Appl Phys A (2010) 100: 461–466DOI 10.1007/s00339-010-5718-9

Improved TL-RLC model for terahertz circular split-ringresonators

Sheng Li · Huai-Wu Zhang · Qi-Ye Wen ·Yun-Song Xie · Dong-Bing Tian · Yuan-Xun Li

Received: 18 November 2009 / Accepted: 12 April 2010 / Published online: 7 May 2010© Springer-Verlag 2010

Abstract Three geometries of split-ring resonators (SRRs)including circular geometry and two elliptical geometrieswere fabricated by modern photolithography process. Thesamples were measured by terahertz time-domain transmis-sion spectroscopy. These spectra clearly revealed the lower-frequency LC resonances and the higher-frequency dipoleresonances; meanwhile, the coupling effect between the LCand dipole resonance had played an important role in theoverall response. Comparing the results shows that the LCresonance, dipole resonance, and the coupling effect canbe designed. A simple transmission-line (TL) RLC circuitmodel was used to help us understand this coupling be-havior and the extent of its effects. But the simple modelhas some disadvantage. For better matching TL-RLC modelresults with measured data, an improved model was intro-duced for the circular and y-direction stretched ellipticalgeometry SRRs. Other resonances and modes existing athigher frequencies except the typical LC and dipole reso-nances were revealed by the improved model.

1 Introduction

The left-handed material with simultaneous negative perme-ability and negative permittivity was theoretically predicted

S. Li · H.-W. Zhang (�) · Q.-Y. Wen · Y.-S. Xie · D.-B. Tian ·Y.-X. LiState Key Laboratory of Electronic Thin Film and IntegratedDevices, University of Electronic Science and Technology ofChina, Chengdu 610054, Sichuan, Chinae-mail: hwzhang@uestc.edu.cn

S. Lie-mail: ltriumph@163.comFax: +86-28-83201810

by Veselago [1] decades ago. In the 1999s, Pendry et al.[2] had first introduced two artificial structures (metamate-rials), split-ring resonators (SRRs) and continuous wires, torealize the theory. The first experimental materialization ofPendry’s ideas was made by Smith et al. [3] in 2000. Follow-ing its initial theoretical [1, 2] and experimental [3] intro-duction, metamaterials research has experienced significantgrowth and interests. While metamaterials promise noveldevices and appealing science over very broad frequencybands [4–7], the terahertz (THz) spectrum (0.1–10 THz,1 THz = 1012 Hz, λ = 30 µm–3 mm) represents a partic-ularly interesting region. THz frequencies cannot clearly beattributed to be either on the “electronic” side or on the “op-tics” side. The THz frequency radiation has been proven tobe a fertile region in the electromagnetic spectrum and apowerful tool in scientific research and many applications[8, 9]. Researchers are optimistic about metametrials as can-didates orf overcoming many limitations of THz technologybecause they have the feature of being dynamically tunable,leading to implementation on large scale, etc. [7].

In this paper, three kinds of split-ring resonators, includ-ing circular geometry and two elliptical geometries, wherethe radius is stretched in the x-direction, and the y-directionfor the two sets respectively, were designed and fabricatedby a radio frequency magnetron sputtering method, mod-ern lithography process and plasma etching process. Thesamples were measured by terahertz time-domain spec-troscopy (THz-TDS). The microscale structural variationsin circular split-ring resonators and the resulting impact onthe macroscale effective properties had been investigated.A simple model in the references was used to help us under-stand this coupling behavior and the extent of its effects. Forbetter matching TL-RLC model results with measured data,an improved model was proposed for further understandingthe response of the circular and y-direction stretched ellip-

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Fig. 1 Split-ring resonator unit cells for planar metamaterial samples.Shaded regions indicate metalized areas

tical geometries split-ring resonators. The improved modelalso shows that other resonances and modes exist at higherfrequencies except the typical LC and dipole resonance.

