Optics Communications 459 (2020) 124919

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Optics Communications

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Actively tunable toroidal excitations in graphene based terahertzmetamaterialsAngana Bhattacharya a,∗, Koijam Monika Devi a, Tho Nguyen b, Gagan Kumar a

a Department of Physics, Indian Institute of Technology Guwahati, Guwahati 781039, Assam, Indiab Department of Physics and Astronomy, Franklin College of Arts and Sciences, University of Georgia, USA

A R T I C L E I N F O

Keywords:MetamaterialToroidal resonanceTerahertzGraphene

A B S T R A C T

Toroidal dipoles, unlike electric and magnetic dipoles, are formed due to currents flowing on the surfaceof a torus along it’s meridians. We investigate the toroidal dipole response in a graphene based infinite 2Darray of terahertz (THz) metamaterial (MM) and report the active tuning of the toroidal resonances. Eachmetamolecule in our proposed MM configuration consists of two joint split ring resonators (SRRs) made upof graphene material with two gaps in each resonator. We examine actively tunable response of the toroidalexcitations with respect to a change in the Fermi energy of graphene material. The toroidal effect is elaboratedusing surface current and magnetic field profiles at the resonance frequency of the metamolecule structure. Themodulation of the metamaterial’s toroidal response and effect on Q factor is further examined by changing therelaxation time of graphene. The excitation of toroidal resonances at terahertz frequencies and their dynamiccontrol could be significant in designing efficient tunable modulators, filters and sensors.

1. Introduction

Metamaterials (MMs) are artificially designed structures with theunit cell being lesser than the wavelength of incident light. The prop-erties of these sub-wavelength structures can be tailored by the mod-ification of the geometry of the metamolecules. By controlling theresponse and symmetry of the structures, several novel phenomenacould be accomplished which are not found in naturally existing mate-rials e.g. negative refractive index, superlensing, cloaking, etc. [1–4].Such phenomena and related applications make use of the electric andmagnetic moments exhibited by these materials. However, recentlyattention has been drawn to a new kind of moment known as toroidaldipole moment. Toroidal dipoles were first introduced by Zel’dovichin 1957 in the context of nuclear and particle physics [5]. They arethe most primitive member of the toroidal multipole family [6]. Usu-ally, a toroidal dipole can be explained by currents flowing along themeridians of a torus on it’s surface. Because of each current loop wemay envision a magnetic moment. The number of current loops onthe meridians of the torus results in a number of magnetic momentsarranged in a loop. It can thus be visualized as magnetic momentsarranged head to tail along a loop. The direction of the toroidaldipole can then be stated as pointing outward along the symmetryaxis [6,7]. The interaction energy of these toroidal dipoles dependson the time derivative of electromagnetic fields unlike the electricand magnetic multipoles whose energy depend on the fields. Thesetoroidal moments also happen to be suppressed in most materials

∗ Corresponding author.E-mail address: angana18@iitg.ac.in (A. Bhattacharya).

found in nature and hence it becomes difficult to observe toroidaleffect. However, in recent times, artificially engineered metamaterialstructures have led to the observation of such effects through carefuldesigning of its constituents. Studies suggest that toroidal excitationscan reduce radiative losses resulting from electric dipole radiation andenhance device performance [8,9]. Most of the investigations till dateon toroidal response have been focused in exploring the characteristicfeature of these toroidal resonances for applications in negative index ofrefraction, cross polarization conversion, backward wave propagationetc. [10,11].

Toroidal excitation in metamaterials have found applications inseveral areas ranging from biosciences to photonics. A wide varietyof study has been made on these excitations [12,13]. Gupta et al.have used high Q toroidal excitations in planar MMs for refractiveindex sensing of analytes [14]. Ahmadivand et al. have recently re-ported the generation of deep ultraviolet light based upon enhancedthird harmonic generation using a toroidal-resonant plasmonic meta-surface [15]. In another study, they have used high Q factor of toroidalresonances to detect Zika-virus (ZIKV) envelope protein using a spe-cific ZIKV antibody [16]. Toroidal resonances have also been usedfor the photodetection applications [17]. Toroidal resonant plasmonicmetamaterials has been utilized to develop an ultrasensitive label-free analytical platform for the detection of specific antibiotics in themid-infrared spectrum [18].

