωc/ωp=2(blue solid line),respectively.Other parameters are ε2=1 for air,d=0.01 mm,and ν=0.

In the next section,we numerically investigate the normal and absorption dispersion relations of SMPPs in the symmetrical waveguide by applying equations(7) and(8).Further,we also discuss the propagation length of the SMPPs by using equation(9).

3.Results and discussion

It is worth emphasizing that the structure considered in the present study is symmetrical,resulting in the dispersion relations of SMPPs excited by the structure being highly symmetrical [49].Therefore,we here only consider the forward-propagating mode,i.e.qr(qi)＞0.In this section,we present and discuss the results of our numerical calculations.We study how the external magnetic field,collision frequency of 2DEG,dielectric constant and the thickness of the background dielectric film,affect the propagation of SMPPs.Considering that the frequency range we are studying is in the GHz range,the electron density of the 2DEG is approximately in the range of 1016-1018m-3,which can be realized in experiments.Other quantities,such as the frequency of incident electromagnetic waveωand the applied magnetic fieldωc,are normalized byωpin our calculation.

We first investigate the influences of the applied magnetic field on the features of the dispersion relation of SMPPs without considering any collisions in plasma(i.e.ν=0).One can find that there is only one dispersion curve of SMPPs without the external magnetic field,seeing the case ofωc/ωp=0 in figure 2.However,two dispersion curves of SMPPs appear when the external magnetic field is introduced,see the cases ofωc/ωp=1 andωc/ωp=2 in figure 2.It is obvious to see that one of the bands,which we will refer to as the lower and higher bands in the following,appears below the bulk plasmon frequency and the other one,which emerges above it.Moreover,we find that the frequency of SMPPs significantly increases along with the increase of the external magnetic field,which is consistent with the high-frequency characteristics of Voigt’s geometry [46,50].Besides,we further find that the dispersion curve of SPPs in the lower band moves downward as a whole while the dispersion curve of SMPPs in the higher band moves upward when increasing the applied magnetic field.The results mean that the penetration ability of SMPPs decreases along with the increasing external magnetic field,which also means that the energy loss decreases correspondingly.Moreover,the plasmon frequency in both lower and higher bands can be tuned by the external magnetic field.

We then study the effect of collision frequency of the 2DEG on the properties of SMPPs,as displayed in figure 3.The normal and absorption dispersion relations of SMPPs are depicted in figures 3(a)and(b),respectively.As the collision frequency of the 2DEG increases,we can observe from figure 3(a) that the dispersion relation of SMPPs gradually moves to the left and gets closer to the light line,indicating that only small wave vectors may excite SMPPs at the same frequency.This result implies that the binding force of charge to SMPPs is weakened.We also find that the higher band disappears when the collision frequency of the 2DEG is high,seeing the case ofν/ωp=0.4 in figure 3(a).Therefore,a stronger external magnetic field is required to show the highfrequency characteristics of SPPs when the collision frequency of the 2DEG is high.From figure 3(b),we can find that the propagation lengthLof SMPPs in the lower band is longer than that in the higher band when the collision frequency of the 2DEG is low,seeing the case ofν/ωp=0.02 in figure 3(b).Moreover,we find that the SMPPs in the lower band lose energy while the SMPPs in the higher band gain energy when the collision frequency of the 2DEG continues to increase and approaches the resonance frequency.Thus,the propagation length of SMPPs in the higher band is greater than that in the lower band.Therefore,one can naturally conclude that tuning the collision frequency of the 2DEG can control the propagation length of SMPPs.

We then turn our attention to the impact of the dielectric constant of the background dielectric film on the features of SMPPs.Figures 4(a) and(b) are plotted for the normal and absorption dispersion relations of SMPPs,respectively.It is worth noting from figure 4(a) that increasing the dielectric constant of the background dielectric film at a lower plasmon frequency region can make the dispersion curve of SMPPs in the higher band disappear,which indicates that the influences of the dielectric constant of background dielectric film are greater than the impact of collision frequency of plasma on the properties of SMPPs.It can be seen from figure 4(b) that the increase in the dielectric constant of background dielectric film makes the propagation length of SMPPs in the highfrequency region shorter.Moreover,increasing the dielectric constant of the background dielectric film makes the dispersion curve of SMPPs in the low-frequency region move downward,and the propagation length of SMPPs first decreases rapidly and then increases slowly.Comparing with the results of our previous work [46],we find that theplasmon in the symmetric waveguide is more stable than that in the asymmetric waveguide.We can easily conclude that the dielectric constant of the background dielectric film plays a crucial role in controlling the propagation of SMPPs in the waveguide structures.