2 Fabrication processes and measurement

All of the metafilms studied in this work are based on theelectric-resonator design [7, 10, 11], because the symme-try of the split-ring resonator can eliminate or minimize themagnetic response. The studied metamaterial samples areshown in Fig. 1. All samples comprised plane periodic res-onator arrays fabricated on intrinsic silicon substrates. Ac-cording to Grischkowsky’s work [12], the intrinsic silicon(Si) with the index of refraction is excellent for THz trans-mission due to its non-dispersive nature and low absorptionin THz region. So here the material for substrate was cho-sen to be Si. We characterized three different samples MM1,MM2, and MM3, all with the same linewidth d = 4 µm,gap spacing g = 2 µm, and gap width w = 6 µm. The ra-dius is the outer ring. The circle sample MM1 had dimen-sion ro = 20 µm with lattice parameters Lx = Ly = 50 µm.The short ellipse sample MM2 had dimensions rx = 24 µm,ry = 11.5 µm with lattice parameters Lx = 60 µm, Ly =26 µm, and the tall ellipse sample MM3 had dimensionsrx = 11.5 µm, ry = 24 µm with lattice parameters Lx =26 µm, Ly = 60 µm.

At first, MM1, MM2, and MM3 pattern are transferred tomask. Aluminum with a 210 nm thickness was deposited ona 500 µm thickness intrinsic silicon wafer by a radio fre-quency magnetron sputtering method. Through a modernlithography process and plasma etching process, the sam-

Fig. 2 Normalized transmission through metafilm samples MM1,MM2, and MM3

ples were fabricated. Experimental characterization was per-formed by THz-TDS. A detailed description of this systemcan be found in Refs. [13, 14]. A linearly polarized THzbeam was focused to a spot approximately 3 mm in fre-quency independent diameter. The total sample area of themetafilm was (1 × 1) cm2 to prevent beam clipping. Mea-surements were conducted in a dry-air environment to mit-igate the effects of water vapor absorption. Transmissionmeasurements were performed on the metafilm samples un-der normal incidence. A bare intrinsic silicon substrate withthe same thickness was also measured as a reference. Sincethe THz measurement is coherent, the time-varying elec-tric field of the transmitted THz radiation following pas-sage through the sample had been directly recorded. With afast Fourier transformation of these time-domain waves, thefrequency-dependent amplitude of THz wave was extractedfrom the time waves. And the frequency-dependent complextransmission coefficient t̃ (ω) = t (ω)e−jφ(ω) had been ex-tracted.

3 Results and discussion

The normalized frequency dependent amplitude transmis-sion coefficients tMeas(ω) was shown in Fig. 2. These dataare normalized by dividing the measured transmission spec-tra of the metafilms, EMF(ω), by the measured transmissionspectrum of the reference substrate, ER(ω), i.e. tMeas(ω) =|t̃Meas(ω)| = |EMF(ω)/ER(ω)|. Two main resonances aretypically observed in THz SRRs-based planar metamateri-als. The first is the low-frequency LC resonance caused byoscillating currents in the SRRs loops and charge accumu-lation at the gap. The second is the higher-frequency dipole(or cut wire) type electric resonance caused by interactions

Improved TL-RLC model for terahertz circular split-ring resonators 463

between the SRRs sides or components parallel to the in-cident electric field [10, 15–17]. Not only the length of themetal wire (SRRs sides) but also the spacing (or period) be-tween the neighbor resonators determined the dipolar res-onance [18]. In this letter, we mainly discussed the effectof the length of the SRRs sides. Typically, the LC and di-pole resonances are not widely separated, and even partiallyoverlapped in frequency. For many kinds of device, a wideseparation is desirable [10].

Figure 2 shows that the LC resonances are actuallyshifted from their apparent positions of 0.690, 0.855 and0.867 THz for samples MM1, MM2 and MM3. It is gener-ally agreed that the gap provides the majority of the capaci-tance C for the LC resonance, though other factors, such asSRRs spacing, can also contribute to the capacitance C. Theinductive response is related to the area of the SRRs [10,15–17]. In an attempt to minimize variances in the LC res-onances caused by the drastic SRRs reshaping, we set all ofthe rings to the same gap width and gap spacing. Becausethe gap width, the gap spacing and the area of the SRRs ofMM2 and MM3 are the same, their LC resonance frequen-cies are similar.

Figure 2 also shows that the dipole resonances are actu-ally displaced from their apparent positions of 1.73, 2.53 and1.45 THz for samples MM1, MM2 and MM3. It is gener-ally agreed that the lengths of these sides [10, 15–17] deter-mine the frequency of the dipole resonance. The lengths ofthe samples MM1, MM2 and MM3 are obviously changed,therefore, the dipole resonances are distinguished.