In terahertz regime, Gerislioglu et al. recently developed toroidalmeta-modulator with high efficiency and tunability which has potential

https://doi.org/10.1016/j.optcom.2019.124919Received 22 August 2019; Received in revised form 27 October 2019; Accepted 7 November 2019Available online 14 November 20190030-4018/© 2019 Elsevier B.V. All rights reserved.

A. Bhattacharya, K.M. Devi, T. Nguyen et al. Optics Communications 459 (2020) 124919

in communication and filtering applications [19]. In this direction,the active modulation and tuning of toroidal resonances by gating thegraphene monolayer has also been reported [20]. Ahmadivand et al.have reported the use of plasmonic toroidal terahertz resonances inimmunobiosensing [21]. The switching of toroidal resonances and theirultrafast response can be advantageous to design and fabrication ofultrafast photonic components. Recently, Toroidal Metamaterial Switchhas been reported by introducing ultrathin silicon layer beneath thecapacitive gap of planar toroidal resonator [22]. An active response isachieved by generation extra carriers in the silicon layer. The switch-ing response depends upon the carrier response time of the siliconmaterial. Due to similarity in sharp resonances, comparison betweenFano and toroidal resonances have been studied [23,24]. The switch-ing behaviour in structures exhibiting Fano-resonances has also beenwidely pursued. Mun et al. has demonstrated switching between dipoleand Fano resonances without changing the structure. The switching isachieved by the interference of magnetic fields [25]. Plasmonic fanoswitching has been studied as well [26]. In contrast, our focus is toachieve ultrafast switching control of resonances with the MM structuremade up of graphene material.

In 2010, Kaelberer et al. showed direct experimental evidence oftoroidal response in MMs which could not be attributed to electric ormagnetic dipolar response [7]. This toroidal response was also studiedby Marinov et al. in a 3D array of toroidal solenoids which showedsignificant measurable toroidal response in contrast to natural materialswhere toroidal response is hardly observed [10]. A recent theoreticalstudy has also proposed a special class of all dielectric MM capableof supporting toroidal dipolar response in the terahertz region [27].Analytical approach for calculating the transmissivity and reflectivityof thin slabs of materials exhibiting toroidal dipolar excitations has alsobeen developed. Such approach has been able to successfully establisha link between microscopic toroidal excitation and macroscopic scatter-ing characteristics of a material [27]. Strong toroidal dipole responseshave also been experimentally demonstrated in different 2D planar MMstructures [10,11,27–29]. The geometry allows to suppress other multi-poles, thereby allowing toroidal dipolar response to dominantly appearin terahertz region [30]. In this context, Gupta et al. have designed a 2Dplanar MM array supporting toroidal excitation. Since with reductionin size of meta-atom, achieving a 3D structure becomes challenging,the design of the 2D structure is a significant achievement. In spite ofnumerous theoretical and experimental work, most of the studies ontoroidal MMs have been focused only on the passive tuning of thesetoroidal responses by tailoring the metamaterial design. However, anactive tuning of the toroidal response enhances the efficiency of theMM structure in several applications where one needs to have a controlover the response in a desirable manner. In this context, recently,active tuning of MM structures using graphene has garnered a lot ofattention [31–37] due to Graphene’s capability in actively tuning ofresonances through application of external biased voltage. However,active tuning of toroidal resonance in graphene has not been in muchfocus [34,38,39]. The active tuning of the toroidal responses usinggraphene-based MMs can be of immense significance in sensing andother applications. Therefore, there is ample scope to explore the activetuning capability of toroidal response in graphene based MM structuresin order to realize the potential applications.

In this article, we report toroidal dipolar excitations in a planar2D MM structure comprising an array of two joint resonators (SRRs)made of graphene with two split gaps in each resonator. The activetuning and modulation of the MM’s toroidal response is examined byvarying the Fermi energy and relaxation time of the graphene material.The surface current and magnetic field profiles of the MM structureis studied to understand the toroidal effect. The paper is organized asfollows: In Section 2, we have discussed the design of graphene basedplanar metamaterial geometry that results in the toroidal excitationat terahertz frequency. In the next section, we discuss active tuningof toroidal dipole excitations and presents a comprehensive pictureof their dependence on graphene Fermi energy and relaxation time.Then a study on the quality factor of the MM is made. The results aresummarized in the conclusion section.