Figure 3.(a) Normal dispersion relation and(b) absorption dispersion relation of SPPs for different plasma collision frequencies with ν/ωp=0.02(green solid line),ν/ωp=0.05(red solid line)and ν/ωp=0.1(blue solid line),respectively.Other parameters are d=0.01 mm,ωc/ωp=1 and ε2=1 for air.

Figure 4.(a) Normal dispersion relation and(b) absorption dispersion relation of SMPPs for the different dielectric constants of the background film with ε2=1(green solid line)for air,ε2=3.9(red solid line)for PbO2,and ε2=11.9(blue solid line)for Si,respectively.Other parameters are d=0.01 mm,ωc/ωp=1,and ωc/ωp=0.02.

Figure 5.(a)Normal dispersion relation and(b)absorption dispersion relation of SMPPs for different thicknesses of the dielectric film with d=0.01 mm(green solid line),d=0.02 mm(red solid line),and d=0.03 mm(blue solid line),respectively.Other parameters are ε2=1 for air,ωc/ωp=1,and ν=0.02.

Finally,we further explore the effect of thicknesses of background dielectric film on the characteristics of SMPPs,as plotted in figure 5.It is clear that a complete dispersion curve holds in both lower and higher bands with the increase of the thickness of the background dielectric film,as displayed in figure 5(a).We also can find that the external magnetic field has a higher degree of effect on the properties of SMPPs in the high-frequency regions for the symmetrical waveguide structure.Moreover,it does not disappear with the increase of the background dielectric film thickness.From figure 5(b),one can observe that the propagation length of SMPPs becomes longer by increasing the thickness of the background dielectric film.For the same thickness of background dielectric film,the loss for high-frequency electromagnetic waves in conductors is usually greater than that for low-frequency electromagnetic waves due to the strong penetration ability and shorter wavelength for high-frequency electromagnetic waves.Therefore,the propagation length of SMPPs for low-frequency regions is longer than that for high-frequency regions.Combining with our previous work [46],we can show that increasing the thickness of the background dielectric film can make the SMPPs more easily excited and can propagate longer distances,whether the structures are symmetrical or asymmetrical.

Remarkably,we have taken note of recent works[51-54]which have utilized hydrodynamic models to study SMPPs excited by structures similar to that constructed in this work.Additionally,some studies have also investigated quantum effects in the high-density 2DEG within the structure[38,55].Further work and progress are needed on this matter.

4.Conclusion

In summary,we have proposed a symmetric structure to explore the features of SMPPs.The multilayered configuration is a symmetrical waveguide constructed of two semiinfinite high-density 2DEG separated by a thin film in Voigt configuration.We have derived the normal and absorption dispersion relations of SMPPs under TM polarization.We uncover that the applied magnetic field,collision frequency of the 2DEG,the dielectric constant and the thickness of the background film all have significant impacts on the propagation of SMPPs,indicating that we can control the locations and propagation lengths of SMPPs consequently.Due to the high symmetry of the physical geometry,we demonstrate that no external elements can alter the symmetry of the dispersion relations of SMPPs.We reveal that SMPPs can propagate further since their ability to penetrate the medium is diminished and their energy loss is reduced when the external magnetic field increases.Moreover,we demonstrate that the lifetime of the plasmon in the symmetric structure is greater than that of the plasmon in the asymmetric structure.Our study provides valuable insights into the behavior of SPPs in the symmetric structures with 2DEG and opens up new possibilities for the design of plasmonic devices with desired properties.

Acknowledgments

This work is supported by National Natural Science Foundation of China(No.11975175).

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