Figure 2 also evidently shows that the ellipse and circleresonators behave quite differently. And the coupling be-tween the LC and dipole resonance has significant influenceon the overall response. The most obvious difference is theseparation between the LC and dipole resonances. In sam-ple MM2, this separation has increased to about 1.6 THz andthe off-resonance transmission stays above 80% (in ampli-tude) all the way out to 1.5 THz. This is an improvementover the electric circular SRRs design. Sample MM1 imme-diately begins to lose transmission amplitude following theLC resonance. Sample MM3 shows the other extreme, thelong side arms of the electric SRRs push the dipole reso-nance even closer to the LC resonance frequency. This re-sults in a sharp decay in transmission and a strong reshapingdue to resonance coupling [10], the transmission betweenLC resonances and the dipole resonances being no more

than 75% (in amplitude), the transmission decreases sharplyfrom 1.0 THz and reaches the lowest point at dipole reso-nances.

To better understand the shifting and reshaping of the res-onances due to coupling, the simple TL-RLC circuit modelin Refs. [7, 10] that describes the wave propagation throughour samples was used. Figure 3 shows the metamaterial ar-rays by a two coupled lumped-element resonant circuit. R1,L1 and C1 represent the LC resonance, R2, L2 and C2 rep-resent the dipole resonance, M represents the coupling be-tween the LC and dipole resonance, ZSi = 109 � is the im-pedance of the intrinsic silicon substrate and Z0 = 377 � isthe impedance of free space. To match this model with themeasured data, the values of R1, R2, L1, L2, C1, C2 andM shown in Table 1 were obtained by manual adjustmentin Advanced Design System™ 2009 and observation of theresult match to the measured data. The model can also beused to decouple the LC and dipole resonances by settingthe coupling parameter M equal to zero, or to turn off a sin-gle resonance by setting either R1 → ∞ (without R1) orR2 → ∞ (without R2) [10].

The simple TL-RLC model results had been comparedwith the experimental data in Fig. 4. The model of short el-lipse matches with most of the range of the measured databut the models of circle and tall ellipse become inaccurate atthe higher frequencies. The reason is that other resonancesand modes exist at higher frequencies [7, 10]. By decou-pling the LC and dipole resonances, it is explicit that theindividual resonance position and strength are not equiva-lent to their apparent values in the measured data. This isbecause LC-dipole resonance coupling distorts the overallresponse [7, 10]. Table 1 shows the coupling parameter M

Fig. 3 The simple TL-RLC model similar to Fig. 6(b) in Ref. [7] andFig. 3 in Ref. [10]

Table 1 Circuit parameters formatching the TL-RLC model tomeasured data. Resistor valuesare given in ohms, capacitorvalues in femtofarads, andinductors values and couplingcoefficients in picohenries

Structure Circuit component

R1(�) L1 (pH) C1 (fF) R2 (�) L2 (pH) C2 (fF) M (pH)

Short ellipse 15 98 0.352 37.5 16.8 0.2585 −12

Circle 12 131 0.398 12.5 24.05 0.395 −17

Tall ellipse 7.4 69.9 0.452 89 63.8 0.2085 −20

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Fig. 4 The TL-RLC model results and the experimental data

of short ellipse, circle and tall ellipse are −12 pH, −17 pHand −20 pH, respectively. Figure 4 shows that decouplingthe LC and dipole resonance makes the dipole resonance

Fig. 5 The improved TL-RLC model

frequency undergoing a large change, because the high fre-quency is more sensitive to coupling coefficients. Compar-ing the short ellipse to the tall ellipse in Fig. 4, we had seenthat the higher the coupling coefficients the more change ofdipole resonance relatively to the separation between the LCand dipole resonance while decoupling the resonances. It isevident that the LC-dipole resonance coupling plays an im-portant role in the overall response.