2. Metamaterial design and numerical simulations

The excitation of toroidal resonances in metamaterials require anoptimized design of the meta-molecule unit cell. Each unit cell ofour proposed MM structure consists of two split ring resonators, eachhaving two gaps, which have been joined together to form a singleresonator. The schematic of the proposed metamaterial array alongwith the transmission characteristics is illustrated in Fig. 1(a). In theproposed structure, the split ring resonators are made of graphenematerial while the substrate is made of intrinsic silicon having athickness of 10 μm. The periodicity of the structure in x and y directionis taken as 𝑝𝑥 = 24 μm, 𝑝𝑦 = 12 μm while the length and breadthof a single resonator is taken as 𝑙 = 18 μm, 𝑏 = 8 μm, respectively.The width ‘w’ of each strip of resonator is 1 μm and the capacitivegap ‘g’ of each resonator is considered as 1 μm. The distance of thecentral position of each gap from their respective side arms is fixed at4.5 μm. The numerical simulations are performed using the techniqueof finite element frequency domain solver in CST Microwave Studio.The metamaterial geometry is simulated under the unit cell boundaryconditions in the x–y plane. We set open boundary conditions along thedirection of light propagation and chose a mesh size of the order of 𝜆

10 ,where 𝜆 is the wavelength of the incident radiation. One may fabricateproposed graphene structures on a silicon carbide (SiC) substrate byfirst growing graphene using chemical vapour deposition (CVD), andthen printing and lithography followed by appropriate post processingtechniques. Several studies have been reported in fabricating highresolution graphene structures in recent time. [40–42] Fig. 1(b) showsa plot of the transmittance profile of the MM structure at normalincidence versus frequency for the Fermi energy 𝐸𝐹 = 1 eV of thegraphene layer. It is evident from the figure that two dips are observedin the transmittance profile, where the first dip occurs at frequency0.98 THz and the second at 1.33 THz. In the transmittance profile, thefirst dip at frequency 0.98 THz indicates a toroidal excitation, whereassecond dip correspond to dipolar resonance. In our work, we will focusour study only on toroidal excitation.

In order to confirm the toroidal behavior of the resonance, weexamined surface current profile and magnetic field profile of the MMstructure in Fig. 2(a) and (b) respectively. It is apparent from Fig. 2(a)that the flow of current in the two resonators are in the oppositedirections i.e. it flows clock-wise in the right hand side split ringresonator while it flows anti-clock wise in the left hand side resonator.This results in magnetic moment formation, going into the plane andout of the plane in the right and left resonators respectively. End toend formation of magnetic moment indicates toroidal excitation. Themagnetic moments formed as a result of the ring like arrangementof our proposed meta-material structure helps in strengthening of thetoroidal moments [43]. Fig. 2(b) shows the magnetic field profile inthe xz plane of the MM structure. The circular magnetic field is moreprominent in the central arm, as for this configuration, there happensto be more current flowing through the central arm than the other twoarms. This leads to the development of toroidal moment in the MMstructure.

3. Active control and modulation of toroidal resonances

Graphene has an important advantage that it’s Fermi level can bemanipulated by the application of a gate voltage,chemical doping orby photo induced doping. Graphene integrated MM structures couldbe advantageous in making active devices as it’s Fermi energy can betuned. [31,44,45]. To explore the active tuning capability of toroidalresonances in the proposed MM structure, we perform numerical simu-lations for different Fermi energy 𝐸𝐹 . The transmittance characteristicsof the proposed MM structure are examined by varying 𝐸𝐹 from 0.2eV to 1 eV for a fixed relaxation time. These chosen values of Fermienergies have been experimentally reported in previous works [31,45,46]. The transmittance profile along with its corresponding contour

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Fig. 1. (a) Schematic of the proposed terahertz metamaterial structure resulting in toroidal excitation. Silicon substrate is shown in blue color, whereas graphene split rings areshown with gray color. The inset shows a larger view of the metamaterial unit. The unit cell consists of two split ring resonators, each having two gaps, which have been joinedtogether to form a single resonator. 𝑝𝑥 and 𝑝𝑦 represent periodicities of the unit cell in the x and y directions, respectively. 𝑙 and 𝑏 stands for length and breadth the unit cell.Width and split gap of the resonator is represented by 𝑤 and 𝑔. The configuration allows us to excite toroidal resonance in the 𝑦-direction, shown by 𝑇 . (b) Transmittance versusfrequency plot of the MM structure resulting in toroidal excitation for 𝐸𝐹 = 1 eV of graphene. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

