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Chelyabinsk Physical and Mathematical Journal. 2021. Vol. 6, iss. 3. P. 375-383.
DOI: 10.47475/2500-0101-2021-16311
EXCITATION OF SURFACE PLASMON-POLARITONS IN HYBRID GRAPHENE METASURFACE — VANADIUM DIOXIDE NANOSTRUCTURE USING PRISM COUPLING
M.O. Usik1, O.G. Kharitonova1, D.A. Kuzmin1'2'", I.V. Bychkov1'2, V.A. Tolkachev1, V.G. Shavrov3, V.V. Temnov4
1 Chelyabinsk State University, Chelyabinsk, Russia
2South Ural State University (National Research University), Chelyabinsk, Russia 3Kotelnikov Institute of Radio-Engineering and Electronics of RAS, Moscow, Russia 5Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau, France " kuzminda@csu.ru
We have investigated the attenuated total internal reflection in hybrid graphene metasurface — vanadium dioxide nanostructure. This effect is caused by surface plasmon-polaritons excitation in the structure. Phase transition in vanadium dioxide leads to crucial change in it's optical plasmonic properties of the entire structure. We believe that the studied structure with vanadium dioxide as one of the layers can serve as the basis for future self-adjusting structures, which can change their parameters during it's operation.
Keywords: graphene, hyperbolic metasurface, surface plasmon-polaritons, vanadium dioxide.
Introduction
In recent years, layered materials with an optical response have developed enormously. Special attention is paid to structures whose electromagnetic properties are not observed in ordinary materials, but are manifested only in artificially created metamaterials and metasurfaces. Among them, nanomaterials and nanostructures capable of supporting plasmon modes occupy a special place. Such structures include graphene-based metasurfaces and other graphene-based derived structures. It is also worth noting that with certain parameters of the graphene lattice, it is possible to obtain a hyperbolic metasurface, whose properties are significantly different from those of ordinary elliptical metasurfaces, which is of even greater interest to scientists [1-3]. Most authors, considering multilayer structures with graphene layers, most often limit themselves only to simple dielectric media with unchanged characteristics [4; 5]. However, the structures whose parameters may change over time are currently insufficiently studied. Changes in the main characteristics can occur for various reasons, for example, when an external magnetic field is applied or during the phase transition of one or more layers.
In this paper, we theoretically considered the excitation of surface plasmon-polaritons (SPPs) in the vanadium dioxide — Silicon dioxide — hyperbolic metasurface structure (Fig. 1). Vanadium dioxide was chosen as a layer with variable characteristics, since its phase transition occurs at sufficiently low (almost room) temperatures, which has long
The work performed in part under financial support of Russian Foundation for Basic Researches (grants 19-07-00246, 20-07-00466, 20-47-740004, 20-37-70038), numerical calculations were performed with the support of the Ministry of Science and Higher Education of the Russian Federation within the framework of the Russian State Assignment under contract no. 075-00250-20-03.
A
Graphene lattice
Fig. 1. Schematic model of SPPs excitation by attenuated total internal reflection method (A) and graphene-based hyperbolic metasurface (B)
been of interest in the scientific community. The results of this work may pave the way for new methods of nanoscale light control.
1. Theory
For experimental observation of propagating SPPs the number of methods may be used, for example grating structure [6] and metasurfaces [7]. Attenuated total reflection method (in Otto configuration) has been recently used for experimental observation of transverse electric polaritonic modes in graphene at infrared frequencies [8]. The Otto configuration of SPPs excitation method is based on the principle of attenuated total internal reflection. When the incidence angle of exciting wave is large d > 9cr (dcr is a critical angle for total internal reflection condition), photons from semi-infinite prisms are tunneling through layers of vanadium dioxide and dielectric and may couple to plasmons on graphene (see Fig. 1A) [4; 5; 9].
We will suppose that the Si-based prism (dielectric constant is ep = 12) is separated from graphene by the vanadium dioxide gap (dielectric constant is eVO2) with thickness tl and the silicon gap (dielectric constant is es = 4) with thickness t2. Electromagnetic wave incident from the prism under incidence angle is d.
A graphene-based hyperbolic metasurface was chosen as the surface on which plasmons are excited. Such a metasurface has a lot of unique properties, such as the radiation pattern of a point dipole placed on such a surface, the hyperbolic appearance of the isofrequency contour, and the Purcell effect [10; 11]. This metasurface can be obtained using a lattice of graphene strips (Fig. 1B). The conductivity tensor of such a surface has the following form [12; 13]:
The components of such a tensor can be calculated using the following formulas:
LaG ao
WaC + GaG '
L
where G is the separation distance between two consecutive strips, aG is the graphene conductivity and aC is an effective conductivity related to the near-field coupling between adjacent strips, obtained using an electrostatic approach. The conductivity
of homogeneous graphene at a frequency u follows:
aG(u) = aintra(u) + CTinter(u),
2nf, where f is a linear frequency, is as
0intra (u)
2ie2kB T
nK(u>+ir
ln
2 cosh
0inter(u) = Is
1 + 1 arctan
2 n
2kB T /
— ln ■
2k b T
) 2n ^ (Sw-2^ch)2+(2kBT)2_
where T is a given temperature, is a graphene chemical potential, ointra(u) corresponds to the intraband electron-phonon scattering process, and ointer(u) meets the direct interband electron transitions and is leading about the absorption edge hu ~ 2^ch [14]. An effective conductivity oq has the following form [11]:
Oq (u) = -iw£0£eff[ l j ln
(nG csc — I 2L
where £eff is the effective permittivity of the media that embed the ribbons.