Figure 4 also shows that the simple TL-RLC model can-not well coincide with the measured data of circle and tallellipse SRRs at the higher-frequencies region for other reso-nances and modes [7, 10]. The similar drawback also showsup in Fig. 4 of Ref. [7] and Fig. 2 of Ref. [10]. For bet-ter matching the TL-RLC model result with measured data,an improved TL-RLC model as shown in Fig. 5 is usedto help us understand the overall response of the circularand tall elliptical geometries SRRs. For the mismatch at ahigher-frequencies region existing in Fig. 4, we consideredthat other resonance frequency, higher than 3 THz, affectedthe overall response in 0–3 THz. Figure 5 shows the meta-material arrays by a three coupled lumped-element resonantcircuit, R1, L1 and C1 represent the LC resonance, R2, L2

and C2 represent the dipole resonance, R3, L3 and C3 repre-sent other resonance of higher than 3 THz, M1 represents thecoupling coefficient between the LC and dipole resonance,and M2 represents the coupling coefficient between the di-pole and the higher than 3 THz resonance. For simplicity, thedirect coupling coefficient between the LC and the higherthan 3 THz resonance was considered to be small enough,and they can be neglected. The reason is that the frequencyspacing between the LC and the other resonance is large, anddipole resonance exists between the two kinds of resonance.

To match this improved model with the measured data,the values of R1, R2, R3, L1, L2, L3, C1, C2, C3, M1 andM2 shown in Table 2 also were obtained by manual adjust-ment in Advanced Design System™ 2009 and observationof the resulting match to the measured data. In Table 2, thevalues of R3 obviously are much larger than the values of R1

and R2, and a possible reason is that the other resonance fre-quency higher than 3 THz has larger loss. The absolute val-ues of M2 being less than the absolute values of M1 showsthat the higher the frequency interval the smaller the mutual

Improved TL-RLC model for terahertz circular split-ring resonators 465

Table 2 Circuit parameters for matching the improved TL-RLC model to measured data. Resistor values are given in ohms, capacitor values infemtofarads, and inductors values and coupling coefficients in picohenries

Structure Circuit component

R1(�) L1 (pH) C1 (fF) R2 (�) L2 (pH) C2 (fF) R3 (�) L3 (pH) C3 (fF) M1 (pH) M2 (pH)

Circle 17 131 0.398 1 26.4 0.34 262.5 2.9 0.3 −17 −7

Tall ellipse 7.4 69.9 0.452 13 58.8 0.21 140 5.95 0.63 −20 −11

Fig. 6 The improved TL-RLC model results and the experimental dataof circle and tall ellipse SRRs

influence coefficient. The model can also be used to decou-ple the LC, dipole and other resonances by setting the cou-pling parameter M equal to zero, or to turn off the other twosingle resonance by setting R2&R3 → ∞ (without R2&R3),R1&R3 → ∞ (without R1&R3) or R1&R2 → ∞ (withoutR1&R2) [10]. Figure 6 shows that the improved model canbetter match the measured data at higher-frequencies regionthan the simple model. By setting R1&R2 → ∞, no evi-dent resonance can be seen in Fig. 6. From another perspec-tive, it shows that another resonance frequency higher than3 THz affected the overall response, and other resonances

and modes exist at higher frequencies except the typical LCand dipole resonance [10].

4 Conclusion

These metamaterials are measured using transmission spec-troscopy in the THz time domain. Comparing these trans-mission curves, these plots clearly revealed the lower-frequency LC resonances and the higher-frequency dipoleresonances, and the coupling of the LC and dipole reso-nance had significant influence on the overall response. Itshows that the LC and dipole resonances frequency can bedesigned, and the coupling of the LC and dipole resonancefor adjusting the overall response can be regulated. A simpleTL-RLC circuit model as in the references was used to helpus understand this coupling behavior and the extent of its ef-fects. But the simple TL-RLC model has some drawbacks.To better match the model results with measured data, animproved TL-RLC model was introduced for the responseof the circular and tall elliptical geometry SRRs. The im-proved model also shows that other resonances and modesexist at higher frequencies except the typical LC and dipoleresonance. The improved TL-RLC model has better match-ing with experimental data than the simple model, especiallyin the circumstances of the LC and dipole resonance beingrelatively close.

Acknowledgements This work was partly supported by the Na-tional Basic Research Program (973) of China, under Grant No.2007CB310407. Foundation for Innovative Research Group of theNSFC under Grant No. 60721001, the International S&T Coop-eration Program of China under Grant Nos. 2006DFA53410 and2007DFR10250.

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