Fig. 2. (a) Surface current profile in the graphene metamolecule structure having 𝐸𝐹 = 1 eV. The current flows in clockwise direction in the right half of the resonator, whereasit flows in the anti-clockwise direction in the left half as indicated by black arrows. The current directions result in the formation of end to end magnetic dipoles, thus leadingto a toroidal response (b) Magnetic field profile at 𝑓 = 1 THz in the xz plane of the metamaterial unit cell structure indicating toroidal behaviour due to oppositely circulatingcurrents.

Fig. 3. (a) Transmittance versus frequency for increasing values of Fermi energy from 0.2 eV (in black) to 1 eV (in green). There is an evident blue shift in the toroidal resonanceon increasing Fermi energy. (b) Color and contour plot for Fermi energy versus frequency indicating a shift in the toroidal resonance. (c) Variation of toroidal frequency versusFermi energy of graphene layer for 𝜏 = 2 ps. The frequency increases with Fermi energy as 𝜔 ∝ 𝐸𝐹

12 . (For interpretation of the references to color in this figure legend, the reader

is referred to the web version of this article.)

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A. Bhattacharya, K.M. Devi, T. Nguyen et al. Optics Communications 459 (2020) 124919

Fig. 4. Surface current and magnetic field profiles for symmetric metamaterial structure with increasing visibility of toroidal resonance with increase of Fermi energy. (a), (c) and(e) represent surface current profiles corresponding to Fermi energies 0.2 eV, 0.6 eV and 1.0 eV respectively, while (b), (d) and (f) represent respective magnetic field profiles.Oppositely flowing currents in the arms for Fermi 0.6 eV and 1.0 eV at fixed phase angle leads to toroidal response.

plot is depicted in Fig. 3. Fig. 3(a) represents the line plot of thetransmission characteristics of the proposed MM structure. The firstdip in the figure indicates the toroidal excitation. The black solid linedepicts transmittance for Fermi energy 0.2 eV while the orange lineshows transmittance for Fermi energy 0.4 eV. The transmittance forFermi energy 0.6 eV is represented by the yellow solid line. The pinktrace signify the transmittance for 𝐸𝐹 = 0.8 eV and the green linecorresponds to Fermi energy 1 eV. It is evident from the figure thatthere is significant shift in the toroidal resonance as the Fermi energyis increased. This can be further understood from the contour plotshown in Fig. 3(b). It is clearly observed from the contour plot, thatthe frequency of the toroidal resonance exhibits a blue shift with theincrease of the value of Fermi energy of the graphene layer as depictedby the dotted line. The blue shift in the resonance with increasing FermiEnergy can be explained using the formula for carrier concentration,

𝑛 = 1𝜋(𝐸𝐹ℏ𝜈𝐹

)2 (1)

As mentioned in [44], the universal relation between 𝜔 and |𝐸𝐹 |

is described by a scaling of 𝜔 ∝ 𝐸𝐹12 ∝ 𝑛

14 where 𝜔 is the frequency,

𝐸𝐹 the Fermi energy and n is the carrier concentration. On taking asquare root of the Fermi energy variation from 0.2 eV to 1 eV, usingthe above mentioned formula, it is observed that the shift in frequencyof the toroidal resonance with an increase in Fermi energy is wellaccounted. This is depicted in Fig. 3(c). The formula for Eq. (1) fits wellfor the toroidal resonance frequency with Fermi energy, as can be seenin Fig. 3(c) and therefore, it is believed that the toroidal excitationsoriginate from graphene plasmons.