In order to investigate the reflection of electromagnetic wave from the structure, one should solve Maxwell's equations with the corresponding boundary conditions at each interface. For monochromatic wave Ea,±, Ha,± ~ exp[-iut + ika,±r], where u is a
a,-.
angular frequency, and ka± = (kx, ky, ±ka,z) is a wavevector (sign " + " corresponds to the wave propagating along the z-axis, while " — " corresponds to the counter-propagating
,d, a denote
"prism",
wave, a = p,VO2 consequently).
Solving Maxwell's equations in each
medium and taking into account the
boundary conditions of continuity of the
tangential components of fields Ea,±, Ha,±
at each boundary, we may obtain the
system of linear equations for amplitudes
of transmitted and reflected waves in each
medium. For given excitation parameters
(frequency, incident angle, polarization of
incident wave, deformation and chemical
potential of graphene) one may calculate
|Ep-|2
"vanadium dioxide", "dielectric" and
"air",
the reflection coefficient R
|Ep+|2
of the
incident electromagnetic wave from this structure, which indicates what part of the incident wave energy passed into the SPPs excitation.
We should note that in attenuated total internal reflection configuration the electromagnetic wave in vanadium dioxide and air is decaying in z-direction.
In vanadium dioxide, a metal-semiconductor phase transition occurs at temperatures close to room temperature. From the point of view of band theory, VO2 has a partially filled d-shell. The electronic spectrum contains a gap (Mott — Hubbard gap), the width of which depends on various external conditions. This enables performing a phase transition due to changes in temperature, electric field (current
Fig. 2. Linear approximation of experimental datas [18] of temperature dependence of Drude model parameters for VO2
or voltage), incident electromagnetic wave of a certain frequency, elastic stresses, etc. [15-17]. In this paper, we consider the temperature dependence of the phase state of vanadium dioxide. The phase transition of vanadium dioxide from the dielectric monoclinic state to the metallic rutile phase is accompanied by a large and rapid change in the electrical and optical properties [18]. Some of the variable parameters have already been measured over the entire temperature section of the phase transition [19]. Taking this data and applying a linear approximation, we can divide the entire phase transition into 4 sections (see Fig. 2): 1) before the phase transition (T < 336 K); 2) during the phase transition (336 K < T < 347 K); 3) at the end of the phase transition (347 K< T < 352 K); 4) after the phase transition (T>352 K). In this paper, we will not delve much into the behavior of surface plasmons at temperatures degrees below 336 K and above degrees 352 K, since the parameters of vanadium dioxide do not change at these temperatures and its influence remains unchanged at these cases. Temperature ranges 2 and 3 are of particular interest, since they are the cases where dramatic changes in the optical properties of vanadium dioxide occur and its influence on the entire structure changes.
Using the Drude model it is possible to take into account the influence of the optical properties of vanadium during the phase transition [20; 21]:
£vo2 = eo - ^p/M^ + ir-1)),
where up and t are plasma frequency and frequency of electron collisions in the VO2 film, consequently.
2. Results
Despite the fact that the graphene-based hyperbolic metasurface is capable of supporting both TM- and TE-polarized plasmons, in this paper we have limited ourselves to the TM-mode only. All over the calculations we suppose the following parameters:
= 0.5 eV, t1=t2=25 nm, G=35 nm, W=15 nm, L = G + W=50 nm. In order to find out at what parameters of the incident radiation (the angle of incidence and the angle of rotation of the metasurface plane) the plasmons manifest themselves most clearly, a color map of the reflection coefficient in the absence of vanadium dioxide was constructed (silicon took its place). The parameters were searched for at a frequency f=0.5 THz. From the results obtained, it is clear that the strongest excitation of surface plasmons occurs at the angle of incidence в « n/4 and the angle of rotation a « n/4 (see Fig. 3). Also, from the obtained color map, it is possible to fairly accurately distinguish the critical angle of the total internal reflection, up to which there is no transition of the incident electromagnetic radiation into the excitation of surface plasmons (всг « 17deg).
Having found out the most suitable angles of incidence and rotation, we calculated the reflection coefficients for different temperatures of the structure (see Fig. 4). The figure shows that at the dielectric phase of vanadium dioxide and at the beginning of its phase transition, its influence is practically absent and the excitation of surface plasmons occurs in a narrow low-frequency range. Then, as the temperature increases, the influence of vanadium dioxide begins to appear. This is expressed in the expansion of the frequency range of excitation of surface plasmons and its shift to higher temperatures.
At the end of the phase transition and at the complete transition of vanadium dioxide to the metal phase, the influence of VO2 on the structure becomes very strong and the absorption of the electromagnetic wave occurs not due to the excitation of surface plasmons on the graphene metasurface, but due to the absorption of part of the wave by the vanadium dioxide layer.
Fig. 3.
that
Color map of the reflection coefficient in the absence of vanadium dioxide shows plasmons manifest themselves most clearly at the angle of incidence 6 « n/4 and the angle of rotation a « n/4
Fig. 4. Reflection coefficients for different temperatures of the structure
Conclusions
In this paper, we investigated the excitation pattern of surface plasmons in multilayer structures with layers subject to phase transition. We believe that the studied structure with vanadium dioxide as one of the layers can serve as the basis for future self-adjusting structures, that is, structures that can independently change their parameters during operation. We assume that when surface plasmons are excited on the graphene metasurface, the heat generated will be able to heat the vanadium dioxide layer sufficiently that it can begin to change its initial parameters. Then, when its influence is strong enough that the surface plasmons actually cease to be excited, the temperature of the entire system will begin to decrease again, vanadium dioxide will enter its initial phase, and the surface plasmons will reappear on the graphene metasurface. Our research can help to study the behavior of surface plasmons in structures with variable parameters, as well as open a new path in the study of structures whose parameters can be independently adjusted and changed.
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Article received 15.08.2021
Corrections received 31.08.2021
Челябинский физико-математический журнал. 2021. Т. 6, вып. 3. С. 375-383.
УДК 537.634 DOI: 10.47475/2500-0101-2021-16311
ВОЗБУЖДЕНИЕ ПОВЕРХНОСТНЫХ ПЛАЗМОН-ПОЛЯРИТОНОВ В ГИБРИДНОЙ НАНОСТРУКТУРЕ ГРАФЕНОВАЯ ПОВЕРХНОСТЬ — ДИОКСИД ВАНАДИЯ ПРИ ПОМОЩИ ПРИЗМЕННОГО ВВОДА
М. О. Усик1, О. Г. Харитонова1, Д. А. Кузьмин1'2", И. В. Бычков12, В. А. Толкачев1, В. Г. Шавров3, В. В. Темнов4
1 Челябинский государственный университет, Челябинск, Россия 2Южно-Уральский государственный университет, Челябинск, Россия 3Институт радиотехники и электроники им. В. А. Котельникова РАН, Москва, Россия 4Политехническая школа, Парижский политехнический институт, Палезо, Франция " kuzminda@csu.ru
Исследован эффект нарушенного полного внутреннего отражения в гибридной наноструктуре графеновая метаповерхность — диоксид ванадия. Этот эффект вызван возбуждением в структуре поверхностных плазмон-поляритонов. Фазовый переход в диоксиде ванадия приводит к кардинальному изменению его оптических свойств, а также плазмонных свойств всей структуры. Мы полагаем, что исследуемая структура с диоксидом ванадия в качестве одного из слоёв может служить основой для будущих саморегулирующихся структур, которые могут изменять свои параметры в процессе эксплуатации.
Keywords: графен, гиперболические метаповерхности, поверхностные плазмон-поляритоны, диоксид ванадия.
Поступила в редакцию 15.08.2021. После переработки 31.08.2021.
Сведения об авторах
Усик Максим Олегович, аспирант физического факультета, Челябинский государственный университет, Челябинск, Россия.
Харитонова Ольга Глебовна, аспирант физического факультета, Челябинский государственный университет, Челябинск, Россия.
Кузьмин Дмитрий Александрович, кандидат физико-математических наук, доцент кафедры радиофизики и электроники, Челябинский государственный университет; научный сотрудник лаборатории функциональных материалов, Южно-Уральский государственный университет, Челябинск, Россия.
Бычков Игорь Валерьевич, доктор физико-математических наук, профессор, профессор кафедры радиофизики и электроники, Челябинский государственный университет; научный сотрудник лаборатории функциональных материалов, Южно-Уральский государственный университет, Челябинск, Россия.
Толкачев Валентин Андреевич, преподаватель кафедры радиофизики и электроники, Челябинский государственный университет, Челябинск, Россия.
Работа выполнена при частичной финансовой поддержке Российского фонда фундаментальных исследований (гранты 19-07-00246, 20-07-00466, 20-47-740004, 20-37-70038). Численные рас-счёты осуществлены при поддержке Министерства науки и высшего образования РФ в рамках государственного задания, договор № 075-00250-20-03.
Ш^авров Владимир Григорьевич, доктор физико-математических наук, заведующий лабораторией магнитных явлений в микроэлектронике Института радиотехники и электроники им. В. А. Котельникова РАН, Москва, Россия.
Темнов Василий Владимирович, доктор физико-математических наук, профессор, Политехническая школа, Парижский политехнический институт, Палезо, Франция.