To further understand the effects of the variation of Fermi energyon the toroidal response of MM structure, the surface current and themagnetic field profile is examined. Fig. 4 represents the magnetic fieldprofile along with the corresponding surface current profile for 𝐸𝐹 =0.2 eV, 0.6 eV and 1 eV for a fixed phase of 153.75 degree. Fig. 4(a)and (b) represents the surface current and the magnetic field profile for

𝐸𝐹 = 0.2 eV. It is evident from Fig. 4(a) and (b), that there is almost notoroidal resonance. As the Fermi energy is increased, a visible formationof toroidal excitation is observed. When the flow of current in the tworesonators are in the opposite direction such that it flows clock-wise inthe one resonator while it flows anti-clock wise in the other resonator,it gives rise to two magnetic monopoles joining end to end and hence atoroidal excitation is obtained. In the right resonator a magnetic dipoleforms pointing out of the plane while in the left resonator, the dipolepoints into the plane. Fig. 4(c) depicts oppositely flowing currentsin the two resonators while a visible formation of toroidal excitationcan be seen from Fig. 4(d). For Fermi energy 1 eV, it is observedfrom Fig. 4(e) and (f) that a uniform distribution of oppositely flowingcurrents occurs in the resonators. This oppositely flowing currents,in turn, leads to the formation of strong toroidal excitation in theMM structure. Therefore, it can be concluded that an effective activetuning of the toroidal resonance can be achieved by varying the Fermienergy of the graphene layer. Thus,as can be seen from the variation ofFermi energy in Fig. 3, the resonanace dips become more prominent onincreasing Fermi energy from 0.2 eV to 1 eV. These particular valueshave been so chosen as they are conventionally experimentally viable,as mentioned before.

This tuning in toroidal resonance on increasing fermi energy is con-firmed by the surface current and magnetic field profile plot in Fig. 4.We further examine the effect of change of relaxation time in graphenematerial on the toroidal excitations. Experimentally viable relaxationtimes as reported in previous works have been chosen [46,47]. Tera-hertz transmittance results for toroidal resonances versus frequency fordifferent relaxation times are shown in Fig. 5.

As the relaxation time (𝜏) is changed, we observe an amplitudemodulation of the toroidal response. In Fig. 5(a), the black line depictsthe transmittance for relaxation time 𝜏 = 1 ps. The red line depictstoroidal excitation for relaxation time 𝜏 = 1.5 ps, the blue line standsfor 𝜏 = 2 ps while the pink line depicts transmission for 𝜏 = 2.5 ps. Forthe change in relaxation time from 1 ps to 2.5 ps, it is observed that

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Fig. 5. Transmittance versus terahertz frequency for toroidal resonance for different values of relaxation times (𝜏 = 1 ps, 1.5 ps, 2 ps, 2.5 ps). The amplitude of toroidal excitationcan be modulated with 𝜏. Inset depicts the Fano fitting of the simulated data 𝜏 = 1 ps. The dotted curve corresponds to simulation, whereas blue curve represent fitting. (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the amplitude of the transmittance dip increases. As can be seen fromthe figure, significant change in amplitude of the transmittance curveis observed on changing the relaxation time. The inset of the figuredepicts the fitting of the toroidal resonance for 1 ps relaxation time witha Fano line shape function (given by Eq. (3)). This is done to evaluatethe Q factor of the resonance as has been described in the next section.The increase in transmittance dip can be understood by the dependenceof conductivity on the relaxation time. According to Drude model, theconductivity of a material can be defined as 𝜎 = 𝑛𝑒2𝜈𝐹 2𝜏∕𝐸𝐹 , where𝜏 is relaxation time, 𝐸𝐹 is Fermi Energy, and 𝜈𝐹 is Fermi velocity.An increase in 𝜏 leads to an increase in 𝜎. The imaginary part of thecomplex wave number 𝑘, is given by [48]

𝑘 ≡ 𝜔√

𝜖𝜇2([√

1 + ( 𝜎𝜖𝜔

)2 − 1])12 (2)

This implies an increase in wavenumber with an increase in 𝜎,which results in a reduced transmittance since it varies as 𝑒−𝛼𝑟, where𝛼 is an absorption coefficient (= 2𝑘). Therefore, increase in relaxationtime leads to a decrease in the amplitude of transmittance dip. A similarexplanation is valid for the increase in transmittance dip amplitudefor increasing Fermi energy. Since 𝜈𝐹 ∝ 𝐸𝐹 , this implies 𝜎 becomesproportional to Fermi Energy. As fermi energy increases, 𝜎 increases,from Eq. (1) and it is seen that k increases as well. This leads toa decrease in the transmittance and decreasing of amplitude of thetransmittance dips for Fermi energy can be explained.

4. Quality factor calculation

In order to appreciate the relevance of the toroidal excitations, wefurther calculate Q-factor of the resonances. The Quality factor (Q) isa dimensionless parameter which is used to quantify the resonator’senergy storage with respect to the energy loss. A high Q factor signifiesless damping, i.e, low rate of loss of energy as compared to the energystored. From the transmittance plots obtained for both fermi energyand relaxation time, the toroidal resonance is fitted with a fittingformula for asymmetric resonances. We calculate the Q-factor of themodes using the procedure discussed below. A number of studies havereported a fitting formula to fit the fano asymmetric resonance [49,50].To further ensure that the toroidal resonance obtained shows fanocharacteristics, the transmittance for varying relaxation time is fittedwith the lineshape fitting formula

𝑇 = |𝑎1 + 𝑖𝑎2 +𝑏

(𝜔 − 𝜔0 + 𝑖𝛾)|

2 (3)

Fig. 6. Variation of Q value with a change in relaxation time and Fermi energy. Thered curve corresponds to the variation of Q-factor with relaxation time (upper x-axis),which remains nearly constant signifying almost no change. The black curve representsthe change of Q-value versus Fermi energy for 𝜏 = 1 ps. Q value decreases with anincrease of Fermi Energy. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

where 𝑎1, 𝑎2 and b are constants, 𝜔0 is the resonant frequency and 𝛾 isthe overall damping rate. The Q value is defined as 𝜔0

2𝛾 . The Q factor

of the dipole resonance for Fermi energy 1 eV is calculated and foundto be 8.82. A decrease in the Q value is observed on increasing theFermi Energy from 0.2 eV to 1 eV. This decrease in Q factor can beattributed to the change in carrier concentration with an increase inFermi energy [51]. As expected the Q-factor decreases sharply in thebeginning and then slowly for large value of 𝐸𝐹 . However,it is observedthat a change in the relaxation time results in the Q value showing veryslight change and can be taken to be constant.

The Q factor obtained for variation in Fermi energy as well asrelaxation time is shown in Fig. 6. The red line plot for right hand yaxis and upper x axis stands for the Q factor variation with relaxationtime. The black line plot for left hand Y axis and lower X axis standsfor the Q factor variation on increase of Fermi energy. An increase inFermi energy results in a decrease in the Q factor. It is observed thata change of relaxation time does not show much effect on the quality

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A. Bhattacharya, K.M. Devi, T. Nguyen et al. Optics Communications 459 (2020) 124919

factor. The Q factor remains constant. This fact can find use in high Qfactor geometries where modulation can be achieved without changingQ values. We further calculated the dephasing time for the toroidalresonance for a given Fermi energy. The dephasing time can be definedas 2ℏ

𝛾 , with 𝛾 being the linewidth of the resonance [52]. Using thisand noting the linewidth for the resonance at Fermi energy 1 eV, thedephasing time comes out to be approximately 17.6 ps. The dephasingtime can be evaluated in a similar fashion for the other resonancescorresponding to their Fermi energy.

5. Conclusions

In conclusion, we reported a graphene based planar metamaterialgeometry capable of supporting toroidal excitations. In the MM struc-ture, the actively tunable response is obtained by changing the Fermienergy of graphene layer. It is observed that an increase in Fermi energyfrom 0.2 eV to 1 eV results in prominent toroidal excitations. It alsocauses a blue shift of the toroidal resonance in the MM structure. Thisblue shift in the resonance scales with Fermi energy and number densityn as 𝜔 ∝ 𝐸𝐹

12 ∝ 𝑛

14 . The magnetic field and surface current profiles

indicate the excitation of toroidal resonances. Further, we observedthat change in relaxation time of graphene layer leads to amplitudemodulation of the toroidal response. An increase in the relaxation timefrom 1 ps to 2.5 ps increases the amplitude of toroidal resonance ofthe MM structure. We have also analyzed the Q-factor of the toroidalresonances by fitting them with the Fano line shape formula. It isobserved that an increase in Fermi energy leads to a decrease in theQ value of the toroidal resonance. Further, a change in relaxationtime has almost null effect on the Q value. This fact can be used inconstructing terahertz devices where signal can be modulated withoutaffecting the quality factor. Such active tuning of MM using grapheneand excitation of toroidal resonances at terahertz frequencies and theirdynamic control could be significant in designing tunable terahertzdevices.